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Understanding Digital ',.. ,
Signal Processing
Richard G. Lyons
.
PRENTICE HALL РТА
А Prentice Hall РТА PubIication
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Lyons, Richard G., 1948
Understanding digita! signa! processing / Richard G. Lyons.
р. ст.
Inc!udes bibliographica! references and index.
ISBN 0-201 6З467 8
1. Signa! processing Digita! techniques.
TK5102.9.L96 1997
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I DEDICATION I
1 dedicate this book to ту two daughters Julie and Meredith, 1 wish 1
could go with уои; to ту mother Ruth for making те finish ту home
work; to ту father Grady who didn't know what he started when he bui1t
that workbench in the basementi to ту brother Ray for improving us аП;
to ту brother Ken who succeeded where 1 failedi to ту sister Nancy for
running interference for us; to JoOO Lennon for not giving ир; to Dr. Laura
Schlessinger for keeping us honesti to ту advisor Glenn Caldwel1 and to
the Iron Riders Motorcycle Club (Niles, СА) who keep те alive. Final1y to
5igi Pardula, without your keeping the place running this book wouldn't
exist.
ICONTENTSI
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
DISCRETE SEQUENCES AND SYSТEMS ........................ 1
1.1 Discrete Sequences and Тheir Notation ...................... 2
1.2 Signa! Amplitude, Magnitude, Power . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Signa! Processing Operationa! Symbo!s .....................10
1.4 Introduction to Discrete Linear Тime Invariant Systems . . . . . . . . 12
1.5 Discrete Linear Systerns ..................................13
1.6 Тime Invariant Systems .................................. 18
1.7 Тhe Commutative Property о! Linear Тime Invariant Systems .. 20
1.8 Ana!yzing Linear Тime Invariant Systerns . . . . . . . . . . . . . . . . . . . 20
2 PERIODIC SAMPLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Aliasing: Signa! Ambiquity in the Frequency Domain ......... 23
2.2 Sampling Low Pass Signa!s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Sampling Bandpass Signa!s .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Spectra1 Inversion in Bandpass Samp!ing . . . . . . . . . . . . . . . . . . . . 43
3 ТНЕ DISCREТE FOURIER TRANSFORM. . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1 Understanding the DFТ Equation . .. . . . . . . . . . . . . . . . . . . . . . . .50
3.2 DFТ Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 DFТ Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 DFТ Magnitиdes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.5 DFТ Frequency Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6 DFТ Shifting Тheorem ................................... 68
3.7 Inverse DFТ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.8 DFТ Leakage ........................................... 71
3.9 Windows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.10 DFТ Scalloping Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.11 DFТ Reso!ution, Zero Stиffing, and Frequency Doтain
Sampling .............................................. 89
3.12 DFТ Processing Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
3.13 Тhe DFТ о! Rectangи1ar Functions ......................... 97
3.14 Тhe DFТ Frequency Response to а Сотр!ех Input . . . . . . . . . . . 119
3.15 Тhe DFT Frequency Response to а Rea! Cosine Input . . . . . . . . . 123
3.16 Тhe DFТ Sing!e Bin Frequency Response to а Rea1
Cosine Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
vii
viii Uпderstaпdiпg D/gita/ Sigпa/ Processiпg
4 ТНЕ FAST FOURIER TRANSFORM ............................ 129
4.1 Re!ationship о! the FFТ to the DFТ . . . . . . . . . . . . . . . . . . . . . . . . 130
4.2 Hints оп Using FFTs in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.3 FFТ Software Programs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.4 Derivation о! the Radix 2 FFТ A!gorithm . . . . . . . . . . . . . . . . . . . 136
4.5 FFT Input/Output Data Index Bit Reversa! . . . . . . . . . . . . . . . . . 145
4.6 Radix 2 FFТ Butterf1y Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5 FINITE IMPULSE RESPONSE FILTERS . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.1 An Introduction to Finite Impu!se Response FIR Filters . . . . . . . 158
5.2 Convo!ution in FIR Filters ............................... 163
5.3 Low Pass FIR Fi1ter Design ..............................174
5.4 Bandpass FIR Fi1ter Design ..............................191
5.5 Highpass FIR Filter Design ..............................193
5.6 Remez Exchange FIR Fi1ter Design Method . . . . . . . . . . . . . . . . . 194
5.7 Ha1f Band FIR Fi!ters ...................................197
5.8 Phase Response о! FIR Filters ............................199
5.9 А Generic Description of Discrete Convo!ution . . . . . . . . . . . . . . 204
6 INFINITE IMPULSE RESPONSE FILTERS. . . . . . . . . . . . . . . . . . . . . . . .219
6.1 An Introduction to Infinite Impulse Response Fi1ters .........220
6.2 The Lap!ace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
6.3 'Пlе z Transform .......................................238
6.4 Impu!se Invariance IIR Filter Design Method. . . . . . . . . . . . . . . . 254
6.5 Bilinear Transform IIR Fi!ter Design Method . . . . . . . . . . . . . . . . 272
6.6 Optimized IIR Filter Design Method . . . . . . . . . . . . . . . . . . . . . . . 284
6.7 Pitfalls in Bui!ding IIR Digita! Filters. . . . . . . . . . . . . . . . . . . . . . . 286
6.8 Cascade and Paralle! Combinations о! Digita! Filters .. . . . . . . . 290
6.9 А Brief Comparison о! IIR and FIR Filters .................. 292
7 ADVANCED SAMPLING TECHNIQUES ...................... 297
7.1 Quadrature Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .297
7.2 Quadrature Samp!ing with Digita! Мixing . . . . . . . . . . . . . . . . . . 301
7.3 Digita! Resampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
8 SIGNAL AVERAGING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .319
8.1 Coherent Averaging ....................................320
8.2 Incoherent Averaging .......... . . . . . . . . . . . . . . . . . . . . . . . . . 327
8.3 Averaging Multip!e Fast Fourier Transforms . . . . . . . . . . . . . . . . 330
8.4 Filtering Aspects о! Time Domain Averaging . . . . . . . . . . . . . . . . 340
8.5 Exponentia! Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .341
Coпteпts
9 DIGITAL DATA FORMATS AND THEIR EFFECTS . . . . . . , , , . , . , . .349
9.1 Fixed Point Binary Formats , . . . , . . , . , , . . . . . . . . . , . . . , . , . . . 349
9.2 Binary Number Precision and Dynarnic Range '........, , . , .356
9,3 Effects о! Finite Fixed Point Binary Word Length ..,...,.,... 357
9.4 F!oating Point Binary Formats . . . . . . , . . . . . . , , . , , . , . , . , . , , . 375
9.5 B!ock F!oating Point Binary Format. . . , . . . . . , . . . . . , . . . , . . , . 381
10 DIGITAL SIGNAL PROCESSING TRICKS '.'.'.'.'...".'..'..385
10.1 Frequency Trans!ation without Mu!tiplication . , . . , , . . . , . , . . . 385
10.2 High Speed Vector Magnitude Approximation . , , . , . . . . . . . . , 400
10,3 Data Windowing Tricks ".........,...,........"....... 406
10.4 Fast Multip!ication of Сотр!ех Numbers . , . , . . . . , . . . , . , . , , .411
10.5 Efficient!y Performing the FFТ о! Rea! Sequences , . , . . , , , . . . , 412
10.6 Calculating the Inverse FFТ Using the Forward FFТ . . . , , . , , , . 425
10.7 Fast FFТ Averaging . . , . , . . . . . . . . . , . . . , . , . . . . . , , . . . . . , , . , 429
10.8 Simp!ified FIR Fi!ter Structure . , . . . . . . . . . , . . . , . . , . , . , . . . . , 430
10.9 Accurate A/D Converter Testing Technique ."..'.".......432
10.10 Fast FIR Fi!tering Using the FFТ. . . , . . . , . . . . . . . . . , . . . , , . . . ,435
10.11 Calcu!ation о! Sines and Cosines о! Consecutive Ang!es ......436
10.12 Generating Normally Distributed Random Data . . , . . . , . , . . . . 438
APPENDIX А. ТНЕ ARITHMETIC OF COMPLEX NUMBERS ..,..,...., 443
А.1 Graphica! Representation о! Rea! and Сотр!ех Numbers . , , . , 443
А.2 Arithmetic Representation о! Сотр!ех Numbers ,........... 444
А.З Arithmetic Operations о! Сотр!ех Numbers '..,......,..,. 446
А.4 Some Practica! Imp!ications о! Using Сотр!ех Numbers . , . . , . 453
APPENDIX В. CLOSED FORM OF А GEOMEТRIC SERIES , . . . . , . . . . . . 455
APPENDIX С. COMPLEX SIGNALS AND NEGATIVE FREQUENCY. . , . 458
с.1 Deve!opment о! Imaginary Numbers . . , . . . . , , . , . . . , . , . , . . ,460
с.2 Representing Rea! Signa!s Using Сотр!ех Phasors , , . . . , . , . , . 462
с.3 Representing Rea! Signa!s Using Negative Frequencies .,...., 467
с.4 Сотр!ех Signa!s and Quadrature Mixing. , . . , , . . , . . , , , . , . . , 471
APPENDIX D. MEAN, VARIANCE, AND STANDARD DEVIATION ....'476
D.1 Statistica! Measures . . . . . , . , . . . , . . . , . , . , . . . , . . , . , , . , . , . . . 476
D.2 Standard Deviation, or RMS, о! а Continuous Sinewave ..,.., 480
D.3 Тhe Mean and Variance о! Random Functions . , . , , . , . . , . , , . , 481
D.4 Тhe Norma! Probabi!ity Density Function ., , . . , . , , . . , . . , . . . 484
х Uпderstaпdiпg Digital Sigпal Processiпg
APPENDIX Е. DECIBELS (dB AND dBm) ...........................486
Е.l Using Logarithms to Determine Relative Signa! Power ....... 486
Е.2 Some Usefu! Decibe! Numbers ........................... 492
Е.3 Absolute Power Using Decibels . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
APPENDIX F. DIGITAL FILTER TERMINOLOGY . . . . . . . . . . . . . . . . . . . . . . . 494
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .507
I PREFACE I
Learning Digital Signal Processing
Learning the fundашеntа1s, and how to speak the language, of digital sig
nal processing does not require profound analytical skills or an extensive
background in mathematics. АН уои need is а litt1e experience with ele
mentary algebra, knowledge of what а sinewave is, this book, and enthu
siasm. This тау sound hard to believe, particularly if you've just f1ipped
through the pages of this book and seen figures and equations that appear
rather complicated. The content here, уои say, looks suspiciously like the
material in technical journals and textbooks, material that is difficu1t to
understand. Wel1, this is not just another book оп digital signal processing.
This book's goal is to gent1y provide explanation fol1owed Ьу illUstra
tion, not so that уои тау understand the material, but that уои must
Ul1derstand the material.+ Remember the first time уои saw two people
playing chess? The game probably appeared to Ье mysterious and con
fusing. As уои now know, по individual chess move is complicated.
Given а litt1e patience, the various chess moves are easy to learn. The
game's complexity comes from deciding what combinations of moves to
make and when to make them. 50 it is with understanding digital signal
processing. First we learn the fundamental rиles and processes and, then,
practice using them in combination.
If learning digital signal processing is so easy, then why does the sub
ject have the reputation of being difficult to understand? The answer lies
partial1y in how the material is typical1y presented in the literatиre. It's
difficult to convey technical information, with its mathematical subtleties,
in written form. It's one thing to write equations, but it's another matter
altogether to explain what those equations really теan from а practical
standpoint, and that's the goal of this book.
Тоо often, written explanation of digital signal processing theory
appears in one of two forms: either mathematical miracles occиr and уои
are simply given а short and swect equation without fиrther explanation,
or уои are engulfed in а flood of complex variable equations and phrases
t "Here we have the opportиnity of expounding more clearly what has already Ьееп said"
(Rene Descartes).
xi
xii Uпderstaпdiпg D/g/ta/ Sigпa/ Processiпg
such as "it is obvious that," "such that W(f) ;:: о V f," and "with judicious
application of the homogeneity property." Authors usually do provide the
needed information, but, too often, the reader must grab а pick and
shovel, put оп а miner's helmet, and try to dig the information out of а
moиntain of mathematical expressions. (This book presents the resu1ts of
several fruitfиl mining expeditions.) How many times have уои followed
the derivation of an equation, after which the author states that he or she
is going to illustrate that equation with а physical example and this
tиrns out to Ье another equation? A1though mathematics is necessary to
describe digital signal processing, I've tried to avoid overwhelming the
reader because а recipe for technical writing that's too rich in equations is
hard for the beginner to digest.+
The intent of this book is expressed in а popular quote from Е. В. White
in the introduction of his Elemeпts о! Style (New York: Macmillan
Publishing,1959):
Will (Strunk) felt that the reader was in serious trouble most о! the
time, а тап f10undering in а swamp, and that it was the duty оЕ any
one attempting to write English to drain the swamp quickly and get
his man ир оп dry ground, or at least throw him а rope.
I've attempted to avoid the traditional instructor student relationship,
but, rather, to make reading this book like talking to а friend while walk
ing in the park. I've used just enough mathematics to develop а fиnda
mental understanding of the theory, and, then, illustrate that theory with
examples.
The Journey
Learning digital signal processing is not something уои accomplish; it's а
joиrney уои take. When уои gain an understanding of some topic, ques
tions arise that cause уои to investigate some other facet of digital signal
processing. Armed with more knowledge, you're likely to begin exploring
fиrther aspects of digital signal processing much like those shown in the
following diagram. Тhis book is yoиr toиr guide dиring the first steps of
yoиr journey.
You don't need а computer to learn the material in this book, but it
would help. Digital signal processing software allows the beginner to ver
t "We need elucidation of the obvioиs more than investigation of the obscure" (Oliver
Wendell Holmes).
Preface xiii
Perlodic sampllng
How сап the spectra 01 sampled
signals Ье analyzed?
How сап the effective sample
rates 01 discrete signals Ье
changed?
How сап spectral
noise Ье reduced (о
enhance signal
"' \
How сап the noise
reduction еНесш 01
averaging Ье analyzed?
Signal averaglng
Figure P-l
Ну signal processing theory through trial and error.+ In particular, soft
ware routines that plot signal data, perform the fast Fourier transform,
and analyze digital fi1ters would Ье very usefиl.
As уои go through the material in this book, don't Ье discouraged if your
иnderstanding comes slowly. As the Greek mathematician Menaechmus
t "One mиst learn Ьу doing the thing; for though you think you know it, you have по cer
tainty until you try it" (Sophocles).
xiv Uпderstaпdiпg Digita/ Sigпa/ Processiпg
curt1y remarked to Alexander the Great, when asked for а quick explana
tion of mathematics, "There is по royal road to mathematics." Menaechmus
was confident in telling Alexander that the only way to learn mathematics
is through carefиl study. The same applies to digital signal processing. Also,
don't worry if уои have to read some of the material оосе. While the con
cepts in this book are not as complicated as qиantum physics, as mysteri
ous as the lyrics of the song "Louie Louie," or as puzz1ing as the assembly
instructions of а metal shed, they do get а litt1e involved. They deserve yoиr
attention and thought. 50 go slow and read the material twice if уои have
to; уои'Н Ье glad уои did. If уои show persistence, to quote а phrase from
5usan В. Anthony, "Pai1иre is impossible."
Coming Attractions
Chapter 1 of this book begins Ьу establishing the notation used through
out the remainder of oиr study. In that chapter, we introduce the concept
of discrete signal sequences, show how they relate to continuous signals,
and Шиstrаtе how those sequences сап Ье depicted in both the Нте and
frequency domains. In аddШоn, Chapter 1 defines the operational sym
bols we'll use to build our signal processing system block diagrams. We
conclude that chapter with а brief introduction to the idea of linear sys
tems and see why linearity enables us to use а number of powerful math
ematical tools in oиr analysis.
Chapter 2 introduces the most frequent1y misиnderstood process in
digital signal processing, periodic sampling. A1though it's straightfor
ward to grasp the concept of sampling а continuous signal, there are
mathematical subtleties in the process that require thoughtful attention.
Beginning gradually with simple examples of low pass sampling and
progressing to the interesting subject of bandpass sampling, Chapter 2
explains and quantifies the frequency domain ambiguity (aliasing) asso
ciated with these important topics. The discussion there high1ights the
power and pitfalls of periodic sampling.
Chapter 3 is devoted to one of the foremost topics in digital signal pro
cessing, the discrete Foиrier transform (DFT). Coverage begins with
detailed examples illustrating the important properties of the DFT and
how to interpret DFT spectral resu1ts, progresses to the topic of windows
used to reduce DFТ leakage, and discusses the processing gain afforded
Ьу the DFТ. The chapter concludes with а detailed discussion of the vari
ous forms of the transform of rectangular fиnctions that the beginner is
likely to encoиnter in the literature. That last topic is included there to
clarify and illustrate the DFТ of both real and complex sinusoids.
Preface xv
Chapter 4 covers the innovation that made the most profoиnd impact
оп the field of digital signal processing, the fast Fourier transform (FFТ).
Тhere we show the relationship of the popular radix 2 FFТ to the DFТ,
quantify the powerful processing advantages gained Ьу using the FFТ,
demonstrate why the FFT fиnctions as it does, and present various FFT
implementation structures. Chapter 4 also includes а list of recommenda
tions to help us when we use the FFТ in practice.
Chapter 5 ushers in the subject of digital fi1tering. Beginning with а
simple low pass finite impulse response (FIR) filter example, we care
fully progress through the analysis of that fi1ter's frequency domain
magnitude and phase response. Next we learn how window functions
affect and сап Ье used to design FIR fi1ters. The methods for converting
low pass FIR fi1ter designs to bandpass and highpass digital fi1ters are
presented, and the popular Remez Exchange (Parks McClellan) FIR ш
ter design technique is introduced and illustrated Ьу example. In that
chapter we acquaint the reader with, and take the mystery out of, the
process called convolution. Proceeding through several simple convolu
tion examples, we conclude Chapter 5 with а discussion of the powerful
convolution theorem and show why it's so useful as а qualitative tool in
understanding digital signal processing.
Chapter 6 introduces а second class of digital fi1ters, infinite impulse
response (ПR) fi1ters. In discussing several methods for the design of
ПR fi1ters, the reader is introduced to the powerful digital signal pro
cessing analysis tool called the z transform. Because the z transform is
so closely related to the continuous Laplace transform, Chapter 6 starts
Ьу gently guiding the reader from the origin, through the properties,
and оп to the utility of the Laplace transform in preparation for learn
ing the z transform. We'll see how IIR fi1ters are designed and imple
mented, and why their performance is so different from FIR fi1ters. То
indicate under what conditions these fi1ters should Ье used, the chap
ter concludes with а qualitative comparison of the key properties of FIR
and ПR fi1ters.
Chapter 7 discusses two important advanced sampling techniques
prominent in digital signal processing, quadrature sampling and digital
resampling. In the chapter we discover why quadrature sampling is so
useful when signal phase must Ье analyzed and preserved, and how this
special sampling process сan circumvent some of the limitations of tradi
tional periodic sampling techniques. Our introduction to digital resam
pling shows how we сап, and when we should, change the effective
sample rate of discrete data after the data has already been digitized.
We've delayed the discussion of digital resampling to this chapter
xvi Uпderstaпd/пg D/gita/ Sigпa/ Processiпg
because some knowledge of low pass digital filters is necessary to under
stand how resampling schemes operate.
Chapter 8 covers the important topic of signal averaging. There we
learn how averaging increases the accuracy of signal measurement
schemes Ьу reducing measurement background noise. This accuracy
enhancement is called processing gain, and the chapter shows how to pre
dict the processing gain associated with averaging signals in both the time
and frequency domains. In addition, the key differences between coherent
and incoherent averaging techniques are explained and demonstrated
with examples. То complete the chapter, the popular scheme known as
exponential averaging is covered in some detai1.
Chapter 9 presents an introduction to the various binary nиmber for
mats that the reader is likely to encounter in modern digital signal pro
cessing. We establish the precision and dynamic range afforded Ьу these
formats along with the inherent pitfalls associated with their use. Our
exploration of the critical subject of binary data word width (in bits) natu
rally leads us to а discussion of the nиmerical resolution limitations of ana
log to digital (A/D) converters and how to determine the optimиm A/D
converter word size for а given application. Тhe problems of data value
overflow roиndoff errors are covered along with а statistical introduction
to the two most popular remedies for overflow, truncation and roиnding.
We end the chapter Ьу covering the interesting subject of floating point
binary formats that allow us to overcome most of the limitations induced
Ьу fixed point binary formats, particularly in reducing the ill effects of data
overflow.
Chapter 10 provides а collection of tricks а! the trade that the profes
sionals often use to make their digital signal processing algorithms more
efficient. Тhose techniques are compiled into а chapter at the end of the
book for two reasons. First, it seems wise to keep our collection of tricks
in one chapter so that we'll know where to find them in the future.
Second, таnу of these schemes require an understanding of the material
from the previous chapters, so the last chapter is an appropriate place to
keep our collection of clever tricks. Exploring these techniques in detail
verifies and reiterates many of the important ideas covered in previous
chapters.
The appendices include а nиmber of topics to help the beginner иnder
stand the mathematics of digital signal processing. А comprehensive
description of the arithmetic of complex numbers is covered in Appendix
А, while Appendix В derives the often used, but seldom explained, closed
form of а geometric series. Appendix С strives to clarify the troubling top
ics of complex signals and negative frequency. The statistical concepts of
Preface xvi
mean, variance, and standard deviation are introduced and illustrated in
Appendix D, and Appendix Е provides а discussion of the origin and util
ity of the logarithmic decibel scale used to improve the magnitude reso
lution of spectral representations. In а slight1y different vein, Appendix F
provides а glossary of the terminology used in the field of digital filters.
Acknowledgments
How do 1 sufficient1y thank the people who helped те write this book? 1
do this Ьу stating that any quality existing herein is due to the following
talented people: for their patient efforts in the unpleasant task of review
ing early versions of the manuscript, 1 ат gratefиl to Sean McCrory, Paul
Chestnut, Paul Капе, John Winter, Terry Daubek, and Robin Wiprud.
Special thanks go to Nancy Silva for her technical and literary guidance,
and encouragement, without which this book would not have been writ
ten. For taking time to help те understand digital signal processing, 1
thank Frank Festini, Harry Glaze, and Dick Sanborn. 1 owe уои people.
Gratitude goes to the reviewers, under the auspices of Addison
Wesley,
whose suggestions improved much of the material. They are Mark
Sullivan, David Goodman, Satyanarayan Namdhari, James Kresse,
Ryerson Gewa1t, David Cullen, Richard Herbert, Мaggie Carr, and anony
mous at A1catel ВеН. Finally, (acknowledge ту good fortune in being аЫе
to work with those talented folks at Addison
Wesley: Rosa Aimee
Gonzalez, Simon Yates, and Tara Herries.
If you're still with те this far into the Preface, 1 end Ьу saying that 1 had
а ЬаН writing this book and hope уои get some value out of reading it.
CHAPTER ONEI
';:'1"
".J : ,I
Discrete Sequences
and Systems
Digital signal processing has never been more prevalent or easier to per-
form, It wasn't that long ago when the fast Fourier transform (РРТ), а
topic we'll discuss in Chapter 4, was а mysterious mathematical process
used only in industrial research centers and universities. Now, amazingly,
the РРТ is readily ауаНаЫе to us аll. It's even а bui1t-in function provided
Ьу inexpensive spreadsheet software for home computers, The availabil
ity of more sophisticated commercial signal processing software now
allows us to analyze and develop complicated signal processing app1ica-
tions rapidly and re1iably, We сап now perform spectral analysis, design
digital fi1ters, develop voice recognition, data communication, and image
compression processes using software that's interactive in both the way
algorithms are defined and how the resulting data are graphically dis-
played, Since the mid-1980s the same integrated circuit technology that
led to affordable home computers has produced powerful and inexpen-
sive hardware development systems оп which to implement our digital
signal processing designs. t Regardless, though, of the ease with which
these new digital signal processing development systems and software
сап Ье applied, we still need а so1id foundation in understanding the
basics of digital signal processing, The purpose of this book is to build
that foundation,
In this chapter we'l1 set the stage for the topics we'll study throughout the
remainder of this book Ьу defining the tепninоlоgу used in digita1 signal
t During а television interview in the early 1990s, а leading computer scientist stated that
had automobile technology made the same strides as the computer industry, we'd аН have
а car that would go а ha1f million miles per hour and get а ha1f million miles per gaHon, Тhe
cost of that car would Ье so low that it would Ье cheaper to throw it away than рау for оnе
day's parking in San Francisco,
2 Discrete Sequeпces aпd Systeтs
processing, illustrating the various ways of graphically representing discrete
signals, estab1ishing the notation used to describe sequences of data values,
presenting the symbols used to depict signal processing operations, and
briefly introducing the concept of а linear discrete system,
1.1 Discrete Sequences and Their Notation
In general, the term sigпal processiпg refers to the science of analyzing
time varying physical processes, As such, signal processing is divided
into two categories, analog signal processing and digital signal process-
ing, The term aпalog is used to describe а waveform that's continuous in
time and сап take оп а continuous range of amplitude values. An ехат-
ple of an analog signal is some voltage that сап Ье applied to an oscillo-
scope resulting in а continuous display as а function of Нте. Analog
signals сап also Ье applied to а conventional spectrum analyzer to deter-
mine their frequency content. The term aпalog appears to have stemmed
from the analog computers used prior to 1980. Тhese computers solved
linear differential equations Ьу means of connecting physical (electronic)
differentiators and integrators using old-style telephone operator patch
cords, That way, а continuous voltage or currentin the actual circuit was
aпalogous to some variable in а differential equation, such as speed, tem
perature, air pressure, etc. (Although the flexibility and speed of modem
day digital computers have since made analog computers obsolete, а
good description of the short-lived utility of analog computers сап Ье
found in reference [1],) Because present-day signal processing of continu-
ous radio-type signals using resistors, capacitors, operational amplifiers,
etc., has nothing to do with analogies, the term aпalog is actually а mis
nomer, The more correct term is coпtiпuous sigпal processiпg for what is
today so commonly called analog signal processing, As such, in this book
we'll minimize the use of the term aпalog sigпals and substitute the phrase
coпtiпuous sigпals whenever appropriate.
The term discrete-tiтe sigпal is used to describe а signal whose inde-
pendent time variable is quantized so that we know only the value of the
signal at discrete instants in time. Тhus а discrete-time signal is not repre-
sented Ьу а continuous waveform but, instead, а sequence of values, In
addition to quantizing time, а discrete-time signal quantizes the signal
amplitude, We сап il1ustrate this concept with an example. Think of а con-
tinuous sinewave with а peak amplitude of 1 at а frequency 10 described
Ьу the equation
хи) = sin(21tfot) .
(Н)
Discrete Sequeпces aпd Their Notatioп 3
Тhe frequency 10 is measured in hertz (Нz), (In physical systems, we usu-
аllу measure frequency in units of hertz, One Hz is а single oscillation, or
cycle, per second, One ki10hertz (kHz) is а thousand Hz, and а megahertz
(MHz) is one million Hz,t) With t in Eq, 1-1 representing time in seconds,
the lot factor has dimensions of cycles, and the complete 2rtfot term is an
angle measured in radians,
Plotting Eq, (1-1), we get the venerable continuous sinewave curve
shown in Figure 1 1(a), If our continuous sinewave represents а physical
vo1tage, we could sarnple it once every t s seconds using an analog to-digital
converter and represent the sinewave as а sequence of discrete values.
Plotting those individual values as dots would give us the discrete wave
form in Figure 1 1(b), We say that Figure 1-1(Ь) is the "discrete time" ver
sion of the continuous signal in Figure 1 1(a), Тhe independentvariable t in
Eq, (1-1) and Figure 1 1(a) is continuous. Тhe independent index variable п
in Figure 1-1(Ь) is discrete and сап have only integer values, Тhat is, index
п is used to identify the individual elements of the discrete sequence in
Figure И(Ь).
Do not Ье tempted to draw lines between the dots in Figure 1 1(b). For
some reason, people (particularly those engineers experienced in working
with continuous signals) want to connect the dots with straight lines, or
the stairstep 1ines shown in Figure 1-1(с). Don't faH into this innocent
looking trap, Connecting the dots сап mislead the beginner into forgetting
that the х(п) sequence is nothing more than а list of numbers, Remember,
х(п) is а discrete-time sequence of individual values, and each value in
that sequence plots as а single dot, It's not that we're ignorant of what 1ies
between the dots of х(п); there is nothing between those dots,
We сan reinforce this discrete time sequence concept Ьу 1isting those
Figure 1-1(Ь) sampled values as follows:
х(О) = О
х(1) = 0.31
х(2) = 0,59
х(3) = 0.81
(1st sequence value, index п = О)
(2nd sequence value, index п = 1)
(3rd sequence value, index п = 2)
(4th sequence value, index п = 3)
and so оп,
(1 2)
t The dimension for frequency used to Ье cycles/second; that' s why the tuning dials of old
radios indicate frequency as kilocycles/second (kcps) or megacycles/second (Mcps). In 1960
the scientific community adopted hertz as the unit of measure for frequency in honor of the
German physicist, Heinrich Hertz, who first demonstrated radio wave transmission and
reception in 1887,
4 Discrete Sequeпces aпd Systeтs
Con!;nous х(
0.5
(а) О
.5
1
Continuous time
variabIe, t
(Ь)
j Dlscre!e х(п) )«7) _t time 7t s seconds
1 ._."../ .8.
. ...... . .
.
05
..
. 1113151719 313335
0 j . I I 11 11 1 1 1 . 11 11 1 1 11 1 . 11 1 1 I 1 11 1 . 11 11 I
13579 212з252729
.. .
5 .. .
. . .
1 _.а 8.
..........-Jo1!
t s
37 39
1 1 .........
Oiscrete time
· index, п
(е)
........
Discrete time
index, п
Figure 1-1 А time-doтoin sinewove: (а) eontinuous woveform representotion;
(Ь) diserete sample representotion; (е) dlserete somples wlth
eonneetlng iines,
where п represents the time index integer seqиence О, 1, 2, 3, etc., and t s is
some constant time period, Those sample valиes сап Ье represented col
lectively, and concisely, Ьу the discrete-time expression
х(п) = sin(21tfoпts) .
(l З)
(Here again, the 21tfoпts term is ап angle measиred in radians.) Notice that
the index п in Eq, (1-2) started with а valиe of О, instead of 1. There's noth-
ing sacred aboиt this; the first valиe of п coиld jиst as wel1 have Ьееп 1, bиt
we start the index п at zero oиt of habit becaиse doing so al10ws иs to
Discrete Sequeпces aпd Their Notatioп 5
х(О), х(1), х(2), х(З), ' , , Discrete у(О), у(1), у(2), у(З), ,"
(а) System
(Ь) х(п) Discrete у(п)
System
Flgure 1-2 With an input applled, а discrete system provldes an output: (а)
the Input and output are sequences of Indivldual values; (Ь) Input
and output using the abbreviated notation of х(п) and у(п),
describe the sinewave starting at time zero. The variable х(п) in Eq, (1 3) is
read as "the sequence х of п." Equations (1 1) and (1 3) describe what are
a1so referred to as tiтe doтain signals because the independent variables,
the continuous time t in Eq. (1 1), and the discrete-time nt s values used in
Eq. (1 3) are measures оЕ time.
With this поНоп of а discrete time signal in mind, let's say that а dis
crete system is а соllесНоп of hardware components, or software routines,
that operate оп а discrete time signal sequence. For example, а discrete
system could Ье а process that gives us а discrete output sequence у(О),
у(l), у(2), etc" when а discrete input sequence of х(О), х(l), х(2), etc., is
applied to the system input as shown in Figure 1 2(a), Again, to keep the
notation concise and still keep track of individual elements of the input
and output sequences, ап abbreviated notation is used as shown in Figure
1 2(b) where п represents the integer sequence 0,1,2,3, etc, Thus, х(п) and
у(п) are general variables that represent two separate sequences of nит
bers, Figure 1 2(b) allows us to describe а system's output with а simple
expression such as
у(п) = 2х(п) 1 ,
(1-4)
Шustrаting Eq. (1 4), if х(п) is the five element sequence: х(О) = 1, х(l) = 3,
х(2) = 5, х(3) = 7, and х(4) = 9, then у(п) is the five element sequence
у(О) = 1, у(l) = 5, у(2) = 9, у(3) = 13, and у( 4) = 17,
The fundamental difference between the way time is represented in
continuous and discrete systems leads to а very important difference in
how we characterize frequency in continuous and discrete systems, То
illustrate, let's reconsider the continuous sinewave in Figure l l(a).
If it represented а vo1tage at the end of а саЫе, we could measure its
6 Discrete Sequeпces aпd Systeтs
frequency Ьу applying it to an oscilloscope, а spectrum analyzer, or а fre
qиency coиnter, We' d have а problem, however, if we were merely given
the list of valиes from Eq. (1-2) and asked to determine the freqиency of
the waveform they represent. We' d graph those discrete valиes, and, sure
enough, we'd recognize а single sinewave as in Figure 1-1(Ь). We сап say
that the sinewave repeats every 20 samples, bиt there's по way to deter-
mine the exact sinewave freqиency from the discrete seqиence valиes
alone, Уои сап probably see the point we're leading to 11ere, If we knew
the time between samples the sample period ts we'd Ье аЫе to deter-
mine the absolиte freqиency of the discrete sinewave, Given that the t s
sample period is, say, 0,05 mil1iseconds/sample, the period of the
sinewave is
, ' d 20 samples 0,05 milliseconds 1 ' 11 ' d
sшеwаvе реrю = . тl lsecon, (1-5)
period sample
Веса иse the freqиency of а sinewave is the reciprocal of its period, we now
know that the sinewave's absolиte freqиency is 1/(1 ms), or 1 kНz, on the
other hand, if we found that the sample period was, in fact, 2 milliseconds,
the discrete samples in Figure 1-1(Ь) woиld represent а sinewave whose
period is 40 mil1iseconds and whose freqиency is 25 Hz, The point here is
that, in discrete systems, absolиte freqиency dеtепrunаtiоп in Hz is depen-
dent оп the sample freqиency fs = 1/ t s ' We'l1 ье reminded of this dependence
throughoиt the rest of this book.
In digital signal processing, we often find it necessary to characterize
the freqиency content of discrete-time domain signals, When we do so,
this freqиency representation takes place in what's called the frequency
doтain, Ву way of example, let's say we have а discrete sinewave
sequence х 1 (п) with an arbitrary freqиency fo Hz as shown оп the left
side of Figure 1-3(а). We сап also describe х 1 (п) as shown оп the right
side of Figиre 1-3(а) Ьу indicating that it has а freqиency of 1, measиred
in иnits offo' and по other freqиency content. Althoиgh we won't dwell
in it jиst now, notice that the freqиency-domain representations in
Figure 1-3 are themselves discrete,
То illиstrate our time- and frequency-domain representations further,
Figure 1-3(Ь) shows another discrete sinewave х 2 (п), whose peak атрН-
tиde is 0,4, with а freqиency of 2fo, The discrete sample valиes of xz{n) are
expressed Ьу the eqиation
х 2 (п) = 0.4 . sin(21t2fonts) ,
(1-6)
Diserete Sequeпees aпd Their Notatioп 7
J, : .. : ::: ... ' ::: . . : . ain
t .., " .,.20 25 зо
(а) О j . /l11 11 1111/1/1 1" /1 , /1/1 11 /l11! ..
5 10 15.' :aTime(n)
. .
.5 ..
. .
. .
1 .....
Х2 (п) in the time domain
0.5
...8. .....
. ' . 10 15 . . 25 зо
(Ь) О j++H-t++.+++++t+.++++н+. 11 1 1 1 11 ..
1 5 ....... 20 ....... Time (п)
...{).5
j х sum (п) . ;" . the tlme domain
1.5
. .-.
1 . 8.
. .
0.5 . : - е.. 20 25 за
(с) О 1 . 11/1 11 [ 111/1 1 " .oi ol 11 1 .. ' 1 '. ' 1 .... 11/1 .. . 1 '. 11
5 10 15 8' , ' .". Tlme (п)
,5 . . ; ; ; ; .
. - Ii
1 .....
1.5
j х,(т) amplltude in the
trequency domain
1 .
0.5 '
О ........-+----- . I!I \. ..,.
о 'о 2fo Зf о 4fo 510 Frequency
r Х2(т) amplitude in the
1 trequency domain
0.5 . 04
О . Ii"""""""'. Ii 1i l ....
о 2fo Зf о 4fo 5'0 Frequency
j X"m(т) aтpli1ude 1" the
1. frequency dоmаlП
05
o. . . . ....
о 'о 2 Зf о 4 510 Frequency
Figure 1-3 Time- ond frequency-domoin graphleol representotions: (о) sinewove of
frequeney fo: (Ь) reduced ompiitude sinewove of frequency 2fo: (е) sum
of the юо sinewoves,
When the two sinewaves, x1(n) and х 2 (п), are added to produce а new
waveform xsum(n), its time-domain equation is
Xsum(n) = x1(n) + х 2 (п) = sin(21tfont.} + 0,4 . sin(21t2font s ) , (1-7)
and its time- and frequency-domain representations are those given in
Figure 1 3(c), We interpret the Xsum(m) frequency-domain depiction, the
spectrum, in Figure 1-3(с) to indicate that Xsum(n) has а frequency сотро-
nent of 10 Hz and а reduced amplitude frequency component of 210 Hz,
Notice three things in Figure 1-3, First, time sequences use lowercase
variable names like the "х" in x1(n), and uppercase symbols for frequency
domain variables such as the "Х" in X1(m), The term X1(m) is read as "the
spectral sequence Х sub one of т," Second, because the X 1 (т) frequency
domain representation of the x1(n) time sequence is itse1f а sequence (а list
8 D/screte Sequeпces aпd Systeтs
of numbers), we use the index "т" to keep track of individual elements in
Х 1 (т), We can list frequency-domain sequences just as we did with the
time sequence in Eq, (1-2), For example Хэит(т) is listed as
Хэит(О) = О
Х эит (1) = 1,0
Х эит (2) = 0.4
Х эит (3) = О
(1st Хэит(т) value, index т = О)
(2nd Хэит(т) value, index т = 1)
(3rd Хэит(т) value, index т = 2)
(4th Хэит(т) value, index т = 3)
and so оп,
where the frequency index т is the integer sequence 0,1,2,3, etc. Тhird,
because the х 1 (п) + х 2 (п) sinewaves have а phase shift of zero degrees rel
ative to each other, we didn't really need to bother depicting this phase
relationship in Хэит(т) in Figure 1-3(с), In genera1, however, phase rela-
tionships in frequency-domain sequences are important, and we'll cover
that subject in Chapters 3 and 5,
А key point to keep in mind here is that we now know three equivalent
ways to describe а discrete-time waveform, Mathematically, we can use а
time-domain equation like Eq. (1-6) for example. We can also represent а
time-domain waveform graphically as we did оп the left side of Figure 1-3,
and we can depict its correspondin discrete, frequency-domain equiva-
lent as that оп the right side of Figure 1-3.
As it turns out, the discrete-time domain signals we're concemed with
are not only quantized in time; their amp1itude values are also quantized.
Because we represent аН digital quantities with binary numbers, there's а
limit to the resolution, or granularity, that we have in representing the val-
ues of discrete numbers, Although signa1 amplitude quantization can Ье
an important consideration we cover that particular topic in Chapter 9
we won't worry about it just now.
1.2 Signal Amplitude, Magnitude, Power
Let' s define two important terms that we'H Ье using throughout this book:
amplitude and magnitude. It's not surprising that, to the layman, these
terrns are typically used interchangeably, When we check our thesaurus,
we find that they are synonymous. t In engineering, however, they mean
t Of course, layman are "other people," То the engineer, the brain surgeon is the layman, То
the brain surgeon, the engineer is the layman.
Sigпa/ Amplitude, Magпitude, Power 9
i
I
I
I
I
I
Flgure 1-4 Mognitude somples, I xj(п) 1, of the time woveform in Flgure 1-3(0).
j IX 1 (n ",, )1
1 """'.....
. .
. .
0.5 · : ;
: : '
: i! i, '1
0 1 . 111111111111
5 10
O,5
.....
. .
. .
.' .
. .
i ' .
II I! i 11 i I i I i 11111 i ......
15 20 25 30 Time (n)
two different things, and we must keep that difference clear in our dis-
cussions, Тhe amplitude of а variable is the measure of how far, and in
what direction, that variable differs from zero. Thus, signal amplitudes
сап Ье either positive or negative, The time-domain sequences in Figure
1-3 presented the sample value amplitudes of three different waveforms,
Notice how some of the individual discrete amplitude values were posi
tive and others were negative,
The magnitude of а variable, оп the other hand, is the measure of how
far, regardless of direction, its quantity differs from zero, 50 magnitudes
are always positive values, Figure 1-4 illustrates how the magnitude of
the x1(n) time sequence in Figиre 1-3(а) is equal to the amplitude, but with
the sign always being positive for the magnitude. We use the modulus
symbol (1 1) to represent the magnitude of x 1 (п), Occasionally, in the 1iter
ature of digital signal processing, we'll find the term magnitude referred to
as the abso/ute va/ue,
When we examine signals in the frequency domain, we'll often Ье
interested in the power level of those signals, The power of а signal is pro-
portional to its amplitude (or magnitude) squared, If we assume that the
proportionality constant is опе, we сап express the power of а sequence
in the time or frequency domains as
xpw/n) = х(п)2 = I х(п) 12 ,
(1-8)
or
Xpw/m) = Х(т)2 = I Х(т) 12 ,
(1-8')
Very often we'll want to know the difference in power levels of two sig-
nals in the frequency domain. Because of the squared nature of power,
1 О Discrete Sequeпces aпd Svsteтs
t Xsum(m) amplltude 'п the
l' frequency domain
0,5 . 0,4
О, I I ' ' ' '\. ..
о 'о 2'0 З'о 4'0 5'0 Frequency
t Xsum(m) power in the
1 I frequency domain
0,5
. 0,16
О, I I , , , '\. ..
о 'о 2'0 З'о 4'0 5(" Frequency
Flgure '-5 Frequency-domaln amplltude and frequency domain power of the
xsum(n) time waveform In Flgure l-З(с),
two signals with moderately different amp1itudes will have а much larger
difference in their relative powers, In Figure 1-3, for example, signal
x1(п)'s amp1itude is 2.5 times the amp1itude of signal х 2 (п), but its power
level is 6.25 that of x 2 (п)'s power level. Тhis is illustrated in Figure 1-5
where both the amp1itude and power of Xsum(m) are shown.
Because of their squared nature, plots of power values often involve
showing both very large and very smal1 values оп the same graph. То
make these plots easier to generate and evaluate, practitioners usual1y
employ the decibel scale as described in Appendix Е.
1.3 Signal Processing Operational Symbols
We'll Ье using block diagrams to graphically depict the way digital signal
processing operations are implemented. Тhose block diagrams will сот-
prise an assortment of fundamental processing symbols, the most common
of which are illustrated and mathematical1y defined in Figure 1 6.
Figure 1 6(a) shows the addition, element for element, of two discrete
sequences to provide а new sequence. If our sequence index п begins at О,
we say that the first output sequence value is equal to the sum of the first
element of the Ь sequence and the first element of the с sequence, or
а(О) = Ь(О) + с(О), Likewise, the second output sequence value is equal to
the sum of the second element of the Ь sequence and the second element
of the с sequence, or а(l) = Ь(I) + с(I). Equation (1-7) is an example of
adding two sequences, Тhe subtraction process in Figure 1-6(Ь) generates
an output sequence that's the element-for element difference of the two
inpиt sequences, There are times when we must ca1culate а sequence
whose elements are the sum of more than two values. Тhis operation,
illustrated in Figure 1-6(с), is cal1ed summation and is very common in
Sigпa/ Processiпg Operatioпa/ Syтbois 11
Adder:
(а)
ь(п) a(n)
с(п)
а(п) = ь(п) + с(п)
(Ь)
Subtractor:
ь(п) а(п) а(п) = ь(п) с(п)
с(п)
(с)
b(n)
Summation: п+З
Ь(n+1) ,
J'. + а(п) а(п) = L b(k) = Ь(п) + Ь(п+1) + Ь(п+2) + Ь(п+З)
Ь(п+2) ... k=n
Ь(n+З)
(d)
Multiplication:
ь(п) а(п) а(п) = Ь(п)с(п) = Ь(п),с(п)
[Sometime5 we "5е а ", "
10 5ignify m"ltiplicalion.]
с(п)
(е)
Unit Delay:
Ь(П) ............... а(п)
а(п) = b(rr--1)
b(n) a(n)
Figure 1-6 Terminology aпd symbols used in digital signal processing bIock dlagrams.
digital signal processing, Notice how the lower and upper limits of the
summation index k in the expression in Figure 1-6(с) tell us exact1y
which elements of the Ь sequence to sum to obtain а given а(п) value,
Because we'l1 encounter summation operations so often, let's make sure
we understand their notation, If we repeat the summation equation
from Figure 1-6(с) here we have
П+З
а(п) = I,b(k) ,
k=п
(1-9)
12 Discrete Sequeпces oпd Systems
This means that
when п = О, index k goes from О to 3, so
when п = 1, index k goes from 1 to 4, so
when п = 2, index k goes from 2 to 5, so
when п = 3, index k goes from 3 to 6, so
а(О) = Ь(О) + b(l) + Ь(2) + Ь(3)
a(l) = b(l) + Ь(2) + Ь(3) + Ь(4)
а(2) = Ь(2) + Ь(З) + Ь( 4) + Ь(5)
а(3) = Ь(3) + Ь(4) + Ь(5) + Ь(6)
and so оп,
(ИО)
We'll begin using summation operations in eamest when we discuss dig-
ital fi1ters in Chapter 5,
ТЬе multip!ication of two sequences is symbolized in Figure 1 6(d),
Multip!ication generates an output sequence that's the e!ement for-element
product of two input sequences: а(О) = Ь(О)с(О), a(l) = b(l)c(l), and so оп,
The last fundamental operation that we'll Ье using is called the unit delay in
Figure 1-6(е), Whi!e we don't need to appreciate its importance at this
point, we'll merely state that the unit delay symbol signifies an operation
where the output sequence а(п) is equa! to а delayed version of the Ь(п)
sequence, For example, а(5) = Ь(4), а(6) = Ь(5), а(7) = Ь(6), etc, As we'll see in
Chapter 6, due to the mathematical techniques used to analyze digital fil-
ters, the unit delay is very often depicted using the term r 1 ,
ТЬе symbols in Figure 1 6 remind us of two important aspects of digi
tal signal processing, First, our processing operations are always per
formed оп sequences of individua! discrete values, and second, the
elementary operations themselves are very simple, It's interesting that,
regardless of how complicated they appear to Ье, the vast majority of dig
ital signal processing a!gorithms сan Ье performed using combinations of
these simp!e operations, If we think of а digital signal processing algo-
rithrn as а recipe, then the symbols in Figure 1 6 are the ingredients.
1.4 Introduction to Discrete Linear
Time-Invariant Systems
In keeping with tradition, we'll introduce the subject of linear time-invari
ant (LТI) systems at this early point in our text. Although аn appreciation
for LТI systems is not essential in studying the next three chapters of this
book, when we begin exploring digital filters, we'll build оп the strict def-
initions of linearity and time invariance. We need to recognize and under
stand the notions of linearity and time invariance not just because the vast
majority of discrete systems used in practice are LТI systems, but also
Discrete Liпear Systeтs 13
because LТI systems are very accommodating when it comes to their
analysis, That's good news for us because we сап use straightforward
methods to predict the performance of any digital signal processing
scheme as long as it's linear and time invariant. Бесаusе linearity and time
invariance are two important system characteristics having very special
properties, we'll discuss them now,
1.5 Discrete Linear Systems
Тhe term /inear defines а special class оЕ systems where the output is the
superposition, or sum, оЕ the individual outputs had the individual
inputs been applied separately to the system. For example, we can say
that the application of an input х 1 (п) to а system results in an output Уl(п).
We symbolize this situation with the following expression:
х 1 (п) resultsin) Уl(п)
(1-11)
Given а different input х 2 (п), the system has а У2(п) output as
Х2(п) resultsin) У2(п) ,
(1-12)
For the system to Ье linear, when its input is the sum х 1 (п) + х 2 (п), its out
put must Ье the sum of the individual outputs so that
Хl(п)+Х2(п) resultsin) Уl(п)+У2(п) ,
(1-13)
One way to paraphrase expression (1-13) is to state that а linear system's
output is the sum of the outputs оЕ its parts, Also, part of this description
of linearity is а proportionality characteristic, Тhis means that if the inputs
are scaled Ьу constant factors С 1 and С 2 then the output sequence parts are
also scaled Ьу those factors as
С1Хl(п) + С2 Х 2(п)
results in
) С 1 Уl(п)+С 2 У2(п) .
(1-14)
In the literature, this proportionality attribute of linear systems in
expression (1-14) is sometimes called the hoтogeneity property. With
these thoughts in mind, then, let's demonstrate the concept of system
linearity,
14 Diserete Sequeпees oпd Systems
1.5.1 Example 01 а Linear System
То illustrate systern linearity, let's say we have the discrete systern shown
in Figure 1-7(а) whose output is defined as
( ) x(п)
у п := ,
2
(1-15)
that is, the output sequence is equal to the negative of the input sequence
with the arnplitude reduced Ьу а factor of two. If we apply an x 1 (п) input
sequence representing а 1 Hz sinewave sarnpled at а rate of 32 sarnples
(а)
Linear
Discrete
System
Inpu! х(n)
(Ь)
! х,(n) t y,(n)
1 ,,"jl,l!'r. 1
.,.;:', .
0.5 ;.: ,! ,i,': : i ,.. 0.5 .. II':I,r.r.
8" :;.::: ,,:', . Ii:; "",'"
о f 8f+НIЖЖ+Ж. lII . ' .. 1 . 1 . 1 . 111 .. 1 .. 11 . 1 . 1 .. о ...'tЖ++tttt++';I .... . . .. IIIIIIIIIIII'"
.i:"":.:.',:.nme t r.::,.,. ,," Tlme
.5 . i: ::: ' ' i .: --0.5 rlj'liL LIi
".!,:.8
1 'i.LIi 1
,,) :: 5 ! f ; ;:::! i, . . : . ,Д;Ъ 1' . .I . ' 1 ; 8 1 .ii 11 ! i r:m .
""V. 'i .' < : '. --0.5. . Ii
1 . . . . . 1
(d)
2 1 ХЗ(n) = х, (n) .. "2(n)
. . ....
1 .!: ' i:!.
1'1: [,. "'!
о f l!Ж,!'..81tЖ' ., .
.. . ..1 i i ,.1i'8., ,: ,!. Тime
1 :'. .
...8 r.. 8
2
I Уз(n)
2
1 а!.. ".
.:8 Ii ;.
О .' ..,""" "ili.L!\'oI!o;...
fТ'Т"П"-.!.!"! 8rтттr. . . rТТТТТ""'II"
!' '.." .,,!j'. Тiтe
1 ..... r.. 8
-2
Ou!pu! у(n) = x(n)/2
t YI (т)
0.5
,
О . ., ...I!J...I!.....,.I!I..,..,..,... ,.,
i '2 4 б 8 10 12 14
....().5 .
t Y2 (т)
0.5
3
О ............................. ...
1 0 . 2 . . '1; '12 '14'
--0.5 .
t Уз(т)
0.5
1 3
О . .,+.,.I!I.I!I.I!I..,.I!I..,.I!I..,.I!I.,.I!I
1 0 2 4 8 8 10 12 14
--О.Б . .
Figure 1-7 Liпear system iпput-to-output relotioпships: (а) system bIoek dlogram
where у(п) = x(п)/2: (Ь) system iпput oпd output with а l-Hz siпewove
opplied: (е) with а 3-Hz siпewove opplied: (d) wlth the sum of l-Hz oпd
3-Hz siпewoves oppiied,
Discrete Liпear Systeтs 15
per cycle, we'llhave а Yl(n) output as shown in the center ofFigure 1-7(Ь),
The frequency-domain spectral amplitude of the Yl(n) output is the plot
оп the right side of Figure 1-7(Ь) indicating that the output comprises а
single tone of peak amplitude equal to o.5 whose frequency is 1 Hz,
Next, applying an х 2 (п) input sequence representing а 3-Hz sinewave, the
system provides а У2(п) output sequence, as shown in the center of Figure
1-7(с), The spectrum of the У2(п) output, У 2 (т), confirming а single 3-Hz
sinewave output is shown оп the right side of Figure 1-7(с), Finally
here's where the linearity comes in Н we apply an хз(п) input sequence
that's the sum of а 1-Hz sinewave and а 3-Hz sinewave, the уз(п) output
is as shown in the center of Figure 1-7(d), Notice how уз(п) is the sample-
for-sample sum of Yl(n) and У2(п). Figure 1-7(d) also shows that the out-
put spectrum Уз(т) is the sum of Y1(т) and У 2 (т), That's linearity,
1.5.2 Example о' а Nonlinear System
It's easy to demonstrate how а nonlinear system yields an output that is
not equal to the sum of Yl(n) and У2(п) when its input is x1(n) + х 2 (п), А
simple example of а nonlinear discrete system is that in Figure 1-8(а)
where the output is the square of the input described Ьу
у(п) = [х(п)]2 ,
(1-16)
We'll use а well-known trigonometric identity and а little algebra to pre-
dict the output of this nonlinear system when the input comprises simple
sinewaves. Following the form of Eq. (1-3), let's describe а sinusoidal
sequence, whose frequency 10 = 1 Hz, Ьу
X 1 (п) = sin(21tfonts) = sin(21t . 1 . nt s ) .
(1-17)
Equation (1-17) describes the x1(n) sequence оп the left side of Figure 1-8(Ь).
Given this x1(n) input sequence, the Yl(n) output of the nonlinear system is
the square of а 1-Hz sinewave, or
Yl(n) = [X 1 (n)]2 = sin(21t . 1 . nt s ) . sin(21t . 1 . nt s ) . (1-18)
We сап simplify our expression for Yl(n) in Eq. (1-18) Ьу using the follow-
ing trigonometric identity:
, ( ) . (fЗ) cos(a fЗ) cos(a + fЗ)
sш а .sш =
2 2
(1-19)
16 Discrete Sequeпces aпd Systeтs
(а)
Input>(n)
о: t : i Th..
(Ь) -о: f .К++IIIЖ+IЖ. 111IТ'111 .
1 ..r..L ".
1 t:C:. :.
o.5 1 i)i. "!ij 'i
(с) : f . . ''!I ''ill !l!l .,.
1 ... r . 8 L8
(d)
2 I Х .. З(П)
... .8.
1 ';. .
.':, i".
О [.; ;j.... ::.:. .
f """". !IЖI-.. t'НtI . "'. .. . . ,
.ili 8!::.Тime
1 : . .
.... ". 8
-2
Nonlinear
Discrete
System
Output у(п) = [>(п)] 2
j Y 1(n)
1 "'.
0.8 IHi
0.6 81'1:':'
0.4 ::!' !,.
0.2 8 1 i!, ::;, ii ,;!;!
I!I;!, i: >,. 8i" <:: ;,".
О "!,ж++-ж-ж+-.-:-,,,ЖТН+Ж+h'"
Тi.,.
df.
.! -
; :;-.
l У1 (т)
1 / zero Н: component
0.5 . 2
О !.."t...,.........,.I!I..,.....,.1!I ,.....
-О.5 ! . 4 6 8 10 12 104 ;:z1
-1
j Y 2(n)
'.'!'....
08 '!'. ; i. 8! ! .
о:в 1 . . . . .: . . . . :' . : ' . ' . ' , . i ... i .
<. О:,' 8';,' . ::,': .
0.4 : -, :: .:' !
.' ;:С: .! ; .:1 ': :8
0.2 ti'\.-ii. :i:\ :i! .-:. i:i
о I!I)i.h"Жn'Н+:I!lнt.'ж+"ни...
Time
1 t У2(т)
05.' 6
О "1JI'1JI"+""'1I!I' 1!I'"1[I'''''''
-о 5 ! 2 4 . 8 10 12 14 Freq
. (н.)
-1
2.5 I з(п) .. . t Уз(m)
1..
1.52 ',1 8;1. .:. 8. 05 t . 4 .
1 i . :. 8 . ' : . 1 :! :. . о о ."t.+.+......,...... ,....
0.5 -1.':\. .'::: ,i1!. .I ! 2 8 10 12 14
О .ЖIН'I.IIН+..ннr.. .
FJgure ) -8 Nonlinear system inpuHo-оutрut relatlonships: (а) system bIoek diagram
where у(п) = (х(п))2; (Ь) system input and output with а '-Hz slnewave
applled; (е) wlth а 3-Hz sinewave applled; (d) wlth the sum 01 '-Нz and 3-Hz
sinewaves app//ed,
Using Eq, (1 19), we сап express Yl(n) as
cos(21t . 1. nt 2л .1. nt )
Yl (п) = s s
2
cos(21t .1. nt s + 2л .1. nt s )
2
= cos(O) cos(41t .1. nt s )
2 2
1 cos(21t . 2. nt.)
2 2
(1-20)
which is shown as the аН positive sequence in the center ofFigure 1 8(b),
Because Eq, (1-19) resu1ts in а frequency sum (а + ) and frequency dif
ference (а ) effect when mи1tiplying two sinиsoids, the Yl(n) oиtput
Discrete Liпear Systeтs 17
sequence will Ье а cosine wave оЕ 2 Hz and а peak amplitude оЕ O.5,
added to а constant value оЕ 1/2. Тhe constant value оЕ 1/2 in Eq. (1 20)
is interpreted as а zero Hz frequency component, as shown in the Yj(т)
spectrum in Figure 1 8(b), We could go through the same algebraic exer
cise to determine that, when а З Нz sinewave х 2 (п) sequence is applied
to this nonlinear system, the output У2(п) would contain а zero Hz com
ponent and а 6 Hz component, as shown in Figure 1 8(c),
System nonlinearity is evident if we apply аn хз(п) sequence compris
ing the sum оЕ а 1 Hz and а З Нz sinewave as shown in Figure 1-8( d). We
сan predict the frequency content оЕ the уз(п) output sequence Ьу using
the algebraic relationship
(а+Ь)2 = а 2 +2аь+ь 2 ,
(1 21)
where а and Ь represent the l Hz and З Нz sinewaves, respectively, From
Eq. (1 19), the а 2 term in Eq. (1 21) generates the zero-Hz and 2-Hz output
sinusoids in Figure 1 8(b), Likewise, the Ь 2 term produces in уз(п) another
zero Hz and the 6 Hz sinusoid in Figure 1-8(с), However, the 2аЬ term
yields additiona12 Hz and 4-Нz sinusoids in уз(п), We сan show this alge-
braically Ьу using Eq, (1 19) and expressing the 2аЬ term in Eq. (1-21) as
2аЬ = 2. sin(21t .1. пt s ) . sin(21t . З. пt s )
= 2 cos(21t .1. пt s 2п. 3 . пt s )
2
2 cos(21t .1. пt s + 2п. 3 . пt s )
2
= cos(21t. 2. пt s ) cos(21t. 4. пt s ) t
(1-22)
Equation (1-22) tells us that two additional sinusoidal components wi11 Ье
present in уз(п) because of the system's nonlinearity, а 2 Hz cosine wave
whose amplitude is +1 and а 4-Hz cosine wave having аn amplitude оЕ
1, Тhese spectral components are i11ustrated in Уз(т) оп the right side оЕ
Figure 1 8(d).
Notice that, when the sum оЕ the two sinewaves is applied to the non
linear system, the output contained sinusoids, Eq. (1-22), that were not
present in either оЕ the outputs when the individual sinewaves alone were
applied. Тhose extra sinusoids were generated Ьу аn interaction оЕ the
t The first term in Eq, (1-22) is cos(21t. nt, БIt, пt,) =cos(41t. пt,) = cos( 21t . 2 ' пt,), However,
because the cosine function is even, cos(--a) = cos(ct), we сап express that fjrst term аз
cos(21t . 2. пt,),
18 D/screte Sequeпces aпd Systeтs
two input sinusoids due to the squaring operation, That's nonlinearity;
expression (1-13) was not satisfied, (E1ectrica1 engineers recognize this
effect of internally generated sinusoids as interтodulation distortion,)
A1though non1inear systems are usually difficult to ana1yze, they are осса-
sionally used in practice. References [2], [3], and [4], for examp1e, describe
their application in non1inear digita1 filters. Again, expressions (1-13) and
(1-14) state that а 1inear system's output resulting from а sum of individ-
ual inputs, is the superposition (sum) of the individua1 outputs, Тhey a1so
stipu1ate that the output sequence Уl(п) depends on1y оп х 1 (п) combined
with the system characteristics, and not оп the other input xz(n), i.e.,
there's по interaction between inputs х 1 (п) and xz(n) at the output of а 1in-
ear system.
1.6 Time-Invariant Systems
А time-invariant system is one where а time delay (or shift) in the input
sequence causes а equiva1ent time de1ay in the system's output sequence,
Keeping in mind that п is just an indexing variable we use to keep track
of our input and output samp1es, 1et's say а system provides an output
у(п) given an input of х(п), or
х(п)
results in
) у(п) .
(1-23)
For а system to Ье оте invariant, with а shifted version of the origina1
х(п) input applied, х'(п), the following applies:
х' (п) = х(п + k) results in ) у' (п) = у(п + k) ,
(1-24)
where k is some integer representing k samp1e period оте de1ays, For а
system to Ье time invariant, expression (1-24) must ho1d true for any inte-
ger va1ue of k and any input sequence.
1.6.1 Example о' а Time-Invariant System
Let's look at а simp1e examp1e of оте invariance illustrated in Figure 1-9,
Assume that our initia1 х(п) input is а unity-amplitude 1-Hz sinewave
sequence with а у(п) output, as shown in Figure 1-9(Ь), Consider а differ-
ent input sequence х'(п), where
х'(п) = х(п+4) ,
(1-25)
Тiтe-/пvariaпt Systeтs
Tlme-Invariant
(а) Input х(п) Discrete Output У(п) = x(п)/2
System
t*) lу(п)
1 ... . о.: "' "
0.5 . 'i11Ij!:j .
. ',i,:i!i :1:8 1821242730 О 3 6 9 12 15 . ! i [ ; [! : ..
(Ь) о , IIIIIIIIIIIIIII I! "! I ! ! I! ! I ! о f" J 1" 1 !!! !! IIJ4 'IIIIIIIIIIIII
о з 6 9 12 15 .! f i, ! ! i :! '. Тiтe ...0.5 ... Ii.. 18 21 24 27 30 Time
..0.51 а';;:; -' -' j 1.
iI i! i !! '.
1 ... I... -,
+. х'(п) 1 1 (п)
1 .. .. о.: " '"
05 f;;:::::: 1I I
! (- ,! i i -' ! : .12 15 18 21 24 27 . О 3 6 9 ._,',!! ,'i": . 30
(с) о' · . III 1111 ""IIII о 1i1'1!,"II , llllilllllll
о з 6 9 .11 ' i ! ' ! i : i i ! i.. 30 Тlmе .;!, '!' ! ..12 15 18 21 2427 -. Time
...0.51 .: i: -' i:: i:. ...0.5 I.......
.;: i ,! '.
1 ....... -,
Flgure 1-9 Тiтe /пvarioпt systeт iпрuНо-оutрut re/otioпships: (а) systeт bIoek
diograт where у(п) = x(п)/2; (Ь) systeт iпput ond output wlth а 1-Hz
siпewove opplied; Се) systeт lпput ond output when а 1-Hz siпewove,
de/oyed Ьу four soтples, Is opplied. Wheп х'(п) = х(п+4), then,
у' (п) = у(п+4) ,
Equation (1-25) tells us that the input sequence х'(п) is equal to sequence
х(п) shifted four samples to the left, that is, х'(О) = х(4), х'(l) = х(5),
х'(2) = х(6), and so оп, as shown оп the left of Figure 1-9(с), The discrete
system is time invariant because the у'(п) output sequence is equal to the
у(п) sequence shifted to the left Ьу four samples, or у'(п) = у(п+4). We
сап see that у'(О) = у(4), y'(l) = у(5), у'(2) = у(6), and so оп, as shown in
Figure 1-9(с). For time-invariant systems, the у time shift is equal to the
х time shift.
Some authors succurnb to the urge to define а time-invariant system as
one whose parameters do not change with time. That definition is incom-
plete and Сan get us in trouble if we're not careful, We'll just stick with the
formal definition that а time-invariant system is one where а time shift in
an inpиt sequence results in an equal time shift in the oиtput seqиence, Ву
the way, time-invariant systems in the literature are often called shift-
invariant systems. t
t An example of а discrete process that's nо! tirne-invariant is the downsampling, or deci-
rnation, process described in Section 7.3,
20 Discrete Sequeпces aпd Systeтs
Input х(п)
LTI
System #1
п)
LTI
System #2
Output у(п)
(а)
/nput х(п)
LTI
System #2
тп)
LTI
System # 1
Output у(п)
(Ь)
Figure 1-10 Linear time-invariont (LTI) systems in series: (о) bIock diogram of two LTI
systems; (Ь) swopping the order of the two systems does not chonge
the resultont output у(п),
1.7 The Commutative Property о' Linear Time-
Invariant Systems
Although we don't substantiate this fact until we reach Section 6.8, it's not
too early to realize that LТI systems have а useful commutative property Ьу
which their sequential order can Ье rearranged with по change in their final
output. This situation is shown in Figure 1-10 where two different LТI вув-
tems are configиred in series. Swapping the order of two cascaded systems
does not alter the final output, Although the intепnеdiаtе data sequences
f( п) and g(n) will ивиаllу not Ье equal, the two pairs of LТI systems will have
identical у(п) output sequences, This commutative characteristic соmев in
hand у for designers of digital filters, ав we'll вее in Chapters 5 and 6,
1.8 Analyzing Linear Time-Invariant Systems
Ав previously stated, LТI systems can Ье analyzed to predict their perfor
mance, Specifically, if we know the unit impulse response of an LТI system,
we сап calcu1ate everything there is to know about the system; that is, the
system's unit impulse response completely characterizes the system. Ву
unit impulse response, we mean the system's time domain output
sequence when the input is а single unity-valued sample (иnit impulse)
preceded and followed Ьу zero-valued samples ав shown in Figure 1-11(Ь).
Клоwing the (unit) impulse response of an LТI system, we сап deter
mine the system's output sequence for any input sequence Ьесаиве the
output is equal to the convolution of the input sequence and the system's
impulse response, Moreover, given an LТI system's time-domain impulse
Refereпces 2
Linear
(а) Input х(п) Time-Invariant Output У(п)
Discrete
System
1'" '""" ,,.. t У(п) impulse response
1 . .
(Ь) :: "unity-vaiued ..mpie t
. .
: i, ...
о .. ............................ .. о ."'!"""'!"'. .""""'I"""!. -.:..e...............
-з о з 6 . 12 l' 18 21 24 Tlme О :: i . ... Time
I I .
Flgure 1-11 LTI system uпit impulse respoпse sequeпces: (о) system bIock diogram;
(Ь) Impulse lпput sequeпce Х(п) ond Impulse respoпse output
sequence у(п),
response, we сап find the system' s frequeпcy respoпse Ьу taking the Fourier
transform in the form of а discrete Fourier traпsforт of that impulse
response[5].
Don't Ье alarmed if you're not exact1y sure what is meant Ьу convolu-
tion, frequency response, or the discrete Fourier transform, We'll intro
duce these subjects and define them slowly and carefully as we need them
in later chapters, The point to keep in mind here is that LТI systems сап
Ье designed and analyzed using а number of straightforward and power-
ful analysis techniques. These techniques will Ьесоmе tools that we'll add
to ош signal processing toolboxes as we joumey through the subject of
digital signal processing.
References
[1] Karplus, W, J" and Soroka, W, W, Analog Methods, Second Edition,
McGraw-Нill, New York, 1959, р, 117,
[2] Mikami, N" Kobayashi, М" and Yokoyama, У, "А New DSP Oriented
Algorithm for Ca1culation оЕ the Square Root Using а Nonlinear Digita!
Filter," IЕЕЕ Trans, оп Signal Processing, Vol, 40, No, 7, July 1992,
[3] Heinen, Р" and Neuvo, У. "FIR-Median Hybrid Filters," IЕЕЕ Trans, оп Acoust,
Speech, and Signal Processing, Vol, ASSP-35, No, 6, June 1987,
[4] Oppenheim, А., Schafer, R" and Stockham, Т, "Nonlinear Filtering оЕ
Multiplied and Convo!ved Signals," Proc, IЕЕЕ, Vol, 56, August 1968,
[5] Pickerd, John, "Impulse-Response Testing Lets а Single Test Оо the Work of
Thousands," EDN, April27, 1995,
I CHAPTER TWO I
Periodic Sampling
Periodic sampling, the process of representing а continuous signal with а
sequence of discrete data values, pervades the field of digital signal pro
cessing. In practice, sampling is performed Ьу applying а continuous sig
nal to an analog-to-digital (A/D) converter whose output is а series of
digital values, Because sampling theory plays an important role in deter-
mining the accuracy and feasibility of any digital signal processing
scheme, we need а solid appreciation for the often misunderstood effects
of periodic sampling. With regard to sampling, the primary concern is just
how fast must а given continuous signal Ье sampled in order to preserve
its information content, We сап sample а continuous signal at any sample
rate we wish, and we'l1 get а series of discrete values but the question is
how well do these values represent the original signal? Let's learn the
answer to that question and, in doing so, explore the various sampling
techniques used in digital signal processing.
2.1 Aliasing: Signal Ambiguity in the
Frequency Domain
Тhere is а frequency-domain ambiguity associated with discrete-time sig
nal samples that does not exist in the continuous signal world, and we сап
appreciate the effects of this uncertainty Ьу understanding the sampled
nature of discrete data. Ву way of example, suppose уои were given the
fol1owing sequence of values,
х(О) = О
x(l) = 0,866
х(2) = 0.866
х(З) = О
х(4) = O,866
х(5) = 0.866
х(6) = О,
23
24 Periodic Soтpliпg
(а) ошf
) . ......
Time
.
(Ь)
Figure 2-1 Frequency ambIgulty: (а) dlscrete-time sequence of values;
(Ь) two dlfferent slnewaves that pass through the polnts of the
dlscrete sequence,
and were told that they represent instantaneous values of а time-domain
sinewave taken at periodic intervals. Next, уои were asked to draw that
sinewave. You'd start Ьу plotting the sequence of values shown Ьу the
dots in Figure 2-1(а). Next, you'd Ье Iikely to draw the sinewave, Шus-
trated Ьу the solid line in Еgше 2-1(Ь), thatpasses throиgh the points rep-
resenting the original sequence,
Another person, however, might draw the sinewave shown Ьу the
shaded line in Figure 2-1(Ь). We see that the original sequence of values
could, with equal validity, represent sampled values of both sinewaves.
Тhe key issиe is that, if the data sequence represented periodic samples of
а sinewave, we cannot unambiguously determine the frequency of the
sinewave from those sample values alone,
Reviewing the mathematical origin of this frequency ambiguity enables
us not only to deal with it, but to use it to our advantage, Let's derive аn
expression for this frequency-domain ambiguity and, then, look at а few
specific examples. Consider the continuous time-domain sinusoida1 signa1
defined as
Affas/пg: S/gпaf Ambigu/ty 'п the Frequeпcy Domaiп 2Б
x(t) = sin(21tfot) ,
(2-1)
Тhis x(t) signal is а garden variety sinewave whose frequency is 10 Hz.
Now let's sample x(t) at а rate of 15 samples/ s, i,e" at regular periods of t s
seconds where t s = 1/1., If we start samp1ing at time t = О, we wi11 obtain
samples at times Ot s ' 1t s ' 2t s ' and so оп, 50, from Eq, (2 1), the first п suc-
cessive samples have the values
Oth sample:
1st sample:
2nd sample:
х(О) = sin(21t/oOt.)
х(l) = sin(21t/он.)
х(2) = sin(21t/02ts)
nth sample:
х(п) = sin(21tlonts) ,
(2-2)
Equation (2 2) defines the value of the nth sample of our х(п) sequence
to Ье equal to the original sinewave at the time instant nt s ' Because two
values of а sinewave are identical if they're separated Ьу an integer mul
tiple of 2п radians, i.e., sin(e) = sin(e+21tm) where m is any integer, we сап
modify Eq, (2-2) as
х(п) = sin(21tf,nt s ) = sin(21tl o nt s + 2пт) = sin(21t(fo + .!!2...)nt s ) (2-3)
nt s
If we let m Ье an integer multiple of п, m = kn, we сап replace the т/ п ratio
in Eq, (2 3) with k so that
х(п) = sin(21t(fo +! )nt s )
t s
(2-4)
Because 1. = 1/ t s , we can equate the х(п) sequences in Eqs, (2-2) and (2-4) as
х(п) = sin(21tlonts) = sin(21t(fo +kfs)nt s ) ,
(2-5)
Тhe 10 and (10 +kfs) factors in Eq, (2-5) are therefore equal, The imp1ication
of Eq. (2-5) is critical, It means that an х(п) sequence of digital sample
values, representing а sinewave of 10 Hz, also exactly represents
sinewaves at other frequencies, namelY, lo + kfs, This is one of the most
important relationships in the field of digital signal processing, It's the
26 Period/c Saтpllпg
thread with which аН sampling schemes are woven. In words, Eq. (2 5)
states that
When sampling at а rate of Is samples/s, if k is any positive or neg-
ative integer, we cannot distinguish between the sampled values of
а sinewave of 10 Hz and а sinewave of <10 +kl s ) Hz.
It's true. No sequence of values stored in а computer, for example, can
unambiguously represent one and only one sinusoid without additional
inforrnation, Тhis fact applies equally to A/D-converter output samples
as well as signal samples generated Ьу computer software routines, Тhe
sampled nature of any sequence of discrete values makes that sequence
also represent an infinite number of different sinusoids.
Equation (2 5) influences аН digital signal processing schemes, It's the
reason that, although we've only shown it for sinewaves, we'll see in
Chapter 3 that the spectrum of any discrete series of sampled values con
tains periodic replications of the original continuous spectrum. Тhe period
between these replicated spectra in the frequency domain will always Ье
1., and the spectral replications repeat аН the way from DC to daylight in
both directions of the frequency spectrum, Тhat's because k in Eq. (2 5) can
Ье any positive or negative integer. (In Chapters 5 and 6, we'lllearn that
Eq, (2 5) is the reason that аН digital filter frequency responses are period
ic in the frequency domain and is crucial to analyzing and designing а
popular type of digital filter known as the infinite impulse response filter,)
То illustrate the effects of Eq. (2 5), let's build оп Figure 2-1 and con
sider the sampling of а 7 kНz sinewave at а sample rate of 6 kHz, Anew
sample is determined every 1/6000 seconds, or once every 167 microsec
onds, and their values are shown as the dots in Figure 2-2(а),
Notice that the sample values would not change at аН Н, instead, we
were sampling а 1 kНz sinewave, In this example f. = 7 kНz,J. = 6 kHz,
and k = 1 in Eq. (2 5), such thatfo+kf. = [7+Н.6)] = 1 kHz, Our problem
is that по processing scheme can determine if the sequence of sampled
values, whose amplitudes are represented Ьу the dots, сате from а 7 kНz
or а l kHz sinusoid, If these amplitude values are applied to а digital
process that detects energy at 1 kHz, the detector output would indicate
energy at 1 kHz, But we know that there is по l kHz tone there-----our
input is а spectrally pure 7 kНz tone, Equation (2-5) is causing а sinusoid,
whose name is 7 kHz, to go Ьу the alias of 1 kHz. Asking someone to
deterrnine which sinewave frequency accounts for the sample values in
Figure 2 2(a) is like asking them "When 1 add two numbers 1 get а sum of
four, What are the two numbers?" Тhe answer is that there are an infinite
number of number pairs that сап add up to four.
Aliasiпg: Sigпai Aтbiguity iп the Frequeпcy Doтaiп 2j
(а)
Time
(Ь)
2 kHz
Time
, Spectral band I
ofintere5t
(с)
3 2
( /2)
о
3 4
( /2)
6 7
('5)
Freq
(kHz)
Figure 2-2 Frequeney ambIguity effeets of Eq, (2-5): (а) sampling а 700kHz sinewave
at а sample rate of 6 kHz; (Ь) sampling а 400kHz sinewave at а sample
rate of 6 kHz; (е) speetral relatioпships showing aliasing of the 7- and
4-kНz slnewaves,
Figure 2 2(b) shows another example of frequency ambiguity, that we'll
саН aliasiпg, where а 4 kHz sinewave could Ье mistaken for а 2-kНz
sinewave. In Figure 2 2(b),fo = 4 kHz,f, = 6 kHz, and k = 1 in Eq, (2-5),
so thatfo+kf. = [4+( 1 . 6)] = 2 kHz. Again, if we examine а seqиence of
numbers representing the dots in Figиre 2-2(Ь), we could not determine if
the sampled sinewave was а 4 kНz tone or а 2-kНz tone. (Althoиgh the
28 Periodic Sampliпg
concept of negative frequencies might seem а bit strange, it provides а
beautifully consistent methodology for predicting the spectral effects of
sampling. Appendix С discusses negative frequencies and how they
relate to real and complex signals.)
Now, if we restrict our spectral band of interest to the frequency range
of tfsl2 Hz, the previous МО examples take оп а special significance. The
frequency 1./2 is an important quantity in sampling theory and is rеfепеd
to Ьу different names in the literature, such as critical Nyquist, half
Nyquist, and folding frequency. А graphical depiction of our мо frequen
су aliasing examples is provided in Figure 2 2(c). We're interested in signal
components that are aliased into the frequency band between f./2 and
+1./2, Notice in Figure 2-2(с) that, within the spectral band of interest (:tЗ
kНz, because fs = 6 kНz), there is energy at 2 kНZ and +1 kНz, aliased
from 4 kНZ and 7 kНz, respectively. Note also that the vertical positions of
the dots in Figure 2 2(c) have по amplitude significance but that their hor
izontal positions indicate which frequencies are related through aliasing.
А general illustration of aliasing is provided in the shark's tooth pattem
in Figure 2 З(а). Note how the peaks of the pattem are located at integer
multiples of fs Нz. The pattem shows how signals residing at the inter
section of а horizontalline and а sloped line will Ье aliased to all of the
intersections of that horizontalline and all other lines with like slopes, Ро!
example, the pattem in Figure 2 З(Ь) shows that our sampling of а 7 kНz
;..., Band 01 .....'.... R l ' .....i4....... R l ' .....;..., R l ' .....'
interest ер Ica........... 1 . ер Ica.........,.. ер Ica!
(а) !\/.
1./2 о 1./2 1. 21. 31. Freq
(Ь) Л&У /.
о 1 3 6 7 12 13 18 19 Freq
(kHz)
Flgure 2.3 Shark's tooth pattern: (а) aliasing at multlples of the sampling
frequency; (Ь) aliaslng ofthe 7-kНz slnewave to 1 kHz, 13 kHz,
and 19 kHz,
Saтpfiпg Low-Pass Sigпafs 29
sinewave at а sample rate of 6 kHz wil1 provide а discrete sequence of
nurnbers whose spectrum ambiguously represents tones at 1 kНz, 7 kНz,
13 kHz, 19 kНz, etc. Let's pause for а moment and let these very impor-
tant concepts soak in а bit, Again, discrete sequence representations of а
continuous signal have unavoidable ambiguities in their frequency
domains, These ambiguities must Ье taken into account in аН practical
digital signal processing algorithms,
ОК, let's review the effects of sampling signals that are more interest
ing than just simple sinusoids,
2.2 Sampling Low-Pass Signals
Consider sampling а continuous real signal whose spectrum is shown in
Figure 2 4(a), Notice that the spectrum is symmetrical about zero Hz,
and the spectral amplitude is zero above +В Hz and below B Hz, i.e" the
signal is baпd-liтited. (From а practical standpoint, the term baпd-liтited
sigпal merely implies that any signal energy outside the range of :!:В Hz
is below the sensitivity of our system.) Given that the signal is sampled
at а rate of fs samples/ s, we сап see the spectral rep1ication effects of sam-
pling in Figure 2 4(b), showing the original spectrum in addition to an
infinite number of replications, whose period of rep1ication is fs Hz,
(Although we stated in Section 1.1 that frequency domain representa-
tions of discrete-time sequences are themselves discrete, the rep1icated
spectra in Figure 2 4(b) are shown as continuous lines, instead of discrete
dots, merely to keep the figure from looking too cluttered. We'H cover the
fuH imp1ications of discrete frequency spectra in Chapter 3,)
Let's step back а moment and understand Figure 2-4 for аН it's worth,
Figure 2 4(a) is the spectrum of а continuous signal, а signal that сап only
exist in one of two forms, Either it's а continuous signal that сап Ье sam-
pled, through A/D conversion, or it is merely an abstract concept such as
а mathematical expression for а signal, It caппot Ье represented in а digi
tal machine in its current band 1imited form, Once the signal is represent-
ed Ьу а sequence of discrete sample values, its spectrum takes the rep1i-
cated form of Figure 2-4(Ь).
The replicated spectra are not just figments of the mathematics; they
exist and have а profound effect оп subsequent digital signal processing,t
The rep1ications тау appear harmless, and it's natural to ask, "Why care
about spectral replications? We're only interested in the frequency band
t Toward the end of Section 5,9, ав аn example of using the convolution theorem, another
derivation of репофс sampling's replicated spectrums will Ье presented,
30 Periodie 8aтpiiпg
Continuous spectrum
(а)
B
о
в
Freq
Discrete spectrum
(Ь)
2f. Freq
2f.
--2f.
2f. Freq
(с)
Flgure 2-4 Spectral replieotions: (о) originol continuous signol spec1rum; (Ь) spec1ro\
replleotlons of the sampled signol when f./2 > В; (е) frequency overlop
ond olloslng when the samp\lng rate is too low becouse './2 < В,
within -z.f/2." Well, if we perform а frequency translation operation or
induce а change in sampling rate through decimation or interpolation,
the spectral replications wil1 shift up or down right in the middle of the
frequency range of interest -z.fs/2 and could cause prob1ems[l]. Let's see
how we сап control the locations о! those spectral replications,
In practical A/D conversion schemes, fs is always greater than 2В to
separate spectral replications at the foldiпg frequeпcies of Ifs/2. Тhis very
important relationship of fs 2В is known as the Nyquist criterion, То
illustrate why the term folding frequency is used, let's lower our sam-
pling frequency to fs = 1.58 Hz, Тhe spectral result of this uпdersaтpliпg
is illиstrated in Figure 2-4(с), The spectral replications are now overlap-
ping the original baseband spectrum centered about zero Hz, Limiting
our attention to the band -z.f/2 Нz, we see two very interesting effects.
First, the lower edge and upper edge of the spectral replications centered
Sampliпg Low Pass Sigпa/s 3 1
at +1. and 1. now Не in our band of interest, This situation is equivalent
to the original spectrum folding to the left at +1./2 and folding to the
right at f/2, Portions of the spectral replications now combine with the
original spectrum, and the result is aliasing errors, The discrete sampled
values associated with the spectrum оЕ Figure 2-4(с) по longer truly rep-
resent the original input signal. The spectral information in the bands of
8 to 8/2 and 8/2 to 8 Hz has been corrupted, We show the amplitude
of the aliased regions in Figиre 2 4(c) as dashed lines because we don't
really know what the amplitudes will Ье if aliasing occurs,
The second effect illustrated Ьу Figure 2-4(с) is that the entire spec-
tral content оЕ the original continuous signal is now residing in the
band of interest between 1./2 and +1./2, This key property was true in
Figure 2 4(b) and will always Ье true, regardless of the original signal
or the sample rate, Тhis effect is particularly important when we're dig-
itizing (A/D converting) continuous signals, It warns us that any sig-
nal energy located above +8 Hz and below 8 Hz in the original con-
tinuous spectrum of Figure 2 4(a) will always end ир in the band of
interest after sampling, regardless of the sample rate. For this reason,
continuous (analog) low pass filters are necessary in practice.
We illustrate this notion Ьу showing а continuous signal оЕ bandwidth
8 accompanied Ьу noise energy in Figиre 2-5(а). Sampling this composite
continuous signal at а rate that's greater than 28 prevents replications of
the signal оЕ interest from overlapping each other, but аН of the noise
energy still ends ир in the range between 1./2 and +1./2 of our discrete
spectrum shown in Figиre 2 5(b), Тhis problem is solved in practice Ьу
(а)
Noise Signal 01 Noise
A /I'Z " K
B О В
...
Freq
(Ь)
/i ft N .
11 . H>i, .
I о I
fs/2 '5/2 (5
...
Freq
:, 1
Figure 2-5 Spectral replications: (а) original continuous signal plus noise
spectrum: (Ь) discrete spectrum with nolse contaminating the
signal of interest,
32 Per/odic Saтp/iпg
Noise Noise
A / 11\ \ А1 ''' /\ 11\ /\ /\ "
B О В Freq I О I r
'э 's/2 's/2's 2'э req
Original \ Analog Low-Pass Filtered \ Dlscrete
continuous continuous signal A/D samples
signal ............. Filter (cutoff frequency = В Hz) I Converter
Ж
B О В Freq
Flgure 2-6 Low-pass analog Illtering prior to sampling at а rate 01 " Hz,
иsing an analog low-pass aпti-aliasiпg filter prior to A/D conversion to
attenuate апу иnwanted signal energy above +В and below B Hz as
shown in Figиre 2-6, An example low pass filter response shape is shown
as the dotted line sиperimposed оп the original continиoиs signal spec-
trит in Figиre 2-6. Notice how the oиtpиt spectrum of the low-pass filter
has Ьеen band-limited, and spectral aliasing is avoided at the oиtpиt of the
A/D converter.
Тhis completes the discussion of simple low-pass sampling, Now let's go
оп to а more advanced sampling topic that's proven so иseful in practice,
2.3 Sampling Bandpass Signals
Althoиgh satisfying the majority of sampling reqиirements, the sampling
of low-pass signals, as in Figиre 2-6, is not the only sampling scheme иsed
in practice. We can иse а techniqиe known as baпdpass saтpliпg to sample
а continиoиs bandpass signal that is centered aboиt some freqиency other
than zero Hz. When а continиoиs inpиt signal's bandwidth and center
frequency permit иs to do so, bandpass sampling not оnlу redиces the
speed reqиirement of A/D converters below that necessary with tradi-
tionallow pass sampling; it also redиces the amount of digital memory
necessary to captиre а given time interval of а continиoиs signal,
Ву way of example, consider sampling the band limited signal shown
in Figиre 2-7(а) centered atfc = 20 MHz, with а bandwidth В = 5 MHz, We
иse the term bandpass sampling for the process of sampling continиoиs
signals whose center freqиencies have Ьеen translated ир from zero Hz.
What we're calling bandpass sampling goes Ьу varioиs other names in the
Sampliпg Baпdpass Sigпa/s 33
Bandpass signal
spectrum
(continuous)
I В=
!5MHzl
(а)
fc
о
(с = Freq
20 MHz
= fc 812 = 17,5 MHz
Bandpass signal
spectrum (discrete)
(Ь)
Freq
»- ; -<
fs
fs
1<
. fs
fs
Flgure 2.7 Baпdpass signal sampllng: (а) original continuous slgnal
spectrum; (Ь) sampled signal spectrum replications when
sample rate is 17,5 MНz,
literature, such as IF sampling, harmonic sampling[2], sub Nyquist sam
pling, and undеrsаmрling[З], In bandpass sampling, we're more con
cemed with а signal's bandwidth than its highest frequency component.
Note that the negative frequency portion оЕ the signal, centered at fc' is the
mirror image оЕ the positive frequency portion as it must Ье for real sig
nals. Our bandpass signal's highest frequency component is 22.5 MHz.
Conforming to the Nyquist criterion (sampling at twice the highest fre
quency content оЕ the signal) implies that the sampling frequency must Ье
а minimum оЕ 45 MHz. Consider the effect if the sample rate is 17.5 MHz
shown in Figure 2 7(b). Note that the original spectral components remain
located at ffc' and spectral replications are located exactly at baseband, i,e"
butting up against еат other at zero Hz, Figure 2 7(b) shows that sam
pling at 45 MHz was unnecessary to avoid aliasing instead we've used
the spectral replicating effects of Eq. (2 5) to our advantage.
Bandpass sampling performs digitization and frequency translation in
а single process, often called saтpliпg traпslatioп, The processes оЕ sam
pling and frequency translation are intimately bound together in the world
оЕ digital signal processing, and every samp1ing operation inherently
results in spectral rep1ications. The inquisitive reader тау ask, "Сап we
sample at some stilllower rate and avoid aliasing?" The answer is yes, but,
to find out how, we have to grind through the derivation of an important
bandpass sampling relationship. our reward, however, will Ье worth the
trouble because here's where bandpass sampling really gets interesting.
34 Periodic Saтpliпg
! (
I
21с- В
'C-B/2
!yl I p с; D1 rYl
-1,' о 1,' 21"
) iB
6 'в'
I r (I.
'с 41 Freq
саrrlе,
frequency
(а) Щl i Y1
-41" -I c -21"
1 . 21с-В )!
(Ь) (.1 [::] f::1 1 ffJl н 1 f::1 l н ' r;1 rfl
-I c О 'с
..
Freq
:(
,
21с+ В
(с) 1 (11) (тi J r1PIl rc;f! 1 (Т' " 1 1 rl1 1
-I c -31," -21." -1," О 1,. 21,,, 31." 'с
) ;
!
.
Freq
Flgure 2-8 Bondposs soтpllng frequency Ilтits: (о) soтple rate (" = (2f c 8)/6;
(Ь) soтple rate Is less thon (,,; (с) тlnlтuт soтple rate (" < (",
Let's assume we have а continuous input bandpass signal of band
width В, Its carrier frequeпcy is fc Hz, i,e" the bandpass signa1 is centered at
fc Hz, and its sampled value spectrum is that shown in Figure 2 8(a). We
сап sample that continuous signal at а rate, say fs, Hz, so the spectral repli
cations of the positive and negative bands, Р and Q, just butt ир against
each other exactly at zero Hz, Тhis situation, depicted in Figure 2 8(a), is
reminiscent of Figure 2 7(b), With an arbitrary nиmber of replications, say
т, in the range of 2fc В, we see that
2{ B
т!s,=2!c B or J.'=....i.L........
S т
(2 6)
In Figure 2 8(a), т = 6 for il1ustrative purposes only. Of coиrse т can Ье
any positive integer so long as fs, is never less than 2В, If the sample rate
fs, is increased, the origina1 spectra (bold) do not shift, but аН the replica
tions will shift. At zero Hz, the Р band will shift to the right, and the Q
band will shift to the left. Тhese replications will overlap and aliasing
occиrs, Тhus, from Eq, (2-6), for an arbitrary т, there is а frequency that
the sample rate must not exceed, or
{, < 2!с в
Js
т
or 2!с В !S' ,
т
(2 7)
Sampliпg Baпdpass Slgпals 35
If we reduce the sample rate below the 1.. value shown in Figure 2-8(а), the
spacing between replications will decrease in the direction of the arrows
in Figure 2 8(b), Again, the original spectra do not shift when the sample
rate is changed, At воте new sample rate 1..., where 1... < 1.., the replication
Р' will just butt ир against the positive original spectrum centered at fc ав
shown in Figure 2-8(с). In this condition, we know that
2!с + В
(т + l)fs.. = 2!с +В or fs.. = ,
т+1
(2 8)
Should 1... Ье decreased in value, Р' will shift further down in frequency
and start to overlap with the positive original spectrum at fc and aliasing
occurs. Therefore, from Eq. (2-8) and for т+ 1, there is а frequency that the
sample rate must always exceed, or
j;.. > 2!с +В
s т+ 1
(2-9)
We сап now сотЫnе Eqs. (2 7) and (2 9) to вау that fs тау Ье chosen any
where in the range between 1... and 1.. to avoid aliasing, or
2!c B fs 2!c+B ,
т т+1
(2 10)
where т is аn arbitrary, positive integer ensиring that 1. 2В. (For this
type of periodic sampling of real signals, known ав real or first order ват-
pling, the Nyquist criterionl. 2В must still Ье satisfied,)
То appreciate the important relationships in Eq, (2 10), let's retиrn to our
bandpass signal example, where Eq, (2 10) enables the generation of Table
2-1. This table tells ив that oиr sample rate сап Ье anywhere in the range of
22.5 to 35 MHz, anywhere in the range of 15 to 17.5 MНz, or anywhere in
the range of 11,25 to 11,66 MHz, Anу sample rate below 11.25 MHz is иnac
ceptable Ьесаиве it will not satisfy Eq, (2-10) ав well ав 1. 2В, The spectra
resulting from several of the sampling rates from Table 2 1 are shown in
Figиre 2 9 for oиr bandpass signal example, Notice in Figure 2-9(f) that
when 1. equals 7.5 MHz (т = 5), we have aliasing problems Ьесаиве neither
the greater than relationships in Eq. (2-10) nor 1. 2В have Ьееn satisfied.
The т = 4 condition is also иnacceptable Ьесаиве 1. 2В is not satisfied, The
last column in Table 2-1 gives the optimum sampling frequency for each
acceptable т value. Optimum sampling frequency is defined here ав that
36 Periodic Saтp/iпg
ТаЫе 2-1 Equatlon (2-1 О) Applled to the Bandpass Slgnal Example
m (2Ec B)/m (2Е с +В)/(m+l) Optimum Sampliпg Rate
1 35,0 MHz 22,5 мнz 22.5 MHz
2 17.5 MHz 15,0 MHz 17,5 MHz
3 11,66 МНz 11,25 MHz 11,25 МНz
4 8,75 MHz 9,0 MНz
5 7,0 MHz 7,5 MHz
frequency where spectral replications do not butt up against each other
except at zero Hz, For example, in the т = 1 range of permissible sampling
frequencies, it is much easier to perform subsequent digital filtering or
other processing оп the signal samples whose spectrum is that of Figure
2-9(Ь), as opposed to the spectrum in Figure 2-9(а),
The reader mау wonder, "Is the optimum sample rate always equal to
the minimum permissible value for!. using Eq, (2-10)?" Тhe answer
depends оп the specific application perhaps there are certain system соn-
straints that must Ье considered, For example, in digital telephony, to sim-
plify the follow-on processing, sample frequencies are chosen to Ье integer
multiples of 8 kНz[4]. Another application-specific factor in choosing the
optirnum!. is the shape of analog anti-aliasing filters[5]. Often, in practice,
high performance A/D converters have their hardware components fiпe
tuпed during manufacture to ensure maximum linearity at high frequen-
cies (>5 MHz), Тheи use at lower frequencies is not recommended.
An interesting way of Шustrаting the nature of Eq, (2-10) is to plot the
minimum sampling rate, (2fc+B)/(т+1), for various values ofm, as а {unc
tion of R defined as
R = highest signal frequency component = !с + в /2 (2-11)
bandwidth В
If we normalize the minimum sample rate from Eq, (2-10) Ьу dividing it
Ьу the bandwidth В, we get а curve whose axes are normalized to the
bandwidth shown as the solid curve in Figure 2 10. This figure shows us
the minimum normalized sample rate as а function of the normalized
highest frequency component in the bandpass signal, Notice that,
regardless of the value of R, the minimum sampling rate need never
exceed 4В and approaches 2В as the carrier frequency increases,
Surprisingly, the minimum acceptable sampling frequency actually
decreases as the bandpass signal's carrier frequency increases. We сап
Sompliпg Boпdposs Sigпo/s 37
fs = 35,0 MHz
(а)
25,O 20,O 15,O 10,O 5,O О 5,0 10,0 15,0 20,0 25,0 MHz
(Ь)
25,O 20,O 15,O 10,O 5,O О 5,0 10,0 15,0 20,0 25,0 MHz
т=2
(с)
25,O 20,O 15,O 1O,O --5,0 О 5,0 10,0 15,0 MHz
20,0 25,0
(d)
25,O 20,O 15,O 10,O 5,O
о
5,0 10,0 15,0 20,0
(е)
25,O 20,O 15,O 10,O 5,O
о
5,0 10,0 15,0 20,0
(1)
25,O 20,O 15,O 10,O --5,0
о
5,0 10,0 15,0 20,0 25,0
Flgure 2-9 Vorious speetrol replleotions from ТаЫе 2 1: (о) " = 35 MHz; (Ь) " = 22,5
MHz; (е) " = 17,5 MHz; (d) " = 15 MHz; (е) " = 11,25 MHz; (f) " = 7,5 MHz,
interpret Figure 2 10 Ьу reconsidering our bandpass signal example
from Figure 2 7 where R = 22.5/5 = 4.5. Тhis R value is indicated Ьу the
dashed 1ine in Figure 2 10 showing that т = 3 and f/ в is 2.25. With В = 5
MHz, then, the minimumfs = 11,25 MHz in agreement with ТаЫе 2-1.
The leftmost line in Figure 2-10 shows the low pass sampling case,
where the sample rate fs must Ье twice the signal's highest frequency
component. So the norma1ized sample rate f/ в is twice the highest fre
quency component over В or 2R.
Figure 2 10 has been prominent in the 1iterature, but its normal pre
sentation enables the reader to jump to the false conclusion that any
38 Periodic Samp//пg
3.0
This line corresponds 10
Ihe low pass sampllng саэе,
Ihal is, fs= 2ис+ 812)
=l
Minimum fs/B
4,0
3,5
2.5
2,25
2,0
;j
"
'ш z ..'
,(>--у "'?- "' 1 d
Z... z.. ...
1,5
1
4 4,5 5 6 7 8
Ralio: highesl frequency component/bandwidlh
10 R
Figure 2-10 Minimum bandpass sampling rate from Eq, (2 10),
sample rate above the minimum shown in the figure will Ье an accept-
аЫе sample rate[6 12], There's а clever way to avoid any misunder-
stаnding[lЗ], If we plot the acceptable ranges of bandpass sample fre
quencies from Eq, (2-10) as а function of R we get the depiction shown
in Figure 2 11. As we saw from Eq, (2 10), ТаЫе 2 1, and Figure 2 9,
acceptable bandpass sample rates are а series of frequency ranges sepa
rated Ьу unacceptable ranges of sample rate frequencies, that is, an
acceptable bandpass sample frequency must Ье above the minimum
shown in Figure 2 10, but cannot Ье just any frequency above that min
imum. Тhe shaded region in Figure 2-11 shows those normalized band
pass sample rates that willlead to spectral aliasing, 5ample rates within
the white regions of Figure 2-11 are acceptable. 50, for bandpass sam
pling, we want our sample rate to Ье in the white wedged areas associ-
ated with some value of т from Eq. (2 10), Let's understand the signifi
саnсе of Figure 2-11 Ьу again using our previous bandpass signal
example from Figure 2 7,
Figure 2-12 shows our bandpass signal example R value (highest fre
quency component/bandwidth) of 4.5 as the dashed verticalline. Because
that line intersects just three white wedged areas, we See that there are
only three frequency regions of acceptable sample rates, and this agrees
with our results from ТаЫе 2-1, Тhe intersection of the R = 4.5 line and the
borders of the white wedged areas are those sample rate frequencies list-
ed in ТаЫе 2 1, 50 Figure 2 11 gives а depiction of bandpass sampling
restrictions much more realistic than Figure 2-10,
Saтpliпg Baпdpass Sigпa/s 39
Sampling rate ('s/8)
14
8
12
10
6
4
2
о
1
234 5 6 7 8 9
Ratio: highest Irequency component/bandwidth
10 R
Flgure 2-11 Reglons 01 acceptabIe bandpass sampllng rates Irom Eq, (2-10),
normallzed to the sample rate over the signal bandwldth и,; В),
A1though Figures 2-11 and 2-12 indicate that we сап иse а sample rate
that lies оп the boиndary between а white and shaded area, these sample
rates should Ье avoided in practice, Nonideal analog bandpass fi1ters,
sample rate clock generator instabilities, and slight imperfections in avail-
аЫе A/D converters make this idea/ case impossible to achieve exact1y. It's
prudent to keep fs somewhat separated from the boиndaries, Consider the
bandpass sampling scenario shown in Figure 2 13, With а typical (non-
ideal) analog bandpass filter, whose frequency response is indicated Ьу
the dashed line, it's prudent to consider the filter's bandwidth not as В,
but as В ь in our equations. Тhat is, we create а gиard band оп either side
of our fiYter so that there сап Ье а small amoиnt of aliasing in the discrete
spectrum withoиt distorting our desired signal, as shown at the bottom of
Figure 2-13.
We can relate this idea of using guard bands to Figure 2-11 Ьу looking
more c10sely at one of the white wedges. As shown in Figure 2 14, we' d like
40 Perlodic Saтpllпg
Sampling rate (1,/8)
з
"
8 = 4,5, 80 (,= 22.5 MHz
8
7
6
5
4
(,
8=3' so t.=15MHz
2
'.
В = 2.25, 80 '.= 11,25 MHz
о
1
2 3 4
Ratio: highest frequency component/bandwidth
5 R
Flgure 2.12 AcceptabIe sample rates for the bandpass slgnal example (В = 5 MHz)
with а value of R = 4.5,
to set oиr sample rate as far down toward the vertex of the white area as we
can lower in the wedge теans а lower sampling rate. However, the clos
er we operate to the boundary of а shaded area, the more naпоw the guard
band must Ье, requiring а sharper analog bandpass filter, as well as the
tighter the tolerance we must impose оп the stability and accиracy of our
A/D clock generator. (Remember, operating оп the boundary between а
white and shaded area in Figиre 2-11 causes spectral replications to butt ир
agamst еаф other.) 50, to Ье safe, we operate at some intermediate point
away from any shaded boundaries as shown in Figиre 2 14. Further analy
sis of how guard band widths and A/D clock parameters relate to the
geometry of Figure 2-14 is available in reference [13]. For this discussion,
we'll just state that it's а good idea to ensure that oиr selected sample rate
does not lie too close to the boundary between а white and shaded area in
Figure 2 11.
There are а couple of ways to make sure we're not operating near а
boundary. One way is to set the sample rate in the middle of а white
wedge for а given value of R. We do this Ьу taking the average between
the maximum and minimum sample rate terms in Eq. (2 10) for а partic
ular value of т, that is, to center the sample rate operating point within а
wedge we use а sample rate of
Sampliпg Baпdpass Sigпa/s 41
L
о )..
Noise в Fllter ,езропзе
""'- I " " т ,
\ >Л; Noise
?1 и l C
Bgb >
Guard band
Freq
; с l [\
Bgb ---------------;
r
Freq
Origlnal
continuous
signal
Bandpass
Filter
ЛI
7
AlIl!Islng
Digital
samples
Filtered
continuous slgnal
Aliaslng
' ' "' " ..---'о
....L..L I ' ] I
о
fs
;
>i<
lrx !
fs
j .\
Freq
"' ", .
>'
Flgure 2-13 Bandpass sampling with aliasing occurring only in the fiiter guard bands,
Guard band width
} AJD clock
tolerance
Ideal operating
point
Flgure 2-14 Typical operating point for (, to compensate for nonideal
hardware,
42 Period/c Sampllпg
!s =.!., [ 2!с B + 2!с +В ] = !С B/2 + !С +В/2 ,
"", 2 m т+1 m т+1
(2-12)
Another way to avoid the boundaries of Figure 2 14 is to use the follow-
ing expression to determine an intermediate fs, operating point:
1
t. = 4!с
" modd'
(2 13)
where modd is an odd integer[14], Of course the choice of modd must ensure
that the Nyquist restriction of fs' > 2В Ье satisfied. We show the resu1ts of
1
Eqs. (2-12) and (2 13) for our bandpass signal example in Figure 2-15.
Sampling rate ('slв)
2
"cnt' = 28,75 MHz
8
7
6
5
4
(.cn!,= 16,25 MHz
з
'. cn!r = 11,46 MHz
о
2 3 4
Ratio: highest frequency component/bandwidth
4,5
5 R
Flgure 2-15 Intermediate "1 and "cntr operating points. from Eqs, (2-12) and (2-1 З).
to avoid operating at the shaded boundaries for the bandpass signal
example, В = 5 MHz and R = 4,5,
Spectro//пversioп iп Boпdposs Soтpliпg 43
2.4 Spectrallnversion in Bandpass Sampling
Some оЕ the permissible fs values from Eq, (2-10) will, although avoiding
aliasing problems, provide а sampled baseband spectrum (located near
zero Hz) that is inverted from the original positive and negative spectral
shapes, that is, the positive baseband will have the inverted shape оЕ the
negative half from the original spectrum, This spectral inversion hap-
pens whenever т, in Eq. (2-10), is an odd integer, as illustrated in
Figures 2 9(b) and 2 9(e), When the original positive spectral bandpass
components are symmetrical about the fc frequency, spectral inversion
presents по problem and any nonaliasing value for f. from Eq, (2 10)
тау Ье chosen, However, if spectral inversion is something to Ье avoid
ed, for example, when single sideband signals are being processed, the
minimum applicable sample rate to avoid spectral inversion is defined
Ьу Eq. (2-10) with the restriction that т is the largest even integer such
thatf. 2В is satisfied. t Using our definition of optimum sampling rate,
the expression that provides the optimum noninverting sampling rates
and avoids spectral replications butting ир against each other, except at
zero Hz, is
; = 2!c B
Js '
о т even
(2 14)
where т еуеп = 2, 4, 6, etc. For our bandpass signal example, Eq. (2-14) and
т = 2 provide an optimum noninverting sample rate оЕ fs = 17.5 MHz, as
о
shown in Figure 2 9(c), In this case, notice that the spectrum translated
toward zero Hz has the same orientation as the original spectrum cen
tered at 20 MHz.
Then again, if spectral inversion is unimportant for your application,
we сап determine the absolute minimum sampling rate without having
to choose various values for т in Eq. (2-10) and creating а table like we
did for Table 2-1. Considering Figure 2 16, the question is "How many
replications оЕ the positive and negative images of bandwidth В сап we
squeeze into the frequency range оЕ 2fc + В without overlap?" That num
ber of replications is
R = frequency span 2!с + В = !с + в / 2
twice the bandwidth 2В В
(2-15)
t Single sideband signals are discussed in Section с.4 of Appendix С.
44 Periodic Saтp//пg
1 -< 2fc+ В >- 1
i 18
8 1
L
\\\
fc о fc Freq
Flgure 2.16 Frequency span of а contlnuous bandpass slgnal,
То avoid overlap, we have to make sure that the number of replications is
an integer less than or equal to R in Eq, (2-15), 50, we can define the inte-
gral number of replications to Ье Rmt where
R int S; R < R int + 1 ,
or
!с +В/2
R int S; < R int + 1 ,
В
(2 16)
With R int replications in the frequency span of 2fc + В, then, the spectral
repetition period, or minimum sample rate f Smin' is
f, = 2!с + В
'",", R int '
(2 17)
In our bandpass signal example, findingf Smin first requires the appro
priate value for l\n! in Eq, (2-16) as
R < 22.5 R 1
in! < in! + ,
5
so R iлt = 4. Then, from Eq, (2 17),fsmiл = (40+5)/4 = 11.25 MНz, which is
the sample rate il1ustrated in Figures 2 9(e) and 2-12,50, we сап use Eq,
(2 17) and avoid using various values for т in Eq. (2 10) and having to cre
ate а table like ТаЫе 2 1, (Ве careful though, Eq. (2 17) places our sampling
rate at the boundary between а white and shaded area of Figure 2 12, and
we have to consider the guard band strategy discussed above,) То recap
the bandpass signal example, sampling at 11,25 MHz, from Eq, (2-17),
Spectra//пversioп iп Baпdpass Saтpliпg 4
avoids aliasing and inverts the spectrum, while sampling at 17.5 MHz,
from Eq, (2-14), avoids aliasing with по spectral inversion,
Now here's some good news, With а little additional digital process-
ing, we сап sample at 11,25 MHz, with its spectral inversion and easily
reinvert the spectrum back to its original orientation. Тhe discrete spec-
trum of any digital signal сап Ье inverted Ьу multiplying the signal's dis-
crete-time samples Ьу а sequence of alternating plus ones and minus
ones (1, 1, 1, 1, etc,), indicated in the literature Ьу the succinct expres
sion ( l)п, This scheme aHows bandpass sampling at the lower rate of Eq,
(2 17) while correcting for spectral inversion, thus avoiding the necessity
of using the higher sample rates from Eq, (2 14), Although mu1tiplying
time samples Ьу ( l)п is explored in detail in Section 10,1, аН we need to
remember at this poirit is the simple rule that multiplication of real sig-
nal samples Ьу ( 1)п is equivalent to multiplying Ьу а cosine whose fre
quency is 1./2, In the frequency domain, this multiplication flips the pos-
itive frequency band of interest, from zero to +1./2 Hz, about 1./ 4 Hz, and
flips the negative frequency band of interest, from 1./2 to zero Hz, about
1./4 Hz as shown in Figure 2 17, Тhe ( l)п sequence is not only used for
inverting the spectra of bandpass sampled sequences; it сап Ье used to
invert the spectra of low-pass sampled signals, Ве aware, however, that,
in the low pass sampling case, any DC (zero Hz) component in the orig
inal continuous signal wШ Ье translated to both +1./2 and 1./2 after
mu1tiplication Ьу ( l)п,
Band 01 Band 01
interest interest
,О l l
12 ',/4 О ',/4 ',/2 Freq
r:=:>
(Ь) ...:......... ..................i<.... ................<..//>
',/2 ',/4 О ',/4 12 Freq
Flgure 2-17 Spectrallnversion through mu\tlplicotion Ьу ( l)П: (о) origlno\ spectrum
of о time-domoin sequence; (Ь) new spectrum of the product of
orlginol time sequence ond the ( l)П sequence,
46 Periodic Sampliпg
B
B .
" ,
'.'" "', " о, "" "
'. ". ". ". '.
'. " "'. "
. ..,.
, . "."' ", "
.. " " " " '
.' ",' ", ""
!
i .
' "
, //:. /'::' ,<::':" .::'
. . . . . ..
>:"'..::.",: ,/. :'.':':,: <--. '..
.." ,,'., . // .
t 'с t
'c 812 'с + 812
..
Freq
'c
о
Figure 2-18 Coпtlпuous sigпai spectrum where baпdpass sampliпg is пot
possibIe because r;, 8/2 < 8,
Now that we have an understanding of bandpass sampling, we' d bet
ter remind ourselves of the situation where bandpass sampling is not pos-
sible, If а continuous bandpass signal's lowest frequency component is
less than the bandwidth, we can't use bandpass sampling, This condition
is shown in Figure 2 18 wherefc В/2 < В. There's по way to squeeze any
spectral replications betweenfc В/2 and zero Hz with bandpass sam
pling, In this case, we'd have to low pass sample the signal in Figure 2-17
at а rate of at least twice the highest frequency component, or
fs ;:: 2ifc + В/2),
We conclude this topic Ьу consolidating in Table 2-2 what we need to
know about bandpass sampling.
ТаЫе 2-2 Bandpass Sampling Relationships
i Requiremeпt Sample Rate Expressioп Coпditioпs
Acceptable ranges of 2!с Б > J; > 2!с + Б т = any positive integer so
fs for bandpass sam- that fs?:: 28,
pling: Eq, (2-10) m s т+l
Samp!e rate in the !с Б/2 !с +Б/2 т = any positive integer so
midd!e of the accept- !scl1tr = + tha t fs cntr ?:: 28,
аЫе samp!e rate m т+l
bands: Eq, (2-12)
Samp!e rate at an J; = 4!с т odd = any positive odd
intermediate point in integer so that fs,?:: 28 ,
the acceptable samp!e ' modd (Spectra! inversion occurs
rate bands: Eq, (2-13) when т odd = 3, 7, 11, etc,)
Optimum samp!e rate = 2fc Б m even = any even positive
to avoid spectra! fs, integer so that fs o ?:: 28,
inversion: Eq, (2-14) т еуеп
I Abso!ute minimum fs = 2!с + Б where
I to avoid aIiasing: Eq, !Smil1 R int !с + в /2
(2-17) R int < R int + 1
I в
Refereпces 47
References
[1] Crochiere, R.E, and Rabiner, L,R. "ОрНmиm PIR Digital Implementations for
Decimation, Interpolation, and Narrow band Filtering," IЕЕЕ Trans, оп
Acoust, Speech, and Sigпal Proc" Vol, ASSP 23, No, 5, October 1975,
[2] Steyskal, Н, "Digital Beamforming Antennas," Microwave Journal, January
1987,
[3] Hill, с. "Тhe Benefits of Undersampling," Electronic Design, July 11, 1994,
[4] Уаm, Е" and Redman, М, "Development of а 60-channel FDM-TDM Trans-
multiplexer," COMSAT Technical Review, Vol, 13, No, 1, Spring 1983,
[5] Ployd, Р., and Taylor, J, "Dual-Channel Space Quadrature-Interferometer
System," Microwave System Designer's Handbook, Fifth Edition, Microwave
Systems News, 1987,
[6] Lyons, R. С, "How Fast Must Уои Sample," Test and Measurement World,
November 1988,
[7] Stremler, F. Introduction to Communication Systeтs, Chapter 3, Second Edition,
Addison Wesley Publishing Со., Reading, Massachusetts, р, 125,
[8] Webb, R. С. "IF Signal Sampling Improves Receiver Detection Accuracy,"
Microwaves & RF, March 1989,
[9] Haykin, 5, Communications Systems, Chapter 7, John Wiley and Sons, New
York, 1983, р, 376,
[10] Feldman, С. В" and Bennett, W. R "Bandwidth and Transmission
Performance," Веи Systeт Tech, Jourпal, Vol, 28, 1989, р, 490.
[11] Panter, Р, F. Modulation Noise, and Spectral Analysis, McGraw-Нi11, New York,
1965, р, 527,
[12] Shanmugam, К. 5, Digital and Analogue Communications Systems, John Wiley
and Sons, New York, 1979, р. 378,
[13] Vaughan, R., Scott, N, and White, о. "Тhe Тheory of Bandpass Sampling,"
IЕЕЕ Trans, оп Sigпal Processing, Vol, 39, No, 9, September 1991, pp,1973 1984.
[14] Xenakis В" and Evans,A. "Vehicle Locator Uses Spread Spectrum Technology,"
RF Desigп, October 1992,
I CHAPTER THREE I
The Discrete Fourier
Transform
'Пlе discrete Fourier transform (DFТ) is one of the two most common, and
powerful, procedures encountered in the field of digital signal processing,
(Digital filtering is the other.) 'Пlе DFТ enables us to analyze, manipиlate,
and synthesize signals in ways not possible with continuous (analog) signal
processing. Even though it' s now used in almost every field of engineering,
we'll see applications for DFТ continue to flourish as its utility becomes
more widely understood. Because of this, а solid understanding of the DFТ
is mandatory for anyone working in the field of digital signal processing.
The DFТ is а mathematical procedure used to determine the harmonic,
or frequency, content of а discrete signal sequence. Although, for our pur
poses, а discrete signal sequence is а set of values obtained Ьу periodic
sampling of а continuous signal in the time domain, we'll find that the
DFТ is useful in analyzing any discrete sequence regardless of what that
sequence actually represents. 'Пlе DFТ's origin, of course, is the continu
ous Fourier transform Х(!) defined as
Х(f) = f x(t)e j21tftdt ,
(3-1)
where x(t) is some continuous time domain signaU
In the field of continuous signal processing, Eq, (3-1) is used to transform
an expression of а continuous time--domain fиnction x(t) into а continuous
frequency-domain fиnction X(fJ. Subsequent evaluation of the Х(!) expres
sion enables us to determine the frequency content of any practical signal
of interest and opens up а wide array of signal analysis and processing
t Fourier is pronounced 'for-ya, In engineering school, we called Eq, (3-1) the "four-year"
transform because it took about four years to do оnе homework problem,
49
50 The Discrete Four/er Traпsforт
possibilities in the fields of engineering and physics, One could argue that
the Fourier transform is the most dominant and widespread mathematical
mechanism available for the analysis of physical systems, (А prominent
quote from Lord Kelvin better states this sentiment: "Fourier's theorem is
not only one of the most beautiful results of modem analysis, but it тау Ье
said to fumish an indispensable instrument in the treatment of nearly every
recondite question in modem physics," Ву the way, the history of Fourier's
original work in harmonic analysis, relating to the problem of heat condиc
tion, is fascinating. References [1] and [2] are good places to start for those
interested in the subject,)
With the advent of the digital computer, the efforts of early digital pro--
cessing pioneers led to the development of the DFТ defined as the discrete
frequency domain sequence Х(т), where
DFТ equation
(exponential Еоrш):
N J
Х(т) = Lx(n)e j21tпm/N
п=О
(3 2)
For our discussion of Eq, (3 2), х(п) is а discrete sequence of time domain
sampled values of the continиous variable хи), Тhe "е" in Eq, (3 2) is, of
course, the base of naturallogarithms and j = Д.
3.1 Understanding the DFТ Equation
Equation (3 2) has а tangled, almost unfriendly, look about it, Not to worry.
After studying this chapter, Eq, (3 2) will Ьесоmе one of our most familiar
and powerful tools in understanding digital signal processing, Let's get
started Ьу expressing Eq, (3 2) in а different way and examining it care
fully, From Euler's relationship j", = COS(0) jSin(0), Eq, (3 2) is equivalent to
DFТ equation N J
(rectanguIar Еоrш): Х(т) = Lx(n)[cos(21tnm / N) j sin(21tnm / N)] , (3 3)
п=О
We have separated the complex exponential of Eq, (3 2) into its real and
imaginary components where
Х(т) = the mth DFТ oиtput component, i,e., Х(О), Х(l), Х(2), Х(3), etc.,
т = the index of the DFT output in the frequency domain,
т = О, 1, 2, 3,. , "N l,
х(п) = the sequence of input samples, х(О), х(l), х(2), х(3), etc"
Uпderstaпdlпg the OFТ Equatioп 51
п = the time domain index оЕ the input samples, п = 0,1,2,3, ' , "N 1,
j = Н, and
N = the number of samples of the input sequence and the number of
frequency points in the DFT oиtput.
Although it looks more complicated than Eq, (3-2), Eq, (3-3) tums out
to Ье easier to understand, (Н you're not too comfortable with it, don't let
the j = н concept bother you too much. It's merely а convenient
abstraction that helps иs compare the phase relationship between vari
ous sinusoidal components of а signal. Appendix С discusses the j oper-
ator in some detail.)t Тhe indices for the input samples (п) and the DFT
output samples (т) always go from О to N l in the standard DFT nota-
tion, Тhis теаns that with N input time-domain sample values, the DFТ
determines the spectral content оЕ the input at N equaHy spaced fre
quency points. The valиe N is an important parameter because it deter-
mines how many input samples are needed, the resolution of the
frequency-domain results, and the amount of processing time necessary
to calculate an N point DFТ.
It's usefиl to see the structure ofEq. (3-3) Ьу eliminating the summation
and writing out аН the terms, For example, when N = 4, п and т both go
from О to 3, and Eq, (3-3) becomes
з
Х(т) = Lx(n)[coS(2тim/ 4) jSin(2т-zm/ 4)]
п=О
(3-4а)
Writing out аН the terms for the first DFТ output term corresponding to
т=О,
Х(О) = х(О) cos(21t. О. О / 4) jx(O) sin(21t. О. О / 4)
+ х(l) cos(21t .1. 0/ 4) jx(l) sin(21t .1. О / 4)
+ х(2) cos(21t. 2. О / 4) jx(2) sin(21t. 2. 0/4)
+ х(3) cos(21t. 3. О / 4) jx(3) sin(21t. 3. 0/4),
(3-4Ь)
For the second DFT output term corresponding to т = 1, Eq. (3-4а)
becomes
t Instead of the letter j, Ье aware that mathematicians often use the letter i to represent the
.[:i operator,
52 The Discrete Foиrier Traпsforт
Х(l) = х(0)СОБ(2п. О .1/ 4) jx(0)sin(21t. О .1/ 4)
+ х(l) СОБ(2п .1.1/ 4) jx(l) sin(21t .1.1/4)
+ х(2) СОБ(2п . 2.1/ 4) jx(2) sin(21t . 2.1/4)
+ х(3) СОБ(2п' 3.1/ 4) jx(3) sin(21t. 3.1/4), (3 4c)
For the third output term corresponding to т = 2, Eq, (3 4a) ЬесотеБ
Х(2) = х(0)СОБ(2п. 0.2/ 4) jx(O) sin(21t. 0.2/4)
+ х(l) СОБ(2п .1. 2 / 4) jx(l) sin(21t .1. 2/4)
+ х(2) cos(21tv2. 2/ 4) jx(2) sin(21t . 2.2/4)
+ х(3) СОБ(2п . 3.2/ 4) jx(3) sin(21t . 3.2/4).
(3 4d)
Finally, for the fourth and last output term corresponding to т = 3, Eq,
(3-4а) ЬесотеБ
Х(3) = х(О) СОБ(2п. 0.3/ 4) jx(O) sin(21t. 0.3/4)
+ х(l) СОБ(2п .1. 3 / 4) jx(l) sin(21t .1. 3/4)
+ х(2) СОБ(2п' 2.3/ 4) jx(2) sin(21t. 2.3/4)
+ х(3) СОБ(2п . 3.3/ 4) jx(3) sin(21t. 3.3/4), (3-4е)
The above multiplication symbol "." in Eq, (3 4) is used merely to separate
the factors in the sine and cosine terms, The pattem in Eq, (3 4b) through
(3 4e) is apparent now, and we сап certainly Бее why it's convenient to use
the summation sign in Eq, (3 3), Each Х(т) DFТ output term is the Бит of
the poiпt for poiпt product between an input sequence of signal values and
а complex sinusoid of the form СОБ(0) jSin(0), The exact frequencies of the
different sinusoids depend оп both the sampling rate fs, at which the orig
inal signal was sampled, and the number of samples N, For example, if we
are sampling а continuous signal at а rate of 500 samples/s and, then, per
forrn а 16 point DFТ оп the sampled data, the fundamental frequency of
the sinusoids is fs/N = 500/16 or 31,25 Hz, The other Х(т) analysis fre
quencies are integral multiples of the fundamental frequency, i.e"
Х(О) = 1st frequency term, with analysis frequency = О . 31,25 = О Hz,
Х(l) = 2nd frequency term, with analysis frequency = 1 . 31.25 = 31.25 Нz,
Х(2) = 3rd frequency term, with analysis frequency = 2 . 31,25 = 62.5 Нz,
Х(3) = 4th frequency term, with analysis frequency = 3 . 31,25 = 93,75 Нz,
Х(15) = 16th frequency term, with analysis frequency = 15 . 31,25 = 468,75 Нz,
Uпderstaпdiпg the OFТ Equatioп 5
Imaginary axis о)
Ximag(m)
Xmag(m)
#
"
This point represents the
complex number
Х(т) = X'eal(m) + jX1mag(m),
о
X'eal(m)
Аеа\ axis
Flgure 3-1 Trlgonometrlc relatlonships of an indlvldual OFТ Х(т) complex
output value,
The N separate DFТ analysis frequencies are
mfs
fanalysis(m)="N '
(3 5)
50, in this example, the Х(О) DFТ term tells из the magnitude of any O-Hz
("ОС") component contained in the input signal, the Х(l) term зресiПез
the magnitude of any Зl,25 Нz component in the input signal, and the
Х(2) term indicates the magnitude of anу 62.5 Hz component in the input
signal, etc. Moreover, аз we'll soon show Ьу example, the DFT output
terms also determine the phase relationship between the various analysis
frequencies contained in an input signal,
Quite often we're interested in both the magnitude and the power
(magnitude squared) contained in each Х(т) term, and the standard def
initions for right triangles apply here аз depicted in Figure 3-1,
If we represent an arbitrary DFТ output value, Х(т), Ьу its real and
imaginary parts
Х(т) = X real (т) + jXimag(m) = Xmag(m) at an angle of Хе(т) , (3 6)
the magnitude of Х(т) is
Xmag(m) = IХ(т)1 = Xreal(т)2+Ximag(m)2
(3 7)
Ву definition, the phase angle of Х(т), Хе (т), is
[ Х. (т) ]
Хе(т) = tan l lmag ,
X rea1 (т)
(3 8)
54 The Discrete Fourier Traпsforт
Тhe power of Х(т), referred to as the power spectrurn, is the magnitude
squared where
Xps(m) = X ma i m )2 = X reai (m)2 + X imag (m)2 ,
(3-9)
3.1.1 DFТ Example 1
Тhe above Eqs. (3-2) and (3 3) will Ьесоте more meaningful Ьу way of an
example, so, let's go through а simple one step by-step, Let's say we want
to sample and perform an 8 point DFT оп а continuous input signal con-
taining components at 1 kНz and 2 kHz, expressed as
Xin(t) = sin(21t.1000.t) + 0.5sin(21t.2000.t+31t/4), (3-10)
То make our example input signal xin(t) а little mOre interesting, we have
the 2-kНz term shifted in phase Ьу 1350 (3п/4 radians) relative to the 1-kНz
sinewave, With а sample rate of fs, we sample the input every 1/ fs = t s sec-
onds, Because N = 8, we need 8 input sample values оп which to perform
the DFТ. 50 the 8-element sequence х(п) is equal to xin(t) sampled at the nt s
instants in tiше so that
х(п) = xin(nt s ) = sin(21t.1000.nt s ) + 0.5sin(21t.2000.nt s +3п/ 4) , (3-11)
If we choose to sample xin(t) at а rate of fs = 8000 samples/ s from Eq, (3-5),
our DFТ results wШ indicate what signal amplitude exists in х(п) at the
analysis frequencies of mfs/ N, or О kНz, 1 kНz, 2 kHz, ' , " 7 kНz, With
fs = 8000 samples/s, our eight х(п) samples are
х(О) = 0.3535, х(l) = 0,3535,
х(2) = 0,6464, х(3) = 1.0607,
х(4) = 0.3535, х(5) = 1.0607,
х(6) = 1,3535, х(7) = ,3535 .
(3 11 ')
These х(п) sample values are the dots plotted оп the solid continuous
xin(t) curve in Figure 3 2(a), (Note that the sum of the sinusoidal terms in
Eq. (3 10), shown as the dashed curves in Figure 3-2(а), is equal to xin(t).)
Now we're ready to apply Eq, (3-3) to determine the DFТ of our х(п)
input, We'll start with m = 1 because the m = О case leads to а special resu1t
that we'll discuss short1y, 50, for m = 1, or the l kНz (mf/N = 1.8000/8)
DFТ frequency term, Eq, (3 3) for this example becomes
Uпderstaпdiпg the оп Equatioп 55
7
Х(l) = LX(n)cos(21tn/8) jX(n)sin(21tn/8)
n=о
(И2)
Next we multiply х(п) Ьу successive points оп the cosine and sine curves
of the first analysis frequency that have а single cycle over our 8 input
samples, In our example, for т = 1, we'll sum the products of the х(п)
sequence with а l kHz cosine wave and а 1-kНz sinewave evaluated at
the angular values of 2пп/8. Тhose analysis sinusoids are shown as the
dashed curves in Figure 3 2(b). Notice how the cosine and sinewaves
have т = 1 complete cycles in our sample interval,
5ubstituting our х(п) sample values into Eq. (3 12) and listing the
cosine terms in the left column and the sine terms in the right column,
we have
Х(l) = 0,3535 . 1,0
+ 0,3535 . 0,707
+ 0,6464 . 0.0
+ 1.0607 . 0.707
+ 0,3535 . 1.0
1,0607 . O,707
1,3535.0.0
0,3535 . 0,707
0.3535
+ 0,250
+ 0.0
0.750
0.3535
+ 0.750
+0,0
0.250
j(O,3535 . 0,0) t-- this is the п = О term
j(0.3535 . 0,707) t-- this is the п = 1 term
j(O,6464 . 1,0) t-- this is the п = 2 term
j(l,0607 . 0.707)
j(O,3535 . 0.0)
jH,0607 . O,707)
k1.3535 . 1.0)
j( 0.3535 . O,707) t-- this is the п = 7 term
+ jO,O
jO,250
jO,6464
jO,750
jO.O
jO,750
j1,3535
jO.250
0.0 j4.0 = 4 L 900,
50 we now see that the input х(п) contains а signal component at а fre
quency of 1 kНz. Using Eq. (3 7), Eq, (3 8), and Eq. (3 9) for our Х(l)
resu1t, X mag (l) = 4, X ps (1) = 16, and X(l)'s phase angle relative to а l kHz
cosine is Х,,(l) = 900,
For the т = 2 frequency term, we correlate х(п) with а 2 kHz cosine wave
and а 2 kНz sinewave. Тhese waves are the dashed curves in Figure 3 2(c),
Notice here that the cosine and sinewaves have т = 2 complete cycles in our
56 The Discrete Fourier Traпsforт
(а)
1'5 !
SIn(21t1000t) / X1n(t)
1 ,. ;:-<:""k
0,5 ./......--/ >_... . \.
. . ,,'" '."
о . -'о ,
.' ....
.."-..--.\
"
"
О,5sin(21t2000t+З"'4)
....',--_... /,:",,"
.. --./ "
"'(),5
1
1,5 о
1'5 !
т=1
1 _'О. ;:-<.
0,5 ><, ......... ........
...........,..... ...... .
о / . ......
п ...
(Ь)
1
/
/
> /./
5 в 7
п
...(),5
(с)
o 4
1'5 !
т=2
1 О" / " /. Х '--О" / ...
" 'о. ,'?"
O,5:...L-..c.... '1' \
8/ ,. \ /. " \
о' , ,
1
/
/
/
./
...(),5
1
\
\
,".
"
.. \, /
''.0 У ' ,". //
. , '-. ./
.
1,5
о
п
1'5 !
m=З
" .............'.... .......,
'/ '\ /,,' ',/
O,5. . .:': { /
о ' \ '
1
\' /
", ,,'\,,, f
I \.
/ / ',IIi,'
1 7 ' 'у' /
/ ., /
."-.......... - - . ,
5 в 7 п
(d)
...(),5
1,5 о
Flgure 3-2 OFТ Example 1: (а) the input slgnal; (Ь) the Input signal and the т = 1
sinusolds; (е) the Input signal and the т = 2 sinusolds; (d) the Input slgnal
and the т = 3 slnusolds,
Uпderstaпdiпg the DFT Equatioп 5:
sample interval in Figure 3 2(c). Substituting our х(п) sample values in Eq,
(3--3), for т = 2, gives
Х(2) = 0,3535 . 1.0
+ 0,3535 . 0.0
+ 0.6464 . 1.0
+ 1,0607 .0.0
+ 0.3535 . 1.0
l,0607 ' 0.0
1,3535 ' l,O
0.3535 . 0,0
= 0,3535
+0.0
0,6464
o.o
+ 0.3535
+ 0,0
+ 1.3535
o,o
j(0.3535 . 0.0)
j(0.3535 . 1.0)
j(O,6464 . 0,0)
j(l,0607 . l,O)
j(0.3535 . 0.0)
k1.0607 . 1,0)
jH.3535 . 0.0)
kO,3535 . l,O)
+ jO.O
jO,3535
jO.O
+ j1,0607
jO.O
+ j1,0607
jO,O
jO.3535
1.414 + j1.414 = 2 L 450,
Here our input х(п) contains а signal at а frequency of 2 kНz whose rela
tive amplitude is 2, and whose phase angle relative to а 2 kHz cosine is
450, For the т = 3 frequency term, we correlate х(п) with а 3 kHz cosine
wave and а 3 kНz sinewave, Тhese waves are the dashed curves in Figure
3 2(d). Again, see how the cosine and sinewaves have т = 3 complete
cycles in our sample interval in Figure 3 2(d). Substituting our х(п) sam-
ple values in Eq. (3 3) for т = 3, gives
Х(3) = 0.3535 . 1.0
+ 0,3535 . O,707
+ 0.6464 . 0,0
+ 1,0607 . 0.707
+ 0,3535 . 1.0
1.0607 . 0.707
1.3535 ' 0,0
0,3535 . 0.707
j(O,3535 . 0,0)
j(0.3535 . 0,707)
j(0.6464 ' l,O)
j(l,0607 . 0,707)
j(O,3535 . 0,0)
jH.0607 . 0.707)
jH , 3535 . 1.0)
kO,3535 ' O,707)
58 The Discrete Fourier Troпsform
+ jO,O
jO,250
+ jO,6464
jO.750
jO,O
jO,750
+ j1,3535
jO,250
0.0 jO.O = О L 00.
0,3535
0,250
+ 0,0
+ 0.750
0,3535
0,750
+ 0,0
+ 0,250
Our DFТ indicates that х(п) contained по signal at а frequency of 3 kHz,
Let's continue our DFТ for the т = 4 frequency term using the sinusoids
in Figure 3-3(а),
So Eq, (3-3) is
Х(4) = 0.3535 . 1.0
+ 0,3535 . l,O
+ 0.6464.1,0
+ 1,0607 . 1.0
+ 0.3535 . 1,0
1.0607 . 1.0
1.3535 . 1,0
0,3535 . 1.0
0,3535
0.3535
+ 0,6464
1.0607
+ 0,3535
+ 1.0607
1.3535
+ 0,3535
j(0.3535 . 0,0)
j(O,3535 . 0,0)
j(0.6464 . 0,0)
j(l,0607 . 0,0)
j(O,3535 . 0.0)
jH,0607. 0.0)
k1.3535 . 0.0)
kO,3535 . 0.0)
jO,O
jO.O
jO.O
jO,O
jO.O
jO,O
jO.O
jO,O
= 0,0 jO,O = О L 00,
Ош DFТ for the т = 5 frequency term using the sinusoids in Figure 3-3(Ь)
yields
Х(5) = 0,3535. 1.0
+ 0,3535 . 0.707
+ 0.6464 . 0.0
+ 1.0607 . 0.707
j(O,3535 . 0,0)
j(O,3535 . O,707)
j(O,6464 . 1.0)
j(1.0607 . 0.707)
Uпderstaпdiпg the OFТ Equatioп 59
"(),5
1'5 1
т=4
1 ,А " . .. "'" " (',
'/ \ " х \ ,"; \ " '( \
0,5 '/ \ . ; \ \ " '. 1', \ ( ' \
. . ,( \ ,.,
О " , ',\" \
'; :, \,' "
1, \'; \ 1, ".,' ,
, у, ,,)', \ J / :' ;
',,' \../ ',,' \/ .,' \ / '..' \,,1
.
5 6 7 п....
1
(а)
(Ь)
1,5 о
1'5 1
т=5
1 ' ( '" '_,/',............... .."\...
,\ 'у х ", \
0,5 ,', \ . 1 " '"
. \ . "\ " 1, ·
о' " , , ,\
, \: , ' \,' / \
\ " \ ' \ '
, / "
,\ J " ' , ' ,1 ,
.....J ........ " v
,r,(\
" " \
/ ", \
:' r
: f... \
" \
, " \
1
I ,\, I \ ,
\ \,' У .' \: I
, ,\ J " ,\ I
. .'- '.'"
5 в 7 п....
"(),5
1,5 о
"(),5
1.5 1
т=6
1 ',/ \ :..,;..... ; /I\ ,"'/\
0,5 f : (,..).. " ), \ "'\ f, \ " ' \
. . \ " I \ :.", \ " ,', \
О \, ,\ " "
, \ ' , ' , \ ' , '\"
\:} \:, \ \ '1 \ \ :1 " У ,".
, ' 'J"" '
, ,\' ,:' ','\ " 1 J ',\'
'. \; . 1; .' J "." '.'
.
5 в 7
1
(с)
(d)
1,5 о
1'5 1
т=7
1 -., "! ....,......... /' ., "' 1 '
'J ' ,\ /.' 1 ' \ " ,( \ "\
0,5 1', \ ,' .: }', \', " (.\, '1' \ : ", \
. ,. ,\ , " \' ' , : 1 " , \ ' '" \
О ,\ I " ' ' : '
, \,' , ' \' , "',
' , ' '" '' \ , :' } ", \/ " )! '" \.: .
" " ,'\ '. " 1
" v ...... "
. .
5 в 7
п .....
,', f'.
, \
: }', \
1
" \,'
" \,'
" ,\/
"(),5
1,5 о
п .....
Flgure 3-3 OFТ Example 1: (о) the Input slgnal and the m = 4 sinusolds: (Ь) the Input
and the m = 5 sinusolds; (с) the input and the m = 6 sinusoids; (d) the
input and the m = 7 slnusoids,
60 The Discrete Fourier Traпsforт
+ 0.3535 . 1,O
1.0607 .0,707
1.3535 . 0,0
0.3535 . 0.707
= 0.3535
0.250
+ 0,0
+ 0,750
0.3535
0.750
+ 0,0
+ 0,250
= 0.0 j.O = О LO°,
j(0.3535 . 0,0)
k1,0607. 0.707)
k1.3535 . 1.0)
k0.3535 . 0,707)
jO,O
+ jO.250
jO.6464
+ jO.750
jO.O
+ jO,750
j1.3535
+ jO.250
For the rn = 6 frequency term using the sinusoids in Figure 3 3(c), Eq, (3 3) is
Х(6) =0.3535 . 1.0
+ 0.3535 . 0.0
+ 0,6464 . 1,O
+ 1.0607 . 0,0
+ 0.3535 . 1,0
1,0607. 0.0
1.3535 . 1.0
0.3535 . 0,0
= 0.3535
+ 0,0
0,6464
+ 0,0
+ 0.3535
+ 0.0
+ 1.3535
+ 0,0
j(0.3535 . 0.0)
j(0.3535 . 1,O)
j(0.6464 . 0.0)
j(1,0607 .1,0)
j(0.3535 . 0.0)
k1,0607 . 1.0)
k1.3535 . 0.0)
k0.3535 . 1,0)
jO,O
+ j0.3S35
jO.O
j1.0607
jO,O
j1,0607
jO,O
+ j0.3535
1,414 j1.414 = 2 L 450,
For the rn = 7 frequency term using the sinusoids in Figure 3 3(d), Eq. (3 3)
is
Х(7) = 0.3535 . 1.0
+ 0.3535.0.707
+ 0.6464 . 0,0
+ 1.0607 . O,707
j(0.3535 ' 0.0)
j(0.3535 . 0.707)
j(O.6464 . 1,O)
j(1,0607 . O,707)
+ 0,3535 . 1.0
1.0607 . 0.707
1.3535 . 0,0
0.3535 . 0,707
0.3535
+ 0,250
+ 0.0
0.750
0.3535
+ 0,750
+ 0,0
0,250
Uпderstaпdiпg the DFТ Equatioп 61
j(0.3535 . 0.0)
k1,0607 . 0,707)
k1.3535 . 1.0)
kO,3535 . 0,707)
+ jO.O
+ jO,250
+ jO.6464
+ jO.750
jO,O
+ jO,750
+ j1,3535
+ jO,250
0,0 + j4.0 = 4 L 900,
If we plot the Х(т) output magnitudes as а function of frequency, we get
the magnitude spectrum of the х(п) input sequence, shown in Figure 3 4(a),
Тhe phase angles of the Х(т) output terms are depicted in Figure 3-4(Ь),
Hang in there, we're almost finished with our example. We've saved
the calculation of the m = О frequency term to the end because it has а spe
cial significance. When m = О, we correlate х(п) with cos(O) jsin(O) so that
Eq. (3 3) becomes
N l
Х(О) = L x(n)[cos(O) j sin(O)] ,
n=о
(3 13)
! Example 1: МаgПltudе 01 Х(т)
4 "
3
2 "
1
(а) О I
045 6
i Example 1: Phase 01 Х(т) '" degroes, Х.(т)
00 "
45 1 " 6 '
(Ь) О т " , . ! I т
-45 i 4 5 7 (kHz)
90 "
i
7 (k Z)
j Example 1: Aeal ра" 01 Х(т)
1.5. .
1
0.5
(с) o . .......... .
о 1 2 3 4 5 6 7 (k Z)
Example 1: Imaginary part 01 Х(т)
4 "
2 1" 6 '
(d) о i " I i . I i
2 ,2 4 5 · 7 (k Z)
-4 "
Flgure 3-4 DFТ results from Example 1: (а) magnitude of Х(т); (Ь) phase of Х(т);
(е) real part of Х(т); (d) imaginary part of Х(т),
62 The Discrete Fourier Traпsforт
Because cos(O) = 1, and sin(O) = О,
N l
Х(О) = LX(n) ,
n=о
(3 13')
We сan see that Eq, (3-13') is the sum of the х(п) samples. Тhis sum is, of
course, proportional to the average of х(п), (5pecifically, Х(О) is equal to N
times x(n)'s average value). Тhis makes sense because the Х(О) frequency
term is the nontime-varying (ОС) component of х(п), If Х(О) were nonzero,
this woиld tel1 us that the х(п) sequence is riding оп а DC bias and has some
nonzero average value. For our specific example input from Eq. (3 10), the
sum, however, is zero, The input sequence has по DC component, so we
know that Х(О) will Ье zero, But let's not Ье lazy we'l1 calcиlate Х(О) anу
way just to Ье sure. Evaluating Eq. (3 3) or Eq. (:И3') for т = О, we see that
Х(О) = 0,3535 . 1,0
+ 0.3535 . 1,0
+ 0,6464 . 1,0
+ 1,0607 . 1,0
+ 0,3535 . 1,0
1,0607 . 1.0
1,3535 . 1,0
0.3535 . 1,0
Х(О)= 0.3535
+ 0,3535
+ 0,6464
+ 1.0607
+ 0,3535
1.0607
1.3535
0,3535
1'(0,3535 . 0,0)
1'(0.3535 . 0.0)
1'(0.6464 . 0,0)
1'(1.0607 . 0,0)
1'(0.3535 . 0.0)
1'( l,0607 . 0,0)
1'H.3535 . 0,0)
1'( O,3535 . 0.0)
1'0.0
jO.O
jO.O
jO,O
1'0.0
1'0,0
jO.O
1'0,0
0.0 1'0,0 = О L 00,
50 our х(п) had по DC component, and, thus, its average value is zero.
Notice that Figure 3 4 indicates that xin(t), from Eq. (3-10), has signal
components at 1 kHz (т = 1) and 2 kHz (т = 2). Moreover, the l kHz tone
has а magnitude twice that of the 2-kНz tone, The DFT results depicted in
Figure 3 4 tel1 us exact1y what's the spectral content of the signal defined
Ьу Eqs, (3-10) and (3 11).
DFT Syттefry 63
The perceptive reader should Ье asking two questions at this point, First,
what do those nonzero magnitude values at т = 6 and т = 7 in Figure 3-4(а)
теan? Alsо, why do the magnitudes seem four times larger than we would
expect? We'll answer those good questions shortly. The above 8-point ОРТ
example, although adm.ittedly simple, illustrates two very important char-
acteristics of the ОРТ that we should never forget. First, anу individual Х(т)
output value is nothing more than the sum of the term-by-term products, а
correlation, of an input signal sample sequence with а cosine and а sinewave
whose frequencies зrе т complete cycles in the total sample interval of N
samples. тhis is true по matter what the fs sample rate is and по matter how
large N is in an N-point ОП The second important characteristic of the ОРТ
of real input sarnples is the symmetry of the ОРТ output terms.
3.2 DFT Symmetry
Looking at Figure 3-4(а) again, there is аn obvious symmetry in the ОРТ
resu1ts. Although the standard ОРТ is designed to accept complex input
sequences, most physical ОРТ inputs (such as digitized values of some
continuous signal) are referred to as real, that is, real inputs have nonzero
real sample values, and the imaginary sample values are assumed to Ье
zero. When the input sequence х(п) is real, as it will Ье for аП of our ехат-
ples, the complex ОРТ outputs for т = 1 to т = (N /2) 1 are redundant
with frequency output values for т > (N /2), The mth ОРТ output wi11
have the same magnitude as the (N m)th ОРТ output, The phase angle of
the OPТ's mth output is the negative of the phase angle of the (N m)th
ОРТ output. 50 the mth and (N m)th outputs are related Ьу the following:
Х(т) =1 Х(т) 1 at Х,,(т) degrees
=1 X(N т) 1 at X,,(N т) degrees.
(3-14)
We сап state that when the ОРТ input sequence is real, Х(т) is the сот-
plex conjugate of X(N m), or
Х(т) = X'(N т) ,+
(3-14')
where the superscript' symbol denotes conjugation,
t Using our notation, the complex conjugate of х = а + jb is defined as х' = а jb; that is, we
merely change the sign of the imaginary part of х, In an equivalent fon1\, if х= ei", thenx' = e j",
64 The Discrete Four/er Traпsforт
In our example аЬоуе, notice in Figures 3-4(Ь) and 3 4(d), that Х(5),
Х(б), and Х(7) are the complex conjugates of Х(3), Х(2), and Х(l), respec
tively, Like the DFТ's magnitиde symmetry, the real part of Х(т) has what
is called eveп symmetry, as shown in Figure 3--4(с), while the DFТ's imagi
nary part has odd symmetry, as shown in Figure 3-4(d), Тhis relationship is
what is meant when the DFТ is called conjugate symmetric in the litera
tиre, It mеаns that, if we perform an N point DFТ оп а real input sequence,
we'll get N separate complex DFТ output terms, but оnlу the first N /2
terms are independent. 50 to obtain the DFТ of х(п), we need оnlу compute
the first N /2 values of Х(т) where О т (N /2) 1; the X(N /2) to X(N 1)
DFТ output terms provide по additional information about the spectrum
of the real sequence х(п).
Although Eqs. (3-2) and (3 3) are equivalent, expressing the DFТ in the
exponential form of Eq. (3 2) has а terrific advantage over the form of Eq,
(3 3). Not оnlу does Eq, (3 2) save реп and paper, Eq. (3 2)'s exponentials
are so much easier to manipulate when we're trying to analyze DFТ rela
tionships. Using Eq. (3 2), products of terms Ьесоmе the addition of ехро-
nents and, with due respect to Euler, we don't have all those
trigonometric relationships to memorize. Let's demonstrate this Ьу prov
ing Eq, (3 14) to show the symmetry of the DFТ of real input sequences.
5ubstitиting N m for т in Eq, (3 2), we get the expression for the (N m)th
component of the DFТ:
N l N l
X(N т) = Lx(n)e j21tп(N m)/ N = Lx(n)e j21tпN /N e j21tп( m)/ N
п O п O
N l
= Lx(n)e j21tпej21t11m/N.
п O
(3 15)
Because e j21tп = cos(21tn) jsin(21tn) = 1 for all integer values of п,
N l
X(N т) = Lx(n)e j21tпm/N.
п O
(3 15')
We see that X(N m) in Eq. (3 15') is merely Х(т) in Eq, (3 2) with the sign
reversed оп X(m)'s exponent and that's the definition of the complex
conjugate, This is illustrated Ьу the DFТ output phase angle plot in
Figure 3 4(b) for oиr DFТ Example 1. Try deriving Eq, (3-15') using the
DFТ Uпearity 65
cosines and sines of Eq. (3 3), and уои'll see why the exponential form of
the DFT is so convenient for analytical purposes.
There's an additional symmetry property of the DFT that deserves
mention at this point. In practice, we're occasionally required to deter-
mine the DFТ of real input functions where the input index п is defined
over both positive and negative values. If that real input function is even,
then Х(т) is always real and even; that is, if the real х(п) = x( п), then,
Xreal(m) is in general nonzero and Ximag(m) is zero, Conversely, if the real
input function is odd, х(п) = x( п), then Xrea1(m) is always zero and
Ximag(m) is, in general, nonzero. тhis characteristic of input function sym-
metry is а property that the DFТ shares with the continuous Fourier trans
form, and (don't worry) we'll cover specific examples of it later in Section
3.13 and in Chapter 5.
3.3 DFT Linearity
Тhe DFТ has а very important property known as liпearity, Тhis property
states that the DFТ of the sum of two signals is equal to the sum of the
transforms of each signal; that is, if an input sequence х 1 (п) has а DFT
Х 1 (т) and another input sequence х 2 (п) has а DFТ Х 2 (т), then the DFТ of
the sum of these sequences xsиm(п) = х 1 (п) + х 2 (п) is
Xsum(m) = Х 1 (т) + Х 2 (т) .
(3 16)
Тhis is certainly easy enough to prove, If we plug xsиm(п) into Eq. (3-2) to
get Xsum(m), then
N l
Xsum(m) = L[Xl(п) + x2(п)]e j21tпт/N
п=О
N l N l
= LXl(п)e j21t11т/N + LX2(п)e j21t11т/N =Х 1 (т)+Х 2 (т).
п=О п=О
Without this property of linearity, the DFТ would Ье useless as an analy
tical tool because we could transform only those input signals that contain
а single sinewave. Тhe real-world signals that we want to analyze are much
more complicated than а single sinewave,
66 The Discrete Foиrier Traпsforт
3.4 DFT Magnltudes
The DFT Example 1 results of I Х(l) I = 4 and I Х(2) I = 2 mау puzzle the
reader because our input х(п) signal, from Eq. (3 11), had peak amplitudes
of 1.0 and 0,5, respectively. There's an important point to keep in mind
regarding DFТs defined Ьу Eq. (3-2). When а real input signal contains а
sinewave component of peak amp1itude Ао with an integral number of
cycles over N input samples, the output magnitude of the DFТ for that
particular sinewave is М ' where
M r =AP/2 .
(3 17)
If the DFT input is а complex sinusoid of magnitude Ао (i,e., Ai 21tfl ) with
an integral number of cycles over N samples, the output magnitude of the
DFТ is Мс where
Мс = AoN .
(3 17')
As stated in relation to Eq, (3 13'), if the DFТ input was riding оп а DC
value equal to Do' the magnitude of the DFТ's Х(О) output will Ье DoN,
Looking at the real input case for the 1000 Hz component of Eq. (3 11),
Ао = 1 and N = 8, so that M rea1 = 1.8/2 = 4, as our example shows. Equation
(3 17) mау not Ье so important when we're using software or floating
point hardware to perform DFТs, but if we're implementing the DFТ with
fixed-point hardware, we have to Ье aware that the output сап Ье as large
as N /2 times the peak value of the input. This means that, for real inputs,
hardware memory registers mиst Ье аЫе to hold values as large as N 12
times the maximum amp1itude of the input sample values, We discuss
DFТ output magnitudes in further detai11ater in this chapter. The DFТ
magnitude expressions in Eqs. (3 17) and (3-17') are why we occasional1y
see the DFТ defined in the literature as
1 N l .
Х' (т) = N Lx(n)e /21tItт/N,
п=О
(3-18)
The l/N scale factor in Eq, (3-18) makes the amp1itudes of Х'(т) equal
to half the time domain input sinusoid's peak value at the expense
of the additional division Ьу N computation. Thus, hardware or soft-
ware implementations of the DFT typical1y use Eq, (3-2) as opposed to
Eq, (3-18). Of course, there are always exceptions, There are commercial
software packages using
DFТ Frequeпcy Axis 67
N l
Х"(т) = lN Ix(n)e j21tnт/N,
п:О
and
N l
х(п) =..l.... IX"(m)ej21tптl N
.JN п:О
(3-18')
for the forward and inverse DFTs. (In Section 3.7, we discuss the теan-
ing and significance of the inverse DFт.) Тhe 1/.JN scale factors in Eqs,
(3 18') seem а little strange, but they're used so that there's по scale
change when transforming in either direction. When analyzing signal
spectra in practice, we're normally more interested in the relative magni
tudes rather than absolute magnitudes of the individual DFT outputs, so
scaling factors aren't usually that important to us.
3.5 DFT Frequency Axis
Тhe frequency axis m of the DFТ result in Figure 3 4 deserves our attention
once again. Suppose we hadn't previously seen our DFТ Example 1, were
given the eight input sample values, from Eq, ( 11 '), and asked to реrfопn
an point DFТ оп them, We' d grind through Eq, (3-2) and get the Х(т) val
ues shown in Figure 4. Next we ask, "What's the frequency of the highest
magnitude component in Х(т) in Нz?" Тhe answer is not "1." Тhe answer
depends оп the original sample rate 1.. Without prior knowledge, we have по
idea over what tiше interval the samples were taken, so we don't know the
absolute scale of the Х(т) frequency юds. Тhe correct answer to the question
is to take 1. and plug it into Eq, (3 5) with m = 1. Тhщ if 1. = 8000 samples/ s,
then, the frequency associated with the largest DFТ magnitude tепn is
m/S 1. 8000
fanalysis(m) =N = fam.1YSis(1) = = 1000 Hz,
If we said the sample ratef. was 75 samples/s, we'd know, from Eq, (3 5),
that the frequency associated with the largest magnitude term is now
1.75
fanalysis(l) = 8 = 9.375Hz.
ОК, enough of this just remember that the DFТ's frequency spacing (res-
olution) is f.! N.
То recap what we've learned БО far:
68 The Discrete Fourier Traпsforт
· each DFT output term is the sum of the term by term products of аn
input time domain sequence with sequences representing а sine and
а cosine wave,
· for real inputs, an N point DFТ's output provides only N /2 indepen
dent terms,
· the DFТ is а linear operation,
· the magnitude of the DFT resu1ts are direct1y proportional to N, and
· the DFТ's frequency resolution is fs/N.
It's also important to realize, from Eq. (3 5), that X(N /2), when m = N /2, cor-
responds to half the sample rate, i.e. the folding (Nyquist) frequency М2.
3.6 DFT Shifting Theorem
There's аn important property of the DFТ known as the shifting theorem,
It states that а shift in Нте of а periodic х(п) input sequence manifests
itself as а constant phase shift in the angles associated with the DFT
results. (We won't derive the shifting theorem equation here because its
derivation is included in just about every digital signal processing text
book in print,) If we decide to sample х(п) starting at п equals some inte-
ger k, as opposed to п = О, the DFТ of those time shifted sample values is
Xshifteim) where
Xshifteim) = ej21tkm/N Х(т) .
(3 19)
Equation (3 19) tells us that, if the point where we start samp1ing х(п) is
shifted to the right Ьу k samples, the DFТ output spectrum of Xshifted(m) is
Х(т) with each of X(m)'s complex terms mu1tip1ied Ьу the linear phase
shift ei 2пkт / N , which is merely а phase shift of21tkm/Nradians or 360km/N
degrees. Conversely, if the point where we start sampling х(п) is shifted to
the left Ьу k samples, the spectrum of Xshifted(m) is Х(т) mu1tip1ied Ьу
e j2xkт/N, Let's illustrate Eq, (3 19) with an example,
3.6. 1 DFТ Example 2
Suppose we sampled our DFТ Example 1 input sequence later in time Ьу
k = 3 samples. Figure 3 5 shows the original input Нте function,
xin(t) = sin(21t1000t) + 0.5sin(21t2000t+31t/4) .
DFТ Shiftlпg Theoreт 69
Тhe DFТ in Example 1 was taken
over these eight sample values,
I
' f .. / ;"(tl /
i (Ш'/ \\ / .
1,5 i '-.....
--3 2
1
о
2
3
4
5
6
7 п.....
The DFТ in Example 2 is taken
over these eight sample values,
Flgure 3-5 Comparlson of sampling tlmes between DFT Example 1 and OFТ
Example 2,
We сап see that Figure 3-5 is а continuation of Figure 3-2(а), Our new х(п)
sequence becomes the values represented Ьу the so1id black dots in Figure
3-5 whose values are
х(О) = 1,0607,
х(2) = l,0607,
х(4) = O,3535,
х(6) = 0,3535,
х(l) = 0,3535,
х(3) = 1.3535,
х(3) = 0.3535,
х(7) = 0.6464 ,
(3 20)
Performing the DFТ оп Eq. (3-20), Xshifted(т) is
X'hifte/ m) =
m X'hifte/m)'S X'hifte/ m)' s X'hifte/m)'s X'hifte/m)'S
тagnitude phase real part iтagтary part
О О О О О
1 4 +45 2,8284 28284
2 2 45 1,4142 1.414
3 О о о о
4 О О О О
5 О О О О
6 2 +45 1.4142 1.4142
7 4 45 2.8284 2,828
(3 21)
70 The Diserete Fourier Traпsform
j Example 2' МаgПl!udе 01 Х shlhed (т)
4 . .
з
(а) 2' · '
1
О" " """""" I .....
m
О 3 4 5 7 (kHz)
t Example 2:
45 .
2
(Ь) 0' 1 I
--45
Phas" 01 X Sh11ted (т) in d"g,e"s
. .
4
i
6
! Example 2
3 .
2
(с) !
О.
О
Real ра'! 01 Xshihed (т)
.
I
' ''''''''''.
з 4 5
..
m
(kHz)
7
1'-
m
i (kHz)
.
4t Exa.mPle 2' Imaglnary ра,! 01 X shlfted (т)
2t 2 . 7
(d) О' I I . . . I .-
! 1 · 3 4 5 '" (k Z)
Figure 3-6 OFТ results from Example 2: (а) magnitude of Xshlf1ed(m): (Ь) phase of
XShlf1ed(m); (е) real part of XShlf1ed(m); (d) Imaginary part of Xshif1ed(m),
The values in Eq, (3 21) are illustrated as the dots in Figure 3-6. Notice that
Figure 3 6(a) is identical to Figure 3 4(a), Equation (3 19) told us that the
magnitude of Xshifted(m) should Ье unchanged from that of Х(т), That's а
comforting thought, isn't it? We wouldn't expect the DFТ magnitude of
our original periodic xin(t) to change just because we sampled it over а dif
ferent time interval, The phase of the DFТ result does, however, change
depending оп the instant at which we started to sample xin(t),
Ву looking at the т '= 1 component of Xshifted(m), for example, we сап
double check to see that phase values in Figure 3 6(b) are correct, Using
Eq. (3 19) and remembering that Х(l) from DFТ Example 1 had а magni
tude of 4 at а phase angle of 90 (or п/2 radians), k '= 3 and N = 8 so that
X shifted (l) = ej21tkm/N . Х(l) = ej2 .1/8. 4e j1t12 = 4ej(61t/8 41t/8) = 4e j1t / 4 , (3 22)
50 Xshifted(l) has а magnitude of 4 and а phase angle of те/ 4 or +450, which
is what we set out to prove using Eq, (3 19),
З.7 Inverse DFT
Although the DFТ is the major topic of this chapter, it's appropriate, now,
to introduce the inverse discrete Fourier transform (IDFТ). Typically we
think of the DFT as transforming time domain data into а frequency-
domain representation, Well, we сап reverse this process and obtain the
original time-domain signal Ьу performing the IDFT оп the Х(т) fre-
queI1cy-dоmain values. The standard expressions for the IDFT are
DFTLeakage 71
N l
х(п) = LX(m)ei21tтnIN
т=О
(3-23)
and equally,
N l
х(п) = L X(m)[cos(21tmn / N) + j sin(21tmn / N)]
т=О
(3-23')
Remember the statement we made in Section 3,1 that а discrete time-
domain signal сап Ье considered the sum of various sinusoidal analytical
frequencies and that the Х(т) outputs of the DFТ are а set of N complex val-
ues indicating the magnitиde and phase of each analysis frequency сот-
prising that sum, Equations (3-23) and (3-23') are the mathematical
expressions of that statement. It's very important for the reader to under-
stand this concept, If we perform the IDFТ Ьу plugging our results from
DFТ Example 1 into Eq. (3-23), we'll go from the frequency-domain back to
the time-domain and get our original real Eq, (3-11') х(п) sample values of
х(О) = 0,3535 + jO,O
х(2) = 0,6464 + jO,O
х(4) = 0,3535 + jO.O
х(6) = 1,3535 + jO,O
х(l) = 0.3535 + jO.O
х(3) = 1.0607 + jO.O
х(3) = l,0607 + jO.O
х(7) = O,3535 + jO.O ,
Notice that Eq, (3 23)'s IDFТ expression differs from the DFТ's Eq. (3-2)
only Ьу а 1/ N scale factor and а change in the sign of the exponent, Other
than the magnitude of the results, every characteristic that we've covered,
thus far, regarding the DFТ, also applies to the IDFТ,
3.8 DFT Leakage
Hold оп to your seat now. Here's where the DFТ starts to get really inter-
esting, The two previous DFT examples gave us correct results because
the input х(п) sequences were very carefиlly chosen sinusoids, As it tums
out, the DFТ of sampled real world signals provides frequency domain
results that сап Ье misleading. А characteristic, known as leakage, causes
our DFТ results to Ье only аn approximation of the true spectra of the
original input signals prior to digital sampling. Although there are ways
to minimize leakage, we can't eliminate it entirely, Thus, we need to
understand exactly what effect it has оп our DFT results,
72 The D/screte Fourier Traпsforт
Let's start from the beginning, DFTs are constrained to operate оп а
finite set of N input values sampled at а sample rate of f s ' to produce ап
N-point transform whose discrete outputs are associated with the indi
vidиal analytical frequencies f analysis(т), with
fanalysis(т) = т;; ,where т = 0,1,2,...,N 1,
(3 24)
Equation (3-24), illustrated in DFТ Example 1, тау not seem like а prob
lem, but it is, Тhe DFT produces correct results оnlу when the input data
seqиence contains energy precisely at the analysis frequencies given in
Eq, (3 24), at integral multiples of our fundamental frequency f.lN. If the
inpиt has а signal component at some intermediate frequency between
our analytical frequencies of тfs/N, say l,5f.lN, this input signal will
show ир to some degree in аll of the N output analysis frequencies of our
DFТ! (We typicaHy say that input signal energy shows ир in аН of the
DFТ's output biпs, and we'H see, in а moment, why the phrase "output
bins" is appropriate,) Let's understand the significance of this problem
with another DFТ example.
Assume we're taking а 64-point DFТ of the sequence indicated Ьу the
dots in Figure 3 7(a), The sequence is а sinewave with exactly three
cycles contained in our N = 64 samples, Figure 3 7(b) shows the first half
of the DFT of the input sequence and indicates that the sequence has an
average value of zero (Х(О) = О) and по signal components at апу fre
quency other than the т = 3 frequency, No surprises so far. Figure 3 7(a)
also shows, for example, the т = 4 sinewave analysis frequency, super-
imposed over the input sequence, to remind us that the analytical fre
quencies always have ап integral number of cycles over our total sample
interval of 64 points, Тhe sum of the products of the input sequence and
the т = 4 analysis frequency is zero, (Or we can say, the correlation of the
input sequence, and the т = 4 analysis frequency is zero.) The sum of the
products of this particular three cycle input sequence and апу analysis
frequency other than т = 3 is zero, Continuing with our leakage exaт
ple, the dots in Figure 3-8(а) show ап input sequence having 3,4 cycles
over our N = 64 samples, Because the input sequence does not have ап
integral number of cycles over our 64-sample interval, input energy has
leaked into аН the other DFТ output Ьins as shown in Figure 3 8(b). Тhe
т = 4 Ьin, for example, is not zero because the sum of the products of the
input sequence and the т = 4 analysis frequency is по longer zero, This
is leakage it causes апу input signal whose frequency is not exactly at а
DFТ Ып center to leak into аН of the other DFТ output bins, Moreover,
DFTLeakage 73
(а)
т = 4 analysis frequency
... r /
:i1 j \
. ... . ! , ' .I i. \
!i! i i
;;;; I
.\ l! [! \8
\ l' i : I 11 \
.1 i <! I i i. .
i!;:::I'!:
.1 ... ' . ' \ "' ... : .. !: . ' Time
;:::! i 1-
.' j\:! j j i!
\:!, i8
' \ : : ' ;
8! ,,1 i
..
DFT output magnitude
.
(Ь)
35
30
25
20
15
10
5
о .-..-.+..........................................................................................
о 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 (F q)
Flgure 3-7 64-point OFТ: (а) \nput sequence of three cycles and the m = 4
analysls frequency sinusold: (Ь) OFТ output magnitude,
leakage is an unavoidable fact of lifе when we perform the DFT оп real
world finite-length time sequences,
Now, as the English philosopher Douglas Adams would say, "Don't
panic." Let's take а quick look at the cause of leakage to learn how to pre-
dict and minimize its unpleasant effects. То understand the effects of leak
age, we need to know the amplitude response of а DFТ when the DFТ's
input is an arbitrary, real sinusoid, A1though Sections 3,14 and 3,15 dis-
cuss this issue in detail, for our purposes, here, we'll just say that, for а
real cosine input having k cycles in the N-point input time sequence, the
amplitude response of an N-point DFТ Ьin in terms of the Ьin index т is
approximated Ьу the sinc function
Х(т) '" N . sin[1t(k т)]
2 1t(k т)
(3 25)
74 The D/screte Four/er Traпsforт
(а)
т = 4 analysis frequency
Time
за DFТ output magnitude
25 .
20
15
i
I
I
I i
10 i i
5.,.ill.
uш . I \.
!! l ' 1 . 1 ......
о I I I I . ., .....,.,.,..,..,.,..,..,..,..,..,..,-,...,.J!I-I!I.....
О 2 4 6 8 10 12 14 16 18 20 22 24 26 28 ЗО т
(Freq)
(Ь)
Flgure 3-8 64-polпt DFТ: (а) 3,4 cycles input sequence and the т = 4 analysls
frequency slnusold; (Ь) DFТ output magnltude,
We'll use Eq. (3-25), Шustrаtеd in Figure 3-9(а), to help us determine
how much leakage occurs in DFTs. We сап think of the curve in Figure
3 9(a), comprising а main 10Ье and periodic peaks and valleys known as
sidelobes, as the continuous positive spectrum of аn N point, real cosine
time sequence having k complete cycles in the N-point input Нте inter
val. Тhe DFТ's outputs are discrete samples that reside оп the curves in
Figure 3 9; that is, our DFT output wi11 Ье а sampled version of the соn-
tinuous spectrum. (We show the DFTs magnitude response to а real
input in terms of frequency (Hz) in Figure 3-9(Ь).) When the DFTs input
sequence has exact1y аn integral k number of cycles (centered exactly in
the т = k Ьin), по leakage occurs, as in Figure 3 9, because when the
angle in the numerator of Eq, (3 25) is а nonzero integral mu1tiple of п,
the sine of that angle is zero.
OFТ Leakage 75
(а)
N
2"
Continuous, positive,
frequency spectrum 01
а discrete cosine
sequence
Discrete DFТ sequence defined Ьу
N sin [ 1t ( k --m)]
Х(т) = 2"' 1t(k--m)
/1/ jj
. V . .'- ..... ...............
, ,............. Freq
" (
m=k
k k 1 k+1 k+З k+5
N .
2" . . .
АА/""'.../"'..
II!I . . .
..
Freq
(Hz)
(Ь)
/"'./\
kfslN
(k--З)fslN (k 1)fslN (k+1)fs/N (k+З)fs/N (k+5)fslN
Flgure 3-9 DFТ posltlve frequency response due to оп N-point input sequence
contolning k cycles of о reol cosine: (о) omplitude response os о
functlon of bin Index т; (Ь) mognltude response os о function of
frequency In Hz,
Ву way of example, we can Шustrаtе again what happens when the
input frequency k is not located at а bin center, Assume that а real 8-kНz
sinusoid, having unity amplitude, has been sampled at а rate of fs = 32,000
samples/s, If we take а 32-point DFT of the samples, the DFТ's frequency
resolution, or bin spacing, is f/N = 32,000/32 Hz = 1,0 kHz, We сап pre-
dict the DFТ's magnitude response Ьу centering the input sinusoid's spec-
tral curve at the positive frequency of 8 kНz, as shown in Figure 3-10(а),
Тhe dots show the DFТ's output bin magnitudes,
Again, here's the important point to remember: the DFT output is а
sampled version of the continuous spectral curve in Figure 3 10(а), Тhose
sampled values in the frequency domain, located at тf/N, are the dots
in Figure 3-10(а), Because the input signal frequency is exactly at а DFT
76 The Discrete Fourier Traпsform
(а)
, \16
'''''' ''''" - ш"' / I \
/'...... /". /\ I ' /'-... ............. .......---..
'м............... . ' ' I ............... . . .
3 4 5 6 7 8 9 10 11 12 13 Freq
(kHz)
(\
''''"''''''''''' - о.. .. J , ."
I '
.. 1 i .
...... ,.............1.....'/1 V I i j V 1"-/"1............., .......
4 5 б 7 8 9 10 11 12 13
(Ь)
..
Freq
(kHz)
.
3
( " 14.05
Input frequency = 8,75 kHz i'
5,15 .. i
-'"" l' '
· V' I
.
.
(с)
.
345
б
7
8 9 10 11 12 13 Freq
(kHz)
Flgure 3-10 DFТ bIn posltlve frequency responses: (о) DFТ input
frequency = 8,0 kHz; (Ь) DFТ input frequency = 8,5 kHz;
(с) DFТ Input frequency = 8,75 kHz,
Ып center, the DFT results have only one nonzero value. Stated in
another way, when an input sinusoid has an integral number of cycles
over N time domain input sample values, the DFf outputs reside оп the
continuous spectrum at its peak and exactly at the curve's zero crossing
points. From Eq. (3 25) we know the peak output magnitude is 32/2 = 16,
(Н the rea1 input sinusoid had an amp1itude of 2, the peak of the response
curve would Ье 2 . 32/2, or 32,) Figure 3 10(b) i11ustrates DFf leakage
where the input frequency is 8.5 kНz, and we see that the frequency-
domain samp1ing resu1ts in nonzero magnitudes for a11 DFf output bins.
An 8.75-kНz input sinusoid would resu1t in the leaky DFf output shown
in Figure 3 10(c). If we're sitting at а computer studying leakage Ьу plot
ting the magnitude of DFf output values, of course, we'll get the dots in
Figure 3 10 and won't see the continuous spectral curves.
At this point, the attentive reader should Ье thinking: "If the continuous
spectrиms that we're sampling are symmetrical, why does the DFf output in
DFT Leakage 77
Figure з..8(Ь) look so asymmetrical?" In Figure 3 8(b), the Ьins to the right of
the third Ьin are decreasing in amp1itude faster than the Ьins to the left of the
third Ьin, "And another thing, evaluating the continuous spectrиm's X(тfs)
function at an abscissa value of 0,4 gives а magnitude scale factor of 0.75,
Applying this factor to the DFТ's maximum possible peak magnitude of 32,
we should have а third Ьin magnitude of approximately 32 . 0,75 = 24 but
Figure з..8(Ь) shows that the third bin magnitude is s1ight1y greater than 25,
What's going оп here?" We answer this Ьу remembering what Figure 3 8(b)
really represents, When examining а DFТ output, we're normally interested
Оnlу in the т = О to т = (N/2 l) Ьins. Thus, for our 3.4 cycles per sample
interval example in Figure з..8(Ь), only the first 32 Ьins are shown, WeU, the
DFТ is periodic in the frequency domain as illustrated in Figure 3 11, Upon
examining the DFТ's output for higher and higher frequencies, we end up
going in circles, and the spectrиm repeats itself forever,
The more conventional way to view а DFT output is to uпwrap the
spectrum in Figure 3 11 to get the spectrum in Figure 3 12. Figure 3 12
shows some of the additional replications in the spectrum for the
3.4 cycles per sample interval example, Concerning our DFT output
asymmetry problem, as some of the input 3.4-cycle signal amp1itude leaks
into the 2nd Ьin, the 1st Ьin, and the Oth Ьin, leakage continues into the
1st bin, the 2nd Ьin, the 3rd Ьin, etc, Remember, the 63rd bin is the 1st
Ьin, the 62nd Ьin is the 2nd Ьin, and so оп, These bin equivalencies allow
'!
.
. ,
.
\
т=ЗЗ
. . / m..Э2
._.-.......... ../
......... т:: 31
-.......
..
..
.
.
. ,
.
. . .
..
. ....
10 ./ т
(Fr.q)
4
Flgure 3-11 Cyclic representation of the DFТ's spectral replication when the
DFТ input is 3,4 cycles per sample interval,
78 The Discrete Fourier Traпsforт
Input signal 8150 shows Up 8t
( О 3.4 = --3.4 cycl"slint"rval
Input signal is 8t 3.4
. .( cycleslinterval
Input signaJ 8150 shows Up 8t 64 3.4 = 60.6.
and 64 + 3.4 = 67.4 cycles/interval
r. )
Spectтums repealln Spectrums repealln
thls dlrectlon 8. . ' Ihis dlrection
....-- . 1 :. ., . .8! :.. i .
... rTii 111111 i i 1 i I i 1 i"'H PfТii 111111111 i 11 i i',',
12 10 ---4 2 О 2 4 6 8 10 12 52 54 56 58 60 62 64 66 68 70 72 74 76 т
(Freq)
Figure 3-12 Spectral replicatlon when the DFТ input 15 3,4 cycles per sample interval,
'i'
.
DFТ output magnitude
(with input = 3,4 cycleslinterval)
. т=О..
.....
.
. , . :.
..I!I, ',: : ;!' ..
' 'TT 111 i 1111111 i I i 11 i " '
50 52 54 56 56 60 62 О 2 4 6 6 10 12 14
H4)H2)( 10)(..s) (--6) (--4) ( 2)
.;
(а)
.-
т
(Fr"q)
. .
DFТ output magnitude
(with input = 28,6 cycleslinterval)
(Ь)
т = N/2
. = 32
. J .
'!'
. .. :
. I " ' ..
....... . 1:,,:;:. '.......
I . 1 1 I I I I I I I , I , I I I I I , I I I I I 1 1 . I ...
16 20 22 24 26 26 за 32 з4 36 36 40 42 44 46 т
(Fr"q)
Figure 3-13 DFТ output magnltude: (а) when the DFТ input i5 3.4 cycle5 per
sample Interval; (Ь) when the DFТ Input 15 28,6 cycle5 per sample
Interval,
иs to view the DFТ oиtpиt bins as if they extend into the negative fre
quency range, as shown in Figиre З lЗ(а). Тhe resиlt is that the leakage
wraps arouпd the т = О freqиency Ьin, as well as aroиnd the т = N fre-
quency Ьin. Тhis is not sиrprising becaиse the т = О freqиency is the т = N
DFТ Leakage 79
frequency. Тhe leakage wraparound at the m = О frequency accounts for
the asymmetry about the DFТ's m = 3 bin in Figure 3 8(b).
Recall from the DFT symmetry discussion, when а DFT input
sequence х(п) is real, the DFТ outputs from m = О to m = (N/2 l) are
redundant with frequency bin values for m > (N /2), where N is the DFT
size, The mth DFT output will have the same magnitude as the (N m)th
DFТ output. That is, I Х(т) I = I X(N m) I , What this means is that leak-
age wraparound also occurs about the m = N /2 bin, Тhis сап Ье illus
trated using an input of 28.6 cycles per sample interval (32 3.4) whose
spectrum is shown in Figure 3-13(Ь). Notice the similarity between
Figure 3 IЗ(а) and Figure 3 IЗ(Ь), So the DFТ exhibits leakage wrap-
around about the m = О and m = N /2 bins. Minimum leakage asymme-
try will occur near the N/4th bin аБ shown in Figure 3 14(a) where the
full spectrum of а 16,4 cycles per sample interval input is provided.
Figure 3-14(Ь) shows а close-up view of the first 32 bins of the 16,4 cycles
per sample interval spectrum.
DFТ ou!pu! magni!ude
(wi!h inpu! = 16.4 cycleslin!erval)
25 . .
20
. .:
15
(а) 10
.: i. .i!1I!I
5 .lI!i'ii -:. .II!I !!:\; .
.... i ' i; , , : : : I!"""'I"""" : i : i ! : i i 1....
О ' """IIII' 1 '11111111111111111111111 IIIIIIIIIIIIIIIII' """ ...
О 4 8 12 16 20 24 26 32 36 40 44 46 52 56 60 (:.:u
25
DFТ ou!pu! magni!ude
(wi!h inpu! = 16.4
cycleslin!erval)
.
20
15
.
(Ь) 10
.' .
5 . " .
.... ;:: i............
о ,.,-, , т , , т I I I I I I I I I I I I I I I I I I I I I I ...
О 2 4 6 6 10 12 14 16 18 20 22 24 26 28 30 т
(Fr.q)
m= Nl4= 16
Flgure 3-14 DFТ output magnitude when the DFT input is 16,4 cycles per
sample interval: (а) full output spectrum view; (Ь) close-up view
showing minimlzed leakage asymmetry at frequency т = N/4,
80 The D/screte Four/er Traпsforт
Уои сan read about leakage all day, However, the best way to appreci
ate its effects is to sit down at а computer and use а software program to
take DFТs, in the form of fast Fourier transforms (FFТs), of your personaHy
generated test signals like those in Figures 3 7 and 3 8, Уои сап, then,
experiment with different combinations of input frequencies and various
DFr sizes. You'll Ье аЫе to demonstrate that DFТ leakage effect is trouble-
some because the bins containing low level signals are corrupted Ьу the
sidelobe levels from neighboring bins containing high amplitude signals.
Although there's по way to eliminate leakage completely, an important
technique known as wiпdowiпg is the most common remedy to reduce its
unpleasant effects. Let's look at а few DFТ window examples,
3.9 Windows
Windowing reduces DFT leakage Ьу minimizing the magnitude of Eq,
(3 25)'s sinc function's sin(x)/x sidelobes shown in Figure 3 9. We do this
Ьу forcing the amp1itude of the input time sequence at both the beginning
and the end of the sample interval to go smootbly toward а single common
amp1itude value. Figure 3 15 shows how this process works, If we consider
the infinite duration time signal shown in Figure 3 15(a), а DFТ сап on1y
Ье performed over а finite time sample intervallike that shown in Figure
3 15(c). We сan think of the DFТ input signal in Figure 3-15(с) as the prod
uct of an input signal existing for all time, Figure 3 15(a), and the rectan
gular window whose magnitude is 1 over the sample interval shown in
Нgше 3-15(Ь), Anytime we take the DFТ of а finite-extent input sequence
we are, Ьу default, multiplymg that sequence Ьу а window of аН ones and
effectively multiplymg the input values outside that window Ьу zeros, As
it tums out, Eq. (3--25)'s sinc function's sin(x)/x shape, shown in Figure
3 9, is caused Ьу this rectangular window because the continuous Fourier
transform of the rectangиlar window in Figure 3-15(Ь) is the sinc function.
As we'l1 soon see, it's the rectangular window's abrupt changes
between one and zero that are the cause of the sidelobes in the the sin(x)/x
sinc function, То minimize the spectralleakage caused Ьу those sidelobes,
we have to reduce the sidelobe amplitudes Ьу using window functions
other than the rectangular window. Imagine if we multiplied our DFТ
input, Figure 3-15(с), Ьу the triangular window function shown in Figure
3 15(d) to obtain the windowed input signal shown in Figure 3 15(e),
N otice that the values of our final input signal appear to Ье the same at the
beginning and end of the sample interval in Figure 3 15(e). Тhe reduced
discontinuity decreases the level of relatively high frequency components
in our overall DFТ output; that is, our DFТ Ьin sidelobe levels are reduced
in magnitude using а triangu1ar window. Тhere are other window func-
Windows 81
tions that reduce leakage even more than the triangular window, such as
the Hanning window in Figure 3-15(f), Тhe product of the window in
Figure 3-15(Е) and the input sequence provides the signal shown in Figure
3 15(g) as the input to the DFТ. Another common window fиnction is the
Hamming window shown in Figure 3 15(h), It's much like the Hanning
window, but it's raised оп а pedesta/.
Sampl. !
Interval
/
/
Time
(е)
I Tlme
(а)
'O ..........., J :
RectangUIar
"'ndow
, Sample 1
interval""""""'"
1,0
(Ь)
(1)
Time
Time
I Sampl& ;
lntervaJ !
I .
Tlme (g)
(о)
./
t: "
1.0
1,0
:
(h)
(d)
Тiтe
Time
Flgure 3.15 Minimlzing sample Interval endpoint dlseontinuities: (а) Infinite duration
input sinusoid: (Ь) reetangular wlndow due to finite-time sample
interval; (е) produet of reetangular window and Infinlte-duratlon Input
slnusoid; (d) trlangular wlndow funetlon; (в) produet of triangular
wlndow and Inflnlte-duration Input slnusold; (О Hanning wlndow
funetlon: (ф produet of Hanning wlndow and Infinite-duration input
s\nusoid; (h) Hammlng window funetlon,
82 The Discrete Fourier Troпsform
Вefore we see exact1y how well these windows minimize DFТ leakage,
let's define them mathematically. Assummg that ош original N inри! signal
sam.ples are indexed Ьу п, where О S; п S; N l, we'll call the N time--domain
window coefficients w(n); that ш, an inри! sequence х(п) is multiplied Ьу
the сопеsропding window w(n) coefficients before the DFТ is perfonned.
So the DFТ of the windowed х(п) inри! sequence, Xw(m), takes the fonn of
N l
Хю(т) = Lro(n).x(n)e j27t11m!N.
п=О
(3 26)
То use window functions, we need mathematical expression о! them in
terms о! п. The following expressions define our window function
coefficients:
Rectangular window: w(n) = 1, for п = 0,1,2, . . ., N l. (3-27)
(also called the unifonn,
or boxcar, window)
Triangular window: п
(very similar to the ro(п) = ,for п = О, 1, 2,...,N / 2, and
Bartlett[3], and N/2
Parzen[4,5] windows) =2 .....!!........
N/2'
for п =N /2+1,N /2+2,...,N 1. (3-28)
Hanning window: w(n) = 0.5 0.5соs(21tnIN-1),
(also called the raised for п = 0,1,2,. . ., N l. (3-29)
cosine, Нann, or
von Нann window)
Hamrning window: w(n) = 0.54 0.46cos(21tnIN 1),
for п = О, 1, 2, . . ., N l. ( 30)
lf we plot the w(n) values from Eqs. (3-27) through (3 30), we'd get the
corresponding window functions like those in Figшеs 3 15(b), 3 15(d),
3 15Щ, and 3 15(h).t
t In the literature, the equations for window functions depend оп the range of the sample
index п, We define п to Ье in the range О < п < N 1, Some authors define п to Ье in the range
N /2 < п < N/2, in which case, for example, the expression for the Hanning window would
have а sign change and Ье ш(п) = 0.5 + O,5cos(21t11/N-1),
Windows 83
Тhe rectangular window's amplitude response is the yardstick we nor-
mally use to evaluate another window function's amplitude response; that
is, we typically get an appreciation for а window' s response Ьу comparing
it to the rectangular window that exhibits the magnitude response shown
in Figure 3-9(Ь). Тhe rectangular window's sin(x)/x magnitude response,
I W(m) 1, is repeated in Figure 3 16(a). Also included in Figure 3 16(a) are
the Hamming, Hanning, and triangular window magnitude responses,
(Тhe frequency axis in Figure 3 16 is such that the curves show the response
of а single N point DFТ bin when the various window functions are used.)
We сап see that the last three windows give reduced sidelobe levels relative
to the rectangular window. Because the Hamrning, Hanning, and triangu-
lar windows reduce the time domain signallevels applied to the DFТ, their
main lobe peak values are reduced relative to the rectangular window,
(Because of the near zero w(n) coefficients at the Ьеgiлning and end of the
sample interval, this signallevelloss is called the processing gain, or loss,
of а window.) Ве that as it тау, we're primari1y interested in the windows'
sidelobe levels, which are difficult to see in Figure 3 16(a)'s linear scale. We
will avoid this difficulty Ьу plotting the windows' magnitude responses оп
а logarithmic decibel scale, and normalize each plot so its main lobe peak
values are zero dB. (Appendix Е provides а discussion of the origin and
utiIity of measuring frequency domain responses оп а logarithmic scale
using decibels.) Defining the log magnitude response to Ье I WdB(m) 1, we
get I WdB(m) I Ьу using the expression
( IW(m) 1 )
IW dB (m)I=20.10g 1o IW(O)I .
(3-31)
(Тhe I W(O) I term in the denominator of Eq. (3-31) is the value of W(m) at
the peak of the main lobe when m = О.) Тhe I WdB(m) I curves for the var
ious window functions are shown in Figure 3 16(b). Now we сап really
see how the various window sidelobe responses compare to each other,
Looking at the rectangular window's magnitude response we see that
its main lobe is the most narrow, f/N, However, unfortunately, its first
sidelobe level is only 13 dB below the main lobe peak, which is not so
good, (Notice that we're only showing the positive frequency portion of
the window responses in Figure 3 16,) The triangular window has
reduced sidelobe levels, but the price we've paid is that the triangular
window' s main lobe width is twice as wide as that of the rectangular win
dow's. Тhe various nonrectangular windows' wide main lobes degrade
the windowed DFT's frequency resolution Ьу almost а factor of two.
However, as we'll see, the important benefits of leakage reduction usually
outweigh the loss in DFТ frequency resolution,
84 Тhe Discrete Fourier Troпsform
Linear-scale window magnitude responses, IW(m)1
1"
' Rectangular (dolted)
(а)
Hanning (solid)
...... .....
..8 .
,. ...
.....
-. ..
,.....
fs/N
2fs/N
4fs/N
Freq
Logarithmic-scale window magnitude responses, IW dB (m)1 'п dB
Rectangular (dotted)
Hanning (solid)
.10
-20
(Ь) -30 "
;;'.
"
-40 "
.'.
"
-50
,
I
,
-60
fs/N 2fs/N 4fs/N Freq
Flgure 3-16 Wiпdow magnitude respoпses: (а) I W(m) I оп а lIпear scale; (ь)
I WdB(m) I оп а normalized logarithmic scale,
Notice the further reduction of the first sidelobe level, and the rapid side
lobe rol1-off of the Hanning window. Тhe Hamming window has even
lower first sidelobe levels, but this window's sidelobes roll оН slowly rela-
tive to the Нanning window. Тhis mеans that leakage three or four bins
away from the center Ьin is lower for the Hamming window than for the
Hanning window, and leakage а half dozen or so bins away from the сen-
ter Ьin is lower for the Hanning window than for the Hamming window,
, \
i; \
. I .
5 ., 1; 1\
I l j t : I : '-
I i 1,,: ----......
... ! ! I : ""'. " . . '. . JI JI ", "
о " ' . .-1.t-"';;;"';;;"'::;"';;;A""""
0123456789101112131415161718 2728293031 т
(Fr"q)
Windowed input signal
(а)
1
/
0.8 /. .
0,6 // : i:
04 / , ,1;
. // 1']: II!I
0,2/' 1!il!
. i ..... . . II : I Ц i , ! . : ,! ' :! ,..
., 1;.
--{),4 ...
--{),6
--{),8
1
OFТ output magnitudes
25 .
20
15
(Ь)
10
Windows 85
" , Hanning
. .' wlndow function
.;ij; I
: ! : j i '"
! i i i.;1 ""
i:: 1: : i , .. """ "__
. i : ! i i ! I . ..!!!..........
и!.i 1.1 п
.. . (rlme)
1 :: , ' , "" ,
. i-lll.
)'1 it!
i!il'
.. i i 1. J«п) - [0.5 0.500.(2."'64)1' [.ln(21tЗ.4r>'64)]
'
window function З.4-сусlе slnewave
· Rectangular window response
Hanning window response
Flgure 3-17 Hanning window: (а) 64-sample product of а Hanning wlndow and а
3,4 cycles per sample intervallnput slnewave; (Ь) Hanning DFТ output
response vs, rectangular window DFТ output response,
When we apply the Hanning window to Figure 3-8(a)'s 3,4 cycles per
sample interval example, we end ир with the DFТ input shown in Figure
3 17(a) under the Hanning window envelope, Тhe DFТ outputs for the
windowed waveform are shown in Figure 3 17(b) along with the DFТ
results with по windowing, i,e" the rectangu1ar window, As we expected,
86 The Discrete Fourier Traпsforт
the shape of the Hanning window's response looks broader and has а
lower peak amplitude, but its sidelobe leakage is noticeably reduced from
that of the rectangular window,
We сап demonstrate the benefit of using а window function to help us
detect а low level signal in the presence of а nearby high-level signal,
Let's add 64 samples of а 7 cycles per sample interval sinewave, with а
peak amplitude of only 0.1, to Figure 3-8(a)'s unity amplitude 3.4 cycles
per sample sinewave, When we apply а Hanning window to the sum of
these sinewaves, we get the time--domain input shown in Figure 3 18(a).
Had we not windowed the input data, our DFТ output would Ье the
squares in Figure 3-18(Ь) where DFТ leakage causes the input signal сот-
ponent at т = 7 to Ье barely discernible. However, the DFТ of the win
dowed data shown as the triangles in Figure 3 18(Ь), makes it easier for us
to detect the presence of the т = 7 signal component. From а practical
standpoint, people who use the DFТ to perform real-world signal detec
tion have learned that their overall frequency resolution and signal sensi
tivity are affected much more Ьу the size and shape of their window
function than the mere size of their DFТs.
As we Ьесоте more experienced using window functions оп our DFТ
input data, we'll see how different window functions have their own indi
vidual advantages and disadvantages. Furthermore, regardless of the
window function used, we've decreased the leakage in our DFТ output
from that of the rectangular window, Тhere are тапу different window
functions described in the literature of digital signal processing so many,
in fact, that they've been named after just about everyone in the digital
signal processing business, It's not that clear that there's а great deal of
difference among тапу of these window functions, What we find is that
window selection is а trade-off between main lobe widening, first side--
lobe levels, and how fast the sidelobes decrease with increased frequency,
The use of any particular window depends оп the app1ication[5], and
there are тапу app1ications.
Windows are used to improve DFT spectrum analysis accuracy[6], to
design digital filters[7,8], to simulate antenna radiation patterns, and
even in the hardware world to improve the performance of certain
mechanical force to vo1tage conversion devices[9]. 50 there's plenty of
window information available for those readers seeking further knowl-
edge, (The Mother of аН technical papers оп windows is that Ьу
Harris[10]. А useful paper Ьу Nuttall corrected and extended some por
tions of Harris's paper[l1].) Again, the best way to appreciate window
ing effects is to have access to а computer software package that contains
DFI, or FFТ, routines and start analyzing windowed signals, (Ву the way,
Windows 87
Windowed input signal
Window function З.4 and 7 le sinewaves
0.8
0.6
0.4
0.2
х(п) '" [0.5 O.5cos{2nnl64)j. [siп{2хЗ.4f)'64) +0 1siп{2п7f)'б4))
../ ..
. ';.
'.
(а)
о .... ..... ..
.2 .. i ia
.4 · .. .
. '
'.
.8.8... ...........
. n
. . . · (Time)
...
.6
,8
1
1.2
.
..
DFТ output magnitudes
25
. Rectangular window response
... Hanning window response
20
j:
10
,
I .
У ", ! 1' ..... . . \ .... Sign tC :p;ent
5 '. 1 ' . / .1 } .(
,. '.y<. ..... ..... ... .
о .+....... I I I I I I i" +.:...:=..:::..=.:4.t-- ;::. = ...............
о 1 2 3 4 5 6 7 8 9 10111213141518171819 282930 31 (F q)
15
(Ь)
Flgure 3-18 Increased signal detection sensltivlty afforded using windowing: (а) 64.
sample product of а Hanning window and the sum of а 3,4 cycles and
а 7 cycles per sample Interva/ sinewaves; (Ь) reduced leakage
Hannlng OFТ output response vs, rectangular window OFТ output
response,
while we delayed their discussion unti1 Section 5.3, there are two other
commonly used window functions that сап Ье used to reduce DFТ leak-
age, They're the Chebyshev and Kaiser window functions, that have
adjustable parameters, enabling us to strike а compromise between
widening таin lobe width and reducing sidelobe levels.)
88 The Discrete Foиr/er Traпsform
3.10 DFT Scalloping Loss
Scalloping is the пате used to describe fluctuations in the overall mаgni
tude response of an N point DFf. Although we derive this fact in Section
3,16, for now we'll just say that when по input windowing function is
used, the sin(x)/x shape of the sinc function's magnitude response app1ies
to еаф DFf output Ьin, Figure 3-19(а) shows а DFf's aggregate mаgni
tude response Ьу superimposing several sin(x)/x bin magnitude
responses. t (Because the sinc function's sidelobes are not key to this dis
cussion, we don't show them in Figure 3 19(a).) Notice from Figure 3 19(b)
that the overa11 DFf frequency domain response is indicated Ьу the bold
envelope curve. Тhis rippled curve, a1so called the picket fence effect, illus-
trates the processing 10ss for input frequencies between the Ьin centers,
(а)
rr:r:fW\\
1 ( f. l (+ 1 e
(m+-1)f s (m+-3)f s (m+-5)f s
N N N
(Ь)
':f:
t t t
mfs (m+-2)f s (m+-4)f s
N N N
..
Freq
Flgure 3-19 OFТ bIn mognitude response curves: (а) Indlvlduol sln(x)/x
responses for eoch OFТ bIn; (Ь) equivolent overoll OFТ mognitude
response,
t Perhaps Figure 3 19(а) is why individua! DFf outputs are cal!ed "Ьins," Anу signa! energy
under а sin(x) / х curve will show ир in the eпc/osed storage compartтeпt of that DFf' s output
sample,
DFТ Reso/ution, Zero Sfuffiпg, aпd Frequeпcv-Doтaiп Saтp/iпg 89
From Figure 3 19(b), we сап determine that the magnitude оЕ the DFf
response fluctuates from 1,0, at Ьin center, to 0.637 halfway between bin
centers, If we're interested in DFf output power levels, this envelope rip-
рlе exhibits а scalloping 10ss оЕ almost 4 dB halfway between bin cen-
ters, Figure 3 19 illustrates а DFT output when по window (i.e" а
rectangular window) is used, Because nonrectangular window functions
broaden the DFf's main 10Ье, their use results in а scalloping 10ss that
will not Ье as severe as the rectangular window[10,12]. That is, their
wider таin 10bes overlap more and ЕШ in the valleys оЕ the envelope
curve in Figure 3-19(Ь), For ехатрlе, the scalloping 10ss оЕ а Hanning
window is approximately 0.82, or 1,45 dB, halfway between Ьin centers,
Scalloping 10ss is not, however, а severe problem in practice. Real-world
signals normally have bandwidths that span many frequency bins so that
DFf magnitude response ripples сап go almost unnoticed, Let's 100k at
а scheme called zero stuffiпg that's used to both alleviate scalloping 10ss
effects and to improve the DFT's frequency resolution,
3.11 DFТ Resolution, Zero Stuffing, and
Frequency-Domain Sampling
One popular method used to improve DFf frequency resolution is known
as zero stuffiпg, or zero paddiпg, Тhis process involves the addition оЕ zero-
valued data samples to an original DFf input sequence to increase the
total number of input data samples. Investigating this zero stuffing tech-
nique illustrates the DFf's important property оЕ frequency-domain sam-
pling al1uded to in the discussion оп leakage. When we sample а
continuous time-domain function, having а continuous Fourier transform
(CFf), and take the DFf оЕ those samples, the DFf results in а frequency
domain sampled approximation оЕ the CFf, The more points in our DFT,
the better our DFf output approximates the CFf.
То illustrate this idea, suppose we want to approximate the CFf оЕ the
continuous f(t) function in Figure 3 20(a). Тhis f(t) wаvеfопn extends to
infinity in both directions but is nonzero only over the time interval оЕ Т
seconds, If the nonzero portion оЕ the time fиnction is а sinewave оЕ three
cycles in Т seconds, the magnitude оЕ its CFf is shown in Figure 3 20(b),
(Because the CFf is taken over an infinitely wide time interval, the CFf
has infinitesimally small frequency resolution, resolution so fine grained
that it's continuous.) It's this CFf that we'll approximate with а DFT
Suppose we want to use а 16 point DFf to approximate the CFf оЕ f(t) in
Figure з..20(а), The 16 discrete samples off(t), spanning the three periods оЕ
f(t)'s sinusoid, are those shown оп the left side оЕ Figure 3-21(а), Applying
90 The D/screte Fourier Traпsforт
OQ
OQ
(а)
...
Continuous
time
(Ь)
11Т 2JT З/Т 4/Т 51Т
Continuous
freque псу
Flgure 3-20 Continuous Fourier transform: (а) continuous time domain f(t) of
а truncated sinusoid of frequency 3/Т: (Ь) continuous Fourler
transform of f(t),
those пте samples to а 16-point DFf results in discrete frequency domain
samples, the positive frequency of which are represented Ьу the dots оп
the right side of Figиre 3-21(а), We сап see that the DFf output samples
Figuтe 3-20(b)'s СУТ, If we append (or zero stuff) 16 zeros to the input
sequence and take а 32 point DFf, we get the output shown оп the right
side of Figure 3 21(b), where we've increased our DFf frequency sampling
Ьу а factor of two, Our DFf is sampling the input function's CFf more
often now, Adding 32 more zeros and taking а 64-point DFf, we get the
output shown оп the right side of Figure 3-21(с), The 64-point DFf output
now begins to show the true shape of the CFf, Adding 64 more zeros and
taking а 128 point DFf, we get the output shown оп the right side of
FigUIe 3-21(d), Тhe DFf frequency-domain sampling characteristic is obvi-
ous now, but notice that the Ьin index for the center of the таin lobe is dif-
ferent for each of the DFf outputs in Figure 3-21,
Does this mean we have to redefine the DFf's frequency axis when
using the zero-stuffing technique? Not really, If we perform zero stuffing
оп L nonzero input samples to get а total of N пте samples for an N-point
DFr, the zero-stuffed DFf output bin center frequencies are related to the
original fs Ьу our old friend Eq. (3-5), or
center frequency of the mth Ьin = m/S .
N
(3-32)
OFТ Reso/ufioп Zero Stuffiпg, aпd Frequeпcy-Oomaiп Sampliпg 91
j'
1 . .
. .
05
(а) о. I I I I 1 111 I I I I I I I .
J 9 10' 11 12 131. 15 Тiтe
О 1 2 3 4 5 7 .
. .
5t
1 . .'
(Ь)
1 j :npu! a:Plitud:
. .
0.5 : . .
O,++++++t"+++++++II!I.""..II!I'IfIl!lIl!l.".II!II!I
о '3 5 9 12' 15 18 21 24 27 зо Time
.5 t .. ·
1 . ·
.:E[
(с) о ,iItI-!!*+*I , ...
о ! 5 10 15 20 25 за 35 40 45 50 55 во Tlme
:, (: .. ..
j 'при! amplitude
11.1!1
1'::,
05 f. 1.
(d) O f .:. "
5, ..
1 j8.
'"
121 Time
OFТ magnitude
"',' \
О ,,' I'
О 1 2
.
4
. ......
6 Freq
OFТ magnitud.
.,
,.
.'
OIl!l 1I!I
о 1 2 3 4
.
I " . I!I
7 8 9 10 11 12 Freq
OFТ magnitud.
.'
.
8. .8,
.-. ,. . .-.
о II!I+++ +++ +++++++I!I+++'I!I
о 2 4 6 8 10 12 1. 16 18 20 22 24 Freq
OFТ magnrtud.
,..
. .
. .
.
2 ..... .'.
....r...ra .. .. 8. ........
О 1I/НЖ-,.!IНЖ.жннt+ItЖI.жнr..iжh\'.
о 4 8 12 16 20 24 28 32 36 40 44 48 Freq
Figure 3-21 OFТ frequeпey domaiп samp/iпg: (а) 16 iпput data samp/es aпd
N = 16; (Ь) 16 iпput data samples, 16 stuffed zeros, aпd N = 32; (е) 16
iпput data samples, 48 stuffed zeros, aпd N = 64; (d) 16 iпput data
samples, 112 stuffed zeros, and N = 128,
50 in our Figure 3 21(a) example, we use Eq. (3-32) to show that,
although the zero-stuffed DFT output Ып index of the main lobe
changes as N increases, the zero-stuffed DFТ output frequency associ-
ated with the main lobe remains the same, Тhe fol1owing list shows how
this works:
92 Тhв Discrete Fourier Transforт
Мain lobe peak Frequeпcy о! main lobe
Figure No, located at m = L= N= peak relative to fs =
Figure Иl(а) 3 16 16 3//16
Figure 3-21(Ь) 6 16 32 6.//32 = 3//16
Figure 3-21(с) 12 16 64 12.//64 = 3//16
Hgure И1(d) 24 16 128 24.//128 = 31./16
Do we gain anything Ьу appending more zeros to the input sequence
and taking larger DFТs? Not really, because our 128 point DFТ is sam-
pling the input's CFТ sufficiently now in Figure 3-21(d). Sampling it more
often with а larger DFТ won't improve our understanding of the input's
frequency content. The issue here is that adding zeros to an input
sequence will improve our DFТ's output resolution, but there's а practi-
callimit оп how much we gain Ьу adding more zeros. For our example
here, а 128 point DFТ shows us the detailed content of the input spec
trum. We've hit а law о! diтiпishiпg returns here. Performing а 256 point
or 512-point DFТ, in our case, would serve little purpose. t Тhere's по rea
son to oversaтple this particular input sequence's CFТ. Of course, there's
nothing sacred about stopping at а 128-point DFI. Depending оп the
number of samples in some arbitrary input sequence and the sample rate,
we might, in practice, need to append any number of zeros to get some
desired DFТ frequency resolution.
There are two final points to Ье made concerning zero stuffing. First,
the DFТ magnitude expressions in Eqs, (3-17) and (3 17') don't apply if
zero stuffing is being used, If we perform zero stuffing оп L nonzero
samples of а sinusoid whose frequency is located at а bin center to get а
total of N input samples for an N point DFТ, we must replace the N with
L in Eqs. (3-17) and (3 17') to predict the DFТ's output magnitude for
that particular sinewave, Second, in practical situations, if we want to
perform both zero stuffing and windowing оп а sequence of input data
samples, we must Ье careful not to apply the window to the entire input
including the appended zero valued samples. The window function
must Ье applied only to the original nonzero time samples, otherwise
the stuffed zeros wШ zero out and distort part of the window function,
t Notice that the DFТ sizes (N) we've discussed are powers of 2 (64, 128, 256, 512), That's
ЬесаUБе we actually perforrn DFТs using а special algorithrn known as the fast Fourier ааns-
forrn (FFТ), As we'll see in Chapter 4, the typical implemel1tation of the FFТ requires that N
Ье а power of 2,
OFТ Processiпg Goiп 93
leading to erroneous results. (Section 4.5 gives additional practical
pointers оп performing the ОРТ using the РРТ algorithm to analyze real
wor1d signals.)
То digress slightly, now's а good time to деfinе the term discrete-tiтe
Fourier traпsforт (OТFТ) that the reader тау encounter in the literature,
Тhe OТFТ is the continuous Foиrier transform of an L point discrete time-
дотаin sequence; some authors use the OТFТ to describe many of the
digital signal processing concepts we've covered in this chapter. We can't
perform the ОТРТ оп а computer because it has an infinitely fine fre
quency resolution but we could, if we wished, approximate the OТFТ
Ьу performing an N point DFT оп an L-point discrete time sequence
where N> L, Тhat is, in fact, what we did in Figure 3-21 when we zero
stuffed the original 16 point time sequence, While we don't emphasize
the ОТРТ in this book, keep in mind that when N = L the OТFТ approxi-
mation is identical to the ОРТ.
3.12 DFТ Processing Gain
Тhere are two types of processing gain associated with DPТs. People who
use the ОРТ to detect signal energy етЬеддед in noise often speak of the
DPТ's processiпg gaiп because the ОРТ can риН signals out of background
noise. Тhis is дие to the inherent сопеlаtiоп gain that takes place in any
N point ОРТ. Beyond this natural processing gain, additional iпtegratioп
gaiп is possible when multiple ОРТ outputs are averaged. Let's look at the
DPТ's inherent processing gain first.
3.2.1 Processing Gain 01 а Single DFТ
Тhe concept of the ОРТ having processing gain is straightforward if we
think of а particular ОРТ Ьin output as the output of а narrowband filter.
Because а ОРТ output bin has the amplitude response of the sin(x)/x
function, that bin's outpиt is primarily дие to inpиt energy residing
under, or very near, the bin's таin lobe. It's valid to think of а ОРТ bin as
а kind of baпdpass filter whose band center is located at тfs/N. We know
from Eq. (3 17) that the maximum possible ОРТ output magnitude
increases as the number of points (N) in а ОРТ increases, Also, as N
increases, the OFТ output Ьin таin lobes Ьесоте more narrow, So а OFТ
output Ьin can Ье treated as а bandpass filter whose gain can Ье increased
and whose bandwidth сап Ье reduced Ьу increasing the value of N.
Decreasing а bandpass filter's bandwidth is usefиl in energy detection
94 The Discrete Fourier Traпsforт
о
5
(а) 10
15
20
25
О
5
I
10 15 20
DFТ Ып number
25
30
т
о
5
10
(Ь) 15
20
25
O
35
О
20
40
60 80
DFТ Ып number
100
120
т
о
Bin power in dB
10
t
SNR
(с)
20
O
--40
О
100
200 300
DFТ Ып number
400
500
т
Flgur& 3-22 Single OFТ proeessing goln: (а) N = 64; (Ь) N = 256; (е) N = 1024,
becaиse the frequency resolution improves in addition to the filter's abil-
ity to minimize the amount оЕ background noise that resides within its
passband, We сап demonstrate this Ьу looking at the DFr оЕ а spectral tone
(а constant frequency sinewave) added to random noise, Figure 3-22(а) is
DFТ Processiпg Gaiп 95
а logarithmic plot showing the first 32 outputs of а 64-point DFT when
the input tone is at the center of the DFTs т = 20th bin, The output
power levels (DFT magnitude squared) in Figure 3-22(а) are normal-
ized so that the highest bin output power is set to О dB. Because the
tone's original signal power is below the average noise power level,
the tone is а bit difficu1t to detect when N = 64. (The time domain
noise, used to generate Figure 3-22(а), has an average value of zero,
i,e" по DC bias or amplitude offset,) If we quadruple the number of
input samples and increase the DFT size to N = 256, we сап now see
the tone power raised аЬоуе the average background noise power as
shown for т = 80 in Figure 3-22(Ь), Increasing the DFT's size to
N = 1024 provides additional processing gain to риН the tone further
ир out of the noise as shown in Figure 3-22(с).
То quantify the idea of DFf processing gain, we сan define а signal-to-
noise ratio (5NR) as the DFf's output sigпal-power level over the average
output пoise power level. (In practice, of course, we 1ike to have this ratio as
large as possible,) For several reasons, it's hard to say what any given sin-
gle DFT output 5NR wi11 Ье. That's because we can't exact1y predict the
energy in any given N samples of random noise. Also, if the input signal
frequency is not at bin center, leakage wi11 raise the effective background
noise and reduce the DFf's output 5NR. In addition, anу window being
used wi11 have some effect оп the leakage and, thus, оп the output 5NR.
What we'l1 see is that the DFf's output 5NR increases as N gets larger
because а DFT bin's output noise standard deviation (rтs) value is pro-
portional to Щ, and the DFf's output magnitude for the Ьin containing
the signal tone is proportional to N, More general1y for real inputs, if
N> N', an N-point DFT's output SNR N increases over the N' point DFT
SNR N , Ьу the fol1owing relationship:
SNR N = SNR N , + 20 .10glO( Ш ).
(3 33)
If we increase а DFf's size from N' to N = 2N', from Eq. (3-33), the DFf's
output 5NR increases Ьу 3 dB. 50 we say that а DFf's processing gain
increases Ьу 3 dB whenever N is doubled. Ве aware that we тау double
а DFf's size and get а resu1tant processing gain of less than 3 dB in the
presence of random noise; then again, we тау gain s1ight1y more than
3 dB. That's the nature of random noise, If we perform many DFfs, we'l1
see an average processing gain, shown in Figure 3-23(а), for various input
signal 5NRs, Because we're interested in the slope of the curves in Figure
96 Тhe Discrete Fourier Troпsform
40 OFТ output SNR (dB)
L Inpu! SNR
35 +6 dB
30 i
OdB .
25 6dB
(а) 20
-12 dB
15 . . .". ....0.
-18 dB
10
5
О
О 200 400 600 800 1000 N
40 OFТ output SNR (dB)
Input SNR
=
15
30
25
(Ь) 20
10
. .
5 ..:;:........ - ..! ' .....; '
О
1
10
100
1000
N
Flgure 3.23 DFТ processlng galn vs, number of DFТ points N for various Input signal-
to-nolse ratlos: (а) Ilnear N axls; (Ь) logarithmlc N axls,
3-2З(а), we plot those curves оп а logarithmic scale for N in Figure 3-23(Ь)
where the curves straighten out and Ьесоте linear, Looking at the slope
of the curves in Figure 3-23(Ь), we сап now see the 3 dB increase in pro
cessing gain as N doubles so long as N is greater than 20 or 30 and the sig-
nal is not overwhelmed Ьу noise, There's nothing sacred about the
absolute values оЕ the curves in Figures 3 23(a) and 3 23(b), Тhey were
generated through а simulation of noise and а tone whose frequency was
at а DFТ bin center, Had the tone's frequency been between bin centers,
the processing gain curves would have been shifted downward, but their
Тhe OFТ of Rectaпgufar Fuпctioпs 97
shapes would still Ье the same,t that is, Eq. (3 33) is still valid regardless
of the input tone' s frequency.
3.12.2 Integration Gain Due to Averaging Multiple DFTs
Тheoretically, we could get very large DFТ processing gains Ьу increasing
the DFТ size arbitrarily. Тhe problem is that the number of necessary OFТ
multiplications increases proportionally to N2, and larger DFТs Ьесоте
very computationally intensive. Бесаusе addition is easier and faster to
perform than multiplication, we Сan average the outputs of multiple DFТs
to оЫаin further processing gain and signal detection sensitivity, Тhe sub
ject of averaging mu1tiple OFТ outputs is covered in Section 8,3,
3.13 The DFТ of Rectangular Functions
We conclude this chapter Ьу providing the mathematical detai1s of two
important aspects of the DFТ. First, we obtain the expressions for the DFТ
of а rectangular function (rectangular window), and then we'll use these
resu1ts to illustrate the magnitude response of the DFТ. We're interested
in the DFТ's magnitude response because it provides an a1temate view-
point to understand the leakage that occurs when we use the DFТ as а sig
nal analysis tool,
One of the most prevalent and important computations encountered in
digital signal processing is the DFТ of а rectangular function, We see it in
sampling theory, window functions, discussions of convolution, spectral
analysis, and in the design of digital filters. As common as it is, however,
the literature covering the DFТ of rectangular functions сап Ье confusing
for several reasons for the digital signal processing beginner, Тhe stan-
dard mathematical notation is а bit hard to fol1ow at first, and sometimes
the equations are presented with too little explanation, Compounding the
problem, for the beginner, are the various expressions of this particular
DFТ. In the 1iterature, we're 1ikely to find any one of the following forms
for the DFТ of а rectangular function:
DFТ sin(x) sin(x) sin(Nx / 2)
rect.function = sin(x / N) , or ,or sin(x / 2) .
(3 34)
t The curves would Ье 5hifted downward, indicating а lower SNR, Ьесаи5е leakage would
rai5e the average noise-power level, and 5calloping 1055 would reduce the ОРТ Ьin'5 output
power level,
98 The Dlscrete Four;er Traпsform
In this section we'll show how the forms in Eq. (3 34) were obtained,
see how they're related, and create а kind of Rosetta stone table allowing
us to move back and forth between the various DFf expressions. Take а
deep breath and let's begin our discussion with the definition of а rectan
gular function.
3.13.1 DFT of а Gвnвral Rвctangular Function
А general rectangular function х(п) can Ье defined as N samples contain
ing К unity valued samples as shown in Figure 3 24, The full N point
sequence, х(п), is the rectangular function that we want to transform. We
саll this the general fопn of а rectangular function because the К unity
samples Ьеgin at а arbitrary index value of пo. Let's take the DFf of х(п)
in Figure 3 24 to get our desired Х(т). Using т as our frequency domain
sample index, the expression for an N-point DFf is
N/2
Х(т) = L x(п)e j2"пт/ N.
n=--{N / 2)+1
(3-35)
With х(п) being nonzero only over the range of пo ::; п ::; пo +(K 1), we
can modify the summation limits of Eq. (3 35) to express Х(т) as
пo+(K 1)
Х(т) = L1'e j2"пт/N,
п= пo
(3 36)
because only the К samples contribute to Х(т), That last step is important
because it allows us to eliminate the х(п) terms and make Eq. (3 36) easier
to handle. То keep the following equations from being too messy, let's use
the dummy variable q = 2тcm/N,
Х(п)
I
< к >-
1.............
! : : !. ,1 i! :
......... .... ....... ........ . .-.
о
: .............................
+ п
п= пo+(K 1) Ю2
+
т+1
п= пo
F/gure 3-24 Rectangular functioп of width К samples defined over N samples
where К < N,
Тhe OFТ о( Rectaпgu/ar Fuпctioпs 99
ОК, here's where the algebra comes in. Over our new limits of sum
mation, we eliminate the factor of 1 and Eq, (3 36) becomes
lI,+(K l)
X(q) = > jqll
п= пo
= e jq( lI,) + e jq( lI, +1) + e jq( lI, +2) +.., + e jq( lI, +(K 1))
= e jq( lI')e jOq + e jq( п')e jlq + e jq( п')e j2q +.., + e jq( п, )e jq(K l)
=ejq(п,) '[e jOq +e jlq +e j2q +...+ejq(K l)] .
(3-37)
Тhe series inside the brackets of Eq. (3 37), allows the use of а summation,
such as
K l
X(q)=ejq(lI,) > jpq ,
р=О
(3 38)
Equation (3 38) certainly doesn't look any simpler than Eq, (3 36), but it
is, Equation (3-38) is а geoтetric series and, from the discussion in
Appendix В, it сап Ье evaluated to the closed form of
K l 1 jqK
L jpq е
е .
1 е }q
р=О
(3 39)
We сап now simplify Eq, (3-39) here' s the clever step, If we multiply and
divide the numerator and denominator of Eq, (3 39)'s right-hand side Ьу
the appropriate half angled exponentials, we break the exponentials into
МО parts and get
K l . e jqK/2(ejqK/2 e jqK/2)
'" е Ipq =
L.J e jq/2 ( ejq/2 e jq/2 )
р=о
( e1qK/2 е }qК/2 )
= e lq(K 1)/2 .
(e1q/2 е }q/2)
(3-40)
100 The D/screte Fourler Traпsforт
Let's pause for а moment here to remind ourselves where we're going,
We're trying to get Eq. (3-40) into а usable form because it's part о!
Eq. (З 38) that we're using to evaluate Х(т) in Eq. (3 36) in our quest for
ап understandable expression for the DFT of а rectangular function,
Equation (3 40) looks even more complicated than Eq, (3-39), but
things сап Ье simplified inside the parentheses, From Euler's equation:
sin(,,) = (&О c j 0)/2j, Eq, (3 40) becomes
Ie jpq = e jq(K 1)/2 . 2j sin(qK /2)
р=О 2jsin(q/2)
= e jq(K 1)/2. sin(qK /2) ,
sin(q /2)
(3 41 )
5ubstituting Eq. (3 41) for the summation in Eq, (3-38), our expression for
X(q) becomes
X(q) = ejq(п,) . e iq(K 1)/2 . sin(qK /2)
sin(q /2)
= eiq(n, (K 1)/2). sin(qK /2) ,
sin(q /2)
(3-42)
Returning our dummy variable q to its original value of 2rcm/N,
Х(т) = ei(21tm/N)(п, (K 1)/2) . sin(2rcmK /2N) , 01
sin(2rcm /2N)
Genera1 form of the
Dirich1etkemel:.... Х(т) = e j (21tm/ N)(п, (K 1)/2) . sin(rcmK / N) .
sin(rcm / N)
(3-43)
50 there it is (Whew!). Equation (3 43) is the general expression for the
DFТ of the rectangи1ar function as shown in Figure 3 24. Our Х(т) is а
,-",.,..
Тhв ОП 01 Rectangu/ar Jffn6t1ons
1.
complex expression (pun intended) where а ratio of sine terxts is the
amplitude of Х(т) and the exponential term is the phase angle дf )((т),;
Тhe ratio of sines factor in Eq. (3-43) lies оп the periodic curve shоwn,ф,"
Figure 3 25(a), and like аН N-point DFf representations, the periodicity of
Х(т) is N. тhis curve is known as the Dirichlet kerne/ (or the aliased siпc
fuпctioп) and has been thorougbly described in the literature[10,13,14],
(It's named after the nineteenth-century German mathematician Peter
Diricblet (pronounced dee-ree-klay'), who studied the convergence of
trigonometric series used to represent arbitrary functions,)
We сап zoom in оп the curve at the т = О point and see more detail in
Figure 3-25(Ь). Тhe dots are shown in Figure 3-25(Ь) to remind us that
the DFf of our rectangular function results in discrete amplitude values
that lie оп the curve. So when we perform DFfs, our discrete results are
sampled values оЕ the continuous sinc function's curve in Figure 3-25(а),
As we'll show later, we're primarily interested in the absolute value, or
magnitude, of the Diricblet kemel in Eq. (3-43). Тhat magnitude I Х(т) I
is shown in Figure 3-25(с). Although we first saw the sinc function's
curve in Figure 3 9 in Section 3,8, where we introduced the topic of DFT
leakage, we'H encounter this curve often in our study of digital signal
processing,
For now, there are just а few things we need to keep in mind concem-
ing the Diricblet kemel, First, the DFf of а rectangular function has а таin
lobe, centered about the т = О point. Тhe peak amplitude of the таin lobe
is К. Тhis peak value makes sense, right? Тhe т = О sample of а DFf Х(О)
is the sum of the original samples, and the sum of К unity valued samples
is К. We can show this in а more substantial way Ьу evaluating Eq. (3-43)
for т = О, А difficulty arises when we plug т = О into Eq. (3-43) because
we end ир with sin(O)/sin(O), which is the indeterminate ratio О/О, Well,
hardcore mathematics to the rescue here. We can use L'Hospital's rule to
take the derivative of the numerator and the denominator of Eq. (3-43),
and theп set т = О to determine the peak value of the magnitude оЕ the
Diricblet kemel. tt We proceed as
t N was an even number in Figure 3-24 depicting the х(n), Had N been an odd number, the
limits оп the summation in Eq, (3-35) would have been (N I)/2 n (N I)/2, Using these
altemate limits would have led us to exactly the saтe Х(т) as in Eq, (3-43),
tt L'Hospital is pronounced 'lб-рё-tOl, like ЬаЬу doll,
1 \:,
"
,,'
)'0
:;./ .
102 The D/serete Four/er Traпsform
I Х(т) I =.!!.... Х(т) = d[sin(1tmK / N)] / dm
т'" О dm d[sin(1tm/ N)]/ dm
cos(1tmK / N) . d(1tmK / N) / dm
cos(1tm / N) d(1tm / N) / dm
= cos(O) . пК / N = 1. К = К ,
cos(O) 1t / N
(3 44)
which is what we set out to show, (We could have been clever and evalu-
ated Eq. (3-35) with т = О to get the result of Eq. (3-44), Try it, and keep in
mind that e jO = 1,) Had the amplitudes of the nonzero samples of х(п) been
Х(т)
(а)
Х(т)
,.,
!
., ! .
/ i [\
. ; '! \
н.. Л: i i 8, .
. .8"8 /8. " :, : F ;, /8 .. . .-
.:.;....!+.ТНo.+++. / . I I I I I I i "'t'++:'.-+++."""""""'. .""
.8.8 ., , ' t о t \ >.' '....,.' т
т",--N/К т'" N/К
к
(Ь)
1,0
0,8
0,6
0.4
0,2
О
--0,2
(с)
1.0 ..'. к
0.8 Main! [\
0,6 lobe !! i !\
0.4 f I \
.; . ,.
0.2 _8. ._8 .н.'. , ' ' }1 \ i 1 ; \ / 'i'. .'... ,..
о .,., ., ..-+++ +++\ I I I I i i I \'++-+-'.++-+-' I!I , 1"' I!I
,+---- 2N/K I m
Side ))))
lobes
Figure 3-25 The Diriehlet kernel of Х(т): (а) periodle eontlnuous eUNe оп whieh
the Х(т) samples Ile; (Ь) Х(т) amplitudes about the m = О sample;
(е) I Х(т) I magnltudes about the m = О sample,
The OFТ of Recfaпgu/a( Fuпcfioпs 103
different than unity, say some amplitude Ао, then, of course, the peak
value of the Dirichlet kernel would Ье АоК instead of just К. The next
important thing to notice about the Dirichlet kernel is the таin lobe's
width, The first zero crossing of Eq, (3 43) occurs when the numerator's
argument is equal to п, Тhat is, when 7tmK/N = п, 50 the value of т at the
first zero crossing is given Ьу
7tN N
тfirst zero crossing = пК = К
(3 45)
as shown in Figure 3 25(b), Тhus the таin lobe width 2N/K as shown in
Figure 3 25(c), is inversely proportional to к.+
Notice that the таin lobe in Figure 3 25(a) is surrounded Ьу а series of
oscillations, called sidelobes, as in Figure 3 25(c), These sidelobe magni
tudes decrease the farther they're separated from the таin lobe, However,
по matter how far we look away from the main lobe, these sidelobes
never reach zero magnitude-----and they cause а great deal of heartache for
practitioners in digital signal processing, These sidelobes cause high
amplitude signals to overwhelm and hide neighboring low amplitude
signals in spectral analysis, and they complicate the design of digital fil
ters. As we'll see in Chapter 5, the unwanted ripple in the passband and
the poor stopband attenuation in simple digital filters are caused Ьу the
rectangular function's DFT sidelobes, (Тhe development, analysis, and
application of window functions сате about to minimize the ill effects of
those sidelobes in Figure 3 25,)
Let's demonstrate the relationship in Eq, (3 45) Ьу way of а simple
but concrete example, Assume that we're taking а 64-point DFТ of the
64-sample rectangular function, with eleven unity values, shown in
Figure 3 26(a), In this example, N = 64 and К = 11, Taking the 64 point
DFT of the sequence in Figure 3 26(a) resu1ts in an Х(т) whose real and
imaginary parts, Xrea1(m) and Ximag(m), are shown in Figure 3 26(b) and
Figure 3-26(с) respectively, Figure 3 26(b) is а good illustration of how
the real part of the DFT of а real input sequence has even symmetry,
and Figure 3-26(с) confirms that the imaginary part of the DFT of а real
input sequence has odd symmetry, (These symmetry properties were
introduced in 5ection 3,2 and are discussed further in Appendix С,
5ection с.3,)
t Тhis is а fundamental characteristic of Fourier transforms. Тhe narrower the function in
оnе domain, the wider its transform will Ье in the other domain.
104 The Discrete Fourier Traпsform
K
1 t х(п) . " " · П " : . ! , ' " ,, ' , -:.
0.5 ! ' 1: ;: i : :
i i i!:! 1
o ",.,.,...""",.""",, ! i i 1 1II1 I i " """""",.""""",
28 24 20 1б 12 -4 О 4 8 12 1б 20 24 28 32 п
(а)
(Ь)
12 [ 'eel(m J .
10 . I .
8 , '
б.' I .
;,,: I I
4 .1 I 1: 18
2 . I ::: 1. 8
. I I j '1 .
О .tt:t.,....."..,., ".t+++-. .++tI.",., ,.''''IIJI.trtt.
2 20 1б 12 ... -4 О 4 ... 12 1б 20 m
К=11
2 t Х; (т) ·
1 5 mag . II!' .
. 1 . I i 1. · : :' .
: I i ; 111 , I : i I · i'
0.5 .". . 20.[iii. ii:l' 4.il!' 12 1681i!i24 28
(с) 0 i . шt+++t I I ! 1 i 11 1 1 1 ! ! ! , 1 I 11 i i , l ' ,, ' " ' I I i I 11 ! I ! I I i 11 l' !! ф "''''
5 28 24.!1 16 8!!:, ...-4 о, I! 8 1:[1 20 ..8 m
1 8. I : i. . ,1 . : :.
1.5 . .i ..
4 .
Flgure 3.26 DFТ of а reetangular funetion: (а) origlnol funetlon >«п); (Ь) real port of
the DFТ of >«п). X,eai(m); (е) Imaglnary part of the DFТ of >«п). Ximag(m),
(а)
! IX(m) I ... 11
8 I l ' ,.
б .' ·
l' I '
4 · I 1:-
. I I L I i I .
2 .8 . . I I 1 1.. 8.
.. ... ... .... ] i I I I I .. . .-. .-. .
O++4' ' ' lill"IIII llIlilll I . 11 I'ТIIII ' 't+Щ,
28 24 20 1б 12 -4 О 4 8 12 1б 20 24 28 m
1 · -'<0(т)
и.....
(Ь) ,...
1! 28 28 m
2
......
Flgure 3-27 DFТ of а generollzed reetangulor funetion: (а) magnltude I Х(т) 1; (Ь)
phose ongle In rodions,
Although Xrea1(m) and Ximeg(m) tell us everything there is to know
about the DFТ of х(п), it's а bit easier to comprehend the true spectral
nature of Х(т) Ьу viewing its absolute magnitude. Тhis magnitude, fron
Eq. (3 7), is provided in Figure 3-27(а) where the main and sidelobes are
The DFr of Rectoпgu/or Fuпct/oпs 105
clearly evident now, As we expected, the peak value of the main lobe is
11 because we had К = 11 samples in х(п). The width of the main lobe
from Eq, (3 45) is 64/11, or 5.82. Thus, the first positive frequency zero
crossing location lies just below the т = 6 sample of our discrete I Х(т) I
represented Ьу the squares in Figure 3 27(a). The phase angles associated
with I Х(т) I , first introduced in Equations (3 6) and (3-8), are shown in
Figure 3 27(b),
То understand the nature of the DFT of rectangular functions more
fully, let's discuss а few more examples using less general rectangular
functions that are more common in digital signal processing than the х(п)
in Figure 3 24.
3.13.2 DFТ of а Symmetrical Rectangular Function
Equation (3-43) is а bit comp1icated because our original function х(п) was
so general, In practice, special cases of rectangu1ar functions lead to sim-
pler versions of Eq. (3 43), Consider the symmetrical х(п) rectangular
function in Figure 3 28, As shown in Figure 3-28, we often need to deter-
mine the DFf of а rectangular function that's centered about the п = О
index point, In this case, the К unity valued samples begin at
п = пo = (K 1)/2. 50 substituting (K 1)/2 for по in Eq. (3 43), yields
Х(т) = ei(21tт/ N)«K l)/HK l)/2) . sin(1tmK / N)
sin( пт / N)
= e i (2mп/N)(O). sin(1tmK / N)
sin( пт / N)
(3 46)
Because ei° = 1, Eq, (3 46) becomes
Symmetrical form of
the Dirich1et kemel:
Х(т) = sin(1tmK / N) .
sin( пт / N)
(3 47)
...с: к >
1 . . .. . . . . . .. . . x(n)
т ,l ii! JJ I iii'i
. ._. . . ._. ._. I i i' ,; I i
о
-N/2 +1 -п о = -(К-1)/2 п о = (К-1)/2
. ....-........-.............
n
N/2
Flgure 3.28 Rectangular х(п) wlth К samples centered about п = О,
1 06 The D/screte Fourier Traпsform
Equation (З 47) indicates that the DFТ of the syrnmetrical rectangular
function in Figure 3 28 is itse1f а real function; that is, there's по cornplex
exponential in Eq, (3-47), so this particular DFТ contains по imaginary part
or phase terrn. As we stated in Section 3.2, if х(п) is real and even,
х(п) = x( п), then Xrea1(m) is nonzero and X ima (т) is always zero, We
demonstrate this Ьу taking the 64-point DFТ 01 the sequence in Figure
3-29(а), Our х(п) is 11 unity-valued sarnples centered about the п = О index,
Here the DFТ resu1ts in an Х(т) whose real and imaginary parts are shown
in Figure 3-29(Ь) and Figure 3-29(с), respectively, АБ Eq, (3-47) predicted,
(а)
1 t Х(п) ........ ..
05 '
, ,
0.......................... 1111 j i I IIII ...........................
28 24 2a 16 1'2 -8 4 О 4 8' 12 1'6 20 24 28 32 п
(Ь)
12 i xr e al(т)
10
8
6
4 . .
2 ._. 20 .... ....g. . 8 ._. ._.
о ...4+++,.itrt.'1+++ i++++ allllllllll , .++++t Щ'.ttli.' ..
i 28 24 16 12 .... о 4 .... 12 16 26 28 3!! т
...
. .
11
(с)
0.5 ! Ximag(т)
0.4
0.3
0.2
0.1
О ..""""..,.,.,.,.",.,...,...,..".., ,..,...,...,.""..",,
28 24 2a 16 -12 -8 О 4 8 12 16 20 24 28 т
(d)
12 1 IX (т)1
10
8
6
4 . ' 1.
2.. ._. ... .... ,.... '. .... .... ._. ...
о .н., , , IIIIT 1111 , 1111111111" 1111" IIII ' ' '
28 24 2a -16 12 -8 О 4 8 12 16 20 24 28 т
11
(е)
4 Т JI0(т)
з ... ...... ...... 1t
r \....L\....L\........1...ч...:'..:...t.. ....
= 28. 24" ;al' 1.6 12'.:.;I" .... .... \ \ /\ /'\ т
з ..".,,__ ..... ...... ..
Figure 3-29 OFТ of а reetangular funetlon eentered about п = о: (а) orlglnal Х(п) ;
(Ь) eal(m); (е) X/mag(m); (d) magnltude of Х(т); (е) phase angle of
Х(т) In radlans,
The DFТ о' Rectaпgu/ar Fuпctioпs 107
Xrea1(m) is nonzero and Ximag(m) is zero. Тhe magnitude and phase of Х(т)
are depicted in Figure 3 29(d) and Figure 3 29(e),
Notice that the magnitudes in Figure 3 27(a) and Figure 3 29(d) are
identical. This verifies the very important shifting theorem of the DFТ;
that is, the magnitude I Х(т) I depends only оп the number of nonzero
samples in х(п), К, and not оп their position relative to the п = О index
value. Shifting the К unity valued samples to center them about the п = О
index merely affects the phase angle of Х(т), not its magnitude.
Speaking of phase angles, it's interesting to realize here that even
though Ximag(m) is zero in Figure 3 29(c), the phase angle of Х(т) is not
always zero, In this case, X(m)'s individual phase angles in Figure 3 29(e)
are either +Л, zero, or л radians. With +л and л radians both being equal
to 1, we could easily reconstruct Xrea1(m) from I Х(т) I and the phase
angle Х..(т) if we must. Xrea1(m) is equal to I Х(т) I with the signs of
I Х(т) I 's a1temate sidelobes reversed. t То gain some further appreciation
of how the DFТ of а rectangular function is а sampled version of the
Diricblet kernel, let's increase the number of our nonzero х(п) samples.
Figure 3 30(a) shows а 64 point х(п) where 31 unity valued samples are
centered about the п = О index location. Тhe magnitude of Х(т) is pro
vided in Figure 3 30(b), Ву broadening the х(п) function, i,e., increasing К,
we've narrowed the Diricblet kemel of Х(т). This follows from Eq. (3 45),
right? Тhe kemel's first zero crossing is inversely proportional to К, so, as
we extend the width of К, we squeeze I Х(т) I in toward т = О, In this
1 Т х(n) ' . .'. .. . ' . ' .. .. ' '8 .. i .. . . . ' .. ! . 8 .. .. ' .. ' . .,. . . . п.
(а) 05 т I ! ! i i i I i ! i , I ! ! , i '! i !!; i i , 1 ! i
о".,...,...,..., I i i j 1 i 11111 , i i 1111 i i 11 i I i i I j i 1 i ,...,...,...,...,....
28 24 2O 16 12 -8 -4 О 4 8 12 16 20 24 28 32 п
! IX(m)1 i\
Ь 20 ..
() 15 I \
10 I \
5 ,. ,. f8\J \,.\ 8, '.
О "' T""""""""4' '+' , , , , ,+,;,.,""""'"""""
28 24 20 16 12 -8 4 О 4 8 12 16 20 24 28 m
31
Flgure 3-30 DFТ of а symmetricol rectongular function with 3 1 unlty volues:
(о) origiпal х(п); (Ь) magпitude of Х(т),
t The particular pattem of +It and It values in Figure З-29(е) is an artifact of the software
used to generate that figure, А different software package тау show а different pattern, but
as long as the nonzero phase samples are either +It or --1t, the phase resu1ts will ье correct,
108 Тhe Discrete Foиrier Troпsforт
example, N = 64 and К = 31. From Eq. (3-45) the first positive zero cross-
ing о! Х(т) occurs at 64/31, Or just slightly to the right о! the т = 2 sam
ple in Figure 3 30(b), Also notice that the peak value of I Х(т) I = К = 31,
as mandated Ьу Eq. (3 44).
3.13.3 DFТ of ап АН Qnes Rectangular Functlon
Тhe DFf of а special form о! х(п) is routinely called for, leading to yet
another simplified form of Eq. (3--43), In the literature, we often encounter
а rectangular function where К = N; that is, аН N samples of х(п) are
nonzero, as shown in Figure 3-31, In this case, the N unity va1ued samples
Ьеgin at п = пo = N 1)/2, We obtain the expression for the DFf of the
function in Figure 3--31 Ьу substituting К = N and по = (N 1)/2 in Eq. (3--43)
to get
Х( ) j(21tm / N)[(N 1) / 2 (N 1) / 2] sin( 1tmN / N)
m e .
sin(1tm / N)
] '(2пт/ N)(O ) sin(1tm)
=е . or
sin( 1tm / N) ,
АН Ones form of
the Dirichlet
kernel (Туре 1):
Х(т) = sin(1tm) .
sin(1tm/N)
(3-48)
Equation (3 48) takes the first form of Eq. (3 34) that we alluded to at
the beginning о! tbls Section 3,13.+ Figure 3 32 demonstrates the mеаn-
ing о! Eq, (3 48). Тhe DFT magnitude of the аН ones function, х(п) in
-<
K=N
х(п)
1
-По= -(N-1 )/2
о
n
Figure 3-3' Rectangular functlon with N unlty-valued samples,
t Ву the way, there's nothing official about calling Eq. (3-48) а Туре 1 Dirichlet kernel, We're
using the phrase Туре 1 merely to distinguish Eq. (3-48) from other mathematical expres-
sions for the Dirichlet keme] that we're about to encounter,
Тhe DFr of Reetaпgu/ar Fuпetioпs 1 01
х(п)
1. .. . . .. ,. ...........
(8) 0.5 i i \ i I i ' i i 11 i!! i i
о
..................................
i: I . - 1"
; i
28 24 20 16 12 ...s 4 о 4 8 12 16 20 24 28 32 п
j IX(m) I .. 64
50 11
40 1 \
(Ь) 30 I
J
o........................... ltl ..... ....................
8 2 o -1'6 ;2 ..8. о /2 1 2 28 m
; L X(m), /"\ \
(с) за /
20 У \
19 ,c yY"{ ......,........yyyy ;
-5 4 ..з 2 1 О 1 2 3 4 5
64
...
m
Flgure 3-32 AII ones funetion: (о) reetongular function with N = 64 unity-volued
somples; (Ь) DFТ mognitude of the 011 ones time funetion; (с) elose-up
view of the DFТ mognitude of оп 011 ones time funetion,
Figure 3 32(a), is shown in Figure 3-32(Ь) and Figure 3 32(c). Take note
that if т was continuous Eq. (3 48) describes the shaded curves in
Figure 3 32(b) and Figure 3-32(с). If т is restricted to Ье integers, then
Eq, (3 48) represents the dots in those figures.
Тhe Diricblet kemel of Х(т) in Figure 3 32(b) is now as narrow as it сап
get. Тhe main lobe's first positive zero crossing occurs at the т = 64/64 = 1
sample in Figure 3-32(Ь) and the peak value of I Х(т) I = N = 64, With х(п)
being аН ones, I Х(т) I is zero for аН т :;с О, Тhe sinc function in Eq. (3 48)
is of utmost importanc s we'H see at the end of this chapter, it defines
the overaH DFТ frequency response to an input sinusoidal sequence, and
it's also the amplitude response of а single DFТ Ьin.
Тhe form of Eq, (3 48) aHows us to go one step further to identify the
most common expression for the DFТ of an аН ones rectangular function
found in the literature. То do this, we have to use an approximation prin
ciple found in the mathematics of trigonometry that уои тау have heard
before, It states that when а is smaH, then sin( а) is approximately equal to
а, i,e" sin(a) '" а, Тhis idea comes about when we consider а pie shaped
11 О The Discrete Fourier Traпsforт
(а)
(Ь)
ь
а
а
Figure 3-33 Relotlonships between оп ongle а, line а = sln(a), ond a's chord
Ь: (о) Large ongle ос (Ь) Smoll ongle а,
section of а circle whose radius is 1 as shown in Figure 3-33(а). Тhat sec-
tiOI1 is defined Ьу the length of the arc а measured in radians and а' s
chord Ь. If we draw а right triangle inside the section, we сап say that
а == sin( а). As а gets smaHer the long sides of our triangle Ьесоте almost
paraHel, the length of chord Ь approaches the length of arc а, and the
length of line а approaches the length of Ь. 50, as depicted in Figure
3-33(Ь), when а is smaH, а '" Ь '" а == sin(a), We use this sin(a) '" а approx-
imation when we look at the denominator of Eq. (3-48). When 1tm/N is
small, then sin(1tm/N) is approximately equal to 1tm/N, 50 we сап, when
N is large, state
АН Ones Еопn оЕ the
Dirichlet kemel (Туре 2):
Х(т) '" sin(1tm) = N . sin(1tm) .
mп/N 1tm
(3-49)
It has been shown that when N is larger than, say, 10 in Eq, (3-48), Eq,
(3 49) accurately describes the OFТ's output,t Equation (3-49) is often
normalized Ьу dividing it Ьу N, so we сап express the normalized OFТ of
an аН ones rectangular function as
t We сап Ье comfortable with this resuIt because, if we let К = N, we'l1 see that the peak value
of Х(т) in Eq, (3-49), for т = О, is equal to N, which agrees with Eq, (3-44),
The OFТ of Rectaпgи/ar Fипctioпs 111
АН Ones form of the
DirichIet kemel (Туре 3):
Х(т) '" sin(1tm) .
пт
(3 50)
Equation (3 50), taking the second form of Eq. (3 34) that is so often seen
in the literature, also has the DFТ magnitude shown in Figure 3-32(Ь) and
Figure 3 32(c),
3.13.4 Time and Frequency Axes Associated with
Rectangular Functlons
Let's establish the physical dimensions associated with the п and т index
values. 50 far in our discussion, the п index was merely an integer
enabling us to keep track of individual х(п) sample values. If the п index
represents instants in time, we сan identify the time period separating
adjacent х(п) samples to establish the time scale for the х(п) axis and the
frequency scale for the Х(т) axis, Consider the time-domain rectangular
function given in Figure 3 34(a). That function comprises N time samples
obtained t s seconds apart, and the full sample interval is Nt s seconds. Each
-с:
Nt.
> i
(а)
t.
' , ' . '"" >(9) .
"'п ) """" time = 9t. seconds '
....\ .... ... + '
. . . . . . . ... . ... ' . . . . . . . ... . . .
о 9 t. .econds
fs/2 Hz
( "" /2 = ---rt'.)
[---rt]
,',
Amplitude 01 Х(т) i \ 1/(Nt,;) = './N = (2"'./М
/ \ : . = (",,/М = [",/N]
. \ !<'--
. "- . / . '..,
.. '. ..., ,,8.. .....
':;,+;>' , , _Р ++-+',,,, , III I I I I ' '--+++' ,ц. , ' " ' I . ;H Z
'..8 о 8..' !
, Hz t >' ("'./М
(2"'. ="': radianslsec) I ,"/2 Hz ["' /N]
["' = 2" radians] ("'. /2 = "'.)
т=7 [,,]
Ireq, = 7 'JN Hz
(Ireq, = 7",. /N radianslsec)
[normalized angle = 7", / N radlans]
(Ь)
''<
Flgure 3-34 DFТ time ond frequency oxls dlmensions: (а) tlme-domoln oxls uses
tlme index п: (Ь) vorlous representotions of the DFT's frequency oxls,
112 The Discrete Foиr/er Traпsforт
ТаЫе 3-1 Characterlstlcs of Varlous OFТ Frequency Axls Representatlons
DFT Frequency Х(m) Repetition Frequency
Axis Frequency Reso/ution Interva/ о! Аш
Representation Variable о! Х(m) Х(m) Range
Frequency in нz тl/N fs!N 1. //2 to 1/2
Frequency in т(J)/N or (J)/Nor Ф. or 21tf s ./2 to Ф./2
radians 21tтM N 21tf s /N or 1tfs to rif.
Nonnalized angle 21tт/N 21t/N 21t --те to 1t
in radians
х(п) sample occurs at nt. seconds for some value of п. For example, the
п = 9 sample value, х(9) = О, occurs at 9t. seconds,
The frequency axis of Х(т) can Ье represented in а number of different
ways. Тhree popular types of DFf frequency axis labeling are shown in
Figure 3 34(b) and listed in ТаЫе 3 1. Let's consider each representation
individuaHy.
3.13,4,1 OFТ Frequency Axis in Hz
If we decide to relate Х(т) to the time sample period t., or the sample
ratef. = l/t., then the frequency axis variable is m/Nt. = mf/N. So each
Х(т) sample is associated with а сусНс frequency of mf/N Hz. In this
case, the resolution of Х(т) isf/N. The DFТ repetition period, or peri-
odicity, is fs Hz, as shown in Figure 3 34(b). If we substitute the сусНс
frequency variable mf/N for the generic variable of m/N in Eq. (3-47),
we obtain аn expression for the DFT of а symmetrical rectangular func-
tion, where К < N, in terms of the sample rate f. in Hz. Тhat expression
is
Х(т!.) = sin(п;т!.К/ N) ,
sin( п;т!. / N)
(3-51)
For аn аН ones rectangular function where К = N, the amp1itude normal
ized sin(x) / х approximation in Eq. (3-50) can Ье rewritten in terms of sam
ple rate f. in нz as
X(m f ) '" sin(п;т!.) .
J · п;тfs
(3 52)
Тhe OFТ о' Rectaпgu/ar Fuпctioпs 113
3,13.4,2 OFТ Frequency Axis in Radians/Second
We сап measure X(m)'s frequency in radians/s Ьу defining the time
domain sample rate in radians/s as 0). = 21tfs' 50 each Х(т) sample is asso-
ciated with а radian frequency оЕ mO)/N = 21tmf/Nradians/s. In this case,
X(m)'s resolution is O)/N = 21tf/N radians/s, and the DFT repetition
period is o)s = 21tf. radians/s, as shown Ьу the expressions in parenthesis in
Figure 3-З4(Ь). With 0). = 21tf., then 1tf. = 0)/2. If we substitute 0).12 for rif.
in Eq. (3-51), we obtain an expression for the DFI оЕ а symmetrical rectan-
gular function, where К < N, in terms оЕ the sample rate 0). in radians/s:
Х( ) = sin(тюsК / 2N)
тю s '
sin(тю. /2N)
(3-53)
For an аН ones rectangular function where К = N, the arnplitude normal-
ized sin(x)/x approximation in Eq. (3-50) сап Ье stated in terms оЕ а sarn-
ple rate Ю. in radians/s as
Х( ) '" 2 sin(тю. /2)
тю. .
тю.
(3-54)
3.13,4,3 OFТ Frequency Axis Using а Normalized Angle VariabIe
Маnу authors simplify the notation оЕ their equations Ьу using а normal-
ized variable for the radian frequency Ю. = 21tfs' Ву normalized, we теаn
that the sample rate f. is assumed to Ье equal to 1, and this establishes а
normalized radian frequency Ю. equal to 2х. Тhus, the frequency axis оЕ
Х(т) now becomes а normalized angle Ю, and each Х(т) sample is asso-
ciated with аn angle оЕ тю/N radians. Using this convention, X(m)'s res-
olution is ю/N radians, and the DFI repetition period is ю = 2х radians,
as shown Ьу the expressions in brackets in Figure 3-34(Ь).
Unfortunately the usage оЕ these three different representations of the
DFTs frequency axis is sometimes confusing for the beginner. When
reviewing the literature, the reader can leam to convert between these fre-
quency axis notations Ьу consulting Figure 3-34 and Table 3-1.
3.13.5 Alternate Form of the DFТ of аn AII Ones
Rectangular Function
Using the normalized radian angle notation for the DFI axis from the bot
tom row of Table 3-11eads to another prevalent form оЕ the DFI оЕ the аН
114 The Discrete Fourier Traпsforт
ones rectangular function in Figure 3 31, Letting our normalized discrete
frequency axis variable Ье <От = 21tm/N, then лт = N<Om/2, Substituting
the term N<Om/2 for лт in Eq, (3 48), we obtain
АН Ones form of the
Dirich1et kemel (Туре 4):
Х(ю) = sin(Nm m /2) ,
sin(m m /2)
ТаЫе 3-2 OFТ of Varlous Rectangular Functions
Descriptioп Expressioп
DFf of а genera! rectangular function, where Х(т)= sin(!tmK/N)
К < N, in terms of the integra! frequency т
variable sin(!tm/N)
'e i (2''''1 N)(", (K 1)/2)
DFf of а symmetrica! rectangu!ar function, Х(т) = sin(1tmK/N),
where К < N, in terms of the integra! frequency sin(!tm / N)
variable т
DFf of an аН ones rectangu!ar function in terms sin(!tm)
of the integer frequency variable т (Dirich1et Х(т)= sin(!tm/N)
keme! Туре 1)
DFf of an аН ones rectangu!ar function in terms Х(т) = N sin(!tm).
of the integer frequency variable т (Dirich1et 1tm
keme! Туре 2)
Amplitude norma!ized DFf of аn аН ones Х(т) = sin(!tm) ,
rectangu!ar fиnction in terms of the integra! 1tm
frequency variable т (Dirich1et keme! Туре З)
DFf of а symmetrica! rectangu!ar fиnction, Х(тД = sin( !tm fs K / N),
where К < N, in terms of the samp!e rate fs in нz sin(!tmj, / N)
Amplitude normalized DFf of аn аН ones X(mj,) = sin(1tmf,) ,
rectangu!ar fиnction in terms of the samp!e 1tmj,
rate fs in Hz
DFf of а symmetrica! rectangular fиnction, Х( ) sin( тro,К / 2N)
where К < N, in terms of the samp!e rate ю s тro, sin(mro, /2N) .
in radians/s
Amp1itude normalized DFf of an аН ones Х( ) = 2sin(mro, /2)
rectangu!ar function in terms of the samp!e тro, '
rate ю s in radians/s тro,
DFT of an аН ones rectangu!ar function in Х(ro) = sin(Nro т /2),
terms of the normalized discrete frequency sin(ro", /2)
variable Ют (Dirich1et keme! Туре 4),
(3 55)
(3 46)
(3 47)
(3 48)
(3-49)
(3-50)
(3 51 )
(3 52)
(3 53)
(3-54)
(3 55)
The OFТ of Rectoпgu/or Fuпct/oпs 11 !
Equation (3-55), taking the third form of Eq, (3 34) sometimes seen in the
literature, also has the DFT magnitude shown in Figure 3 32(b) and
Figure 3 32(c).
We've covered so тапу different expressions for the DFT of various
rectangular functions that it's reasonable to compile their various forms in
Table 3 2.
3.13.6 Inverse DFT 01 а General Rectangular Function
Let's think now about computing the inverse DFТ of а rectangular fre-
quency domain function; that is, given а rectangular Х(т) function, find
а time domain function х(п). We сап define the general form of а rectan-
gular frequency domain function, as we did for Figure 3-24, to Ье that
shown in Figure 3-35, The inverse DFТ of the rectangular function Х(т) in
Figure 3 35 is
1 N /2
х(п)= N LX(m)ei27tmп/N,
m= N / 2)+1
(3-56)
The same algebraic acrobatics we used to arrive at Eq. (3 43) сап Ье
applied to Eq. (3 56), guiding us to
( ) J '(27tп/N )( m ( K 1 ) /2) 1 sin(nnK/ N)
хп=е ' . .
N sin(nn / N) ,
(3 57)
for the inverse DFT of the rectangular function in Figure 3 35, The
descriptions we gave for Eq. (3-43) apply equally well to Eq, (3 57) with
the exception of а 1/ N scale factor term and а change in the sign of
Eq, (3 43)'s phase angle, Let's use Eq. (3-57) to take а 64 point inverse
DFТ of the 64 sample rectangular function shown in Figure 3 36(a), The
-< к >-
1 . . . . . . . . . . . . . Х(т)
, i ! ;
. . . . . . . .... . .... i
+ о
N12 + 1 т = тo
+
т= тo+(K 1)
. . .... ..... .
т
Ю2
Flgure 3-35 General rectangular frequency domain function of width К
sampies deftned over N samples where К < N.
116 The Discrete Fourier Traпsform
(а)
1 Т Х(т) ·
05 Т
0.""""""""""""" 1
-28 -24 20 16 12
. ...
11 I ."._,.,.,....,...,...,...,..
8 12 16 20 24 28 32 m
0.2
0.15
0.1
(Ь) 0.05
О
.05
x,eal (п) ... 0,172
. i
. i: i.
l'
. 1; I
i i : i: i i. 8
.'ii1t.....".".",.tt+t. 111 J i I i ; I i .!t11.,,,.,..,.,......i1ii'
20 -16 12 ... О 4 ... 12 16 20 n
: f Xima9(n) ... .1". .[11. .П.
0.01 28 2Ч i , 16 . , !: о ' : : 8 : i ! ! 20.. n
(с) Ь '.. lJ ' i JoiT ll'l 11 iJi'llll( 1 i 11 ' 111 : j I i j] 1l1l.JJ i71: i
.. I 1.
...о,аз · Ii ..
Flgure 3-36 Inverse DFТ of а generol reetongulorfunetion: (а) orlglnol funetion Х(т);
(Ь) reol port, x,eol(п); (е) imoglnory part, X;mog(п),
о 2 1 ,x (n), ... 0172
015 .: .
01 ·
(а) о 05 ' 1.
.. ... .... .... . . 81 ' . . .. .... .... ... .
0+i4.+h4.++-t+" 1111' 1111 ' 1111111111 "IIII"IIII ' ' '
I -28 24 -20 16 -12 О 4 8 12 16 20 24 28 n
(Ь)
:+ х,,(п)
1 t 28
f'\ .....
-з1 .
Figure 3-37lnverse DFТ of а generalized reetongulor funetlon: (а) mognltude
I х(п) I ; (Ь) phase ongle of х(п) In radlons,
inverse DFT of the sequence in Figure 3-36(а) resu1ts in an х(п) whose
real and imaginary parts, x rea1 (n) and Ximag(n), are shown in Figure 3 36(b)
and Figure 3-36(с), respectively, With N = 64 and К = 11 in this example,
we've made the inverse DFT functions in Figure 3-36 easy to compare
Тhe OFТ of Rectaпgu/ar Fuпct/oпs 11 :
with those forward DFT functions in Figure 3-26. 5ее the similarity
between the real parts, Xreal(m) and xrea1(n), in Figure 3 26(b) and Figure
3-36(Ь)? Also notice the sign change between the imaginary parts in
Figure 3-26(с) and Figure 3 36(c),
Тhe magnitude and phase angle of х(п) are shown in Figure 3-37(а)
and Figure 3 37(b), Note the differences in main lobe peak magnitude
between Figure 3 27(a) and Figure 3 37(a). Тhe peak magnitude value in
Figure 3 37(a) is K/N = 11/64, or 0,172. Also notice the sign change
between the phase angles in Figure 3 27(b) and Figure 3 37(b). Тhe illus-
trations in Figures 3 26, 3 27, 3 36, and 3 37 are good examples of the fun
damental duality relationships between the forward and inverse DFТ.
3.13.7 Inverse DFТ of а Symmetrical Rectangular Function
The inverse DFT of the general rectangular function in Figure 3-36 is
not very common in digital signal processing, However, in discussions
concerning digital fi1ters, we will encounter the inverse DFТ of sym
metrical rectangular functions, This inverse DFT process is found in the
window design method of what are called low-pass finite impulse
response (FIR) digital fi1ters. That method begins with the definition of
а symmetrical frequency function, Н(т), such as that in Figure 3-38,
The inverse DFT of the frequency samples in Figure 3 38 is then taken
to determine the time domain coefficients for use in а low pass FlR fil
ter. (Time Domain FlR fi1ter coefficients are typically denoted h(n)
instead of х(п), so we'll use h(n) throughout the remainder of this
inverse DFT discussion,)
In the case of the frequency domain Н(т) in Figure 3-38, the К unity-
valued samples Ьеgin at т = тo = (K 1)/2, 50 plugging (K 1)/2 in for
то in Eq. (3 57), gives
h(n) = e j (27m! N)((K 1)!2 (K 1)!2) ..l. sin(1tnK / N)
N sin( пп / N)
= e j (21tп! N)(O) . .l. sin( ltпK / N) ,
N sin(1tn / N)
(3 58)
Again, because e jO = 1, Eq, (3 58) becomes
h(n) =.l. sin(1tnK / N) ,
N sin(1tn / N)
(3 59)
118 Тhe Diserete Fourier Traпsform
-<
к
.........I!I.
>-
.. Н(т)
....... . ................ ...... .
о
N'2+1 т= ПЬ= К 1)/2
.......... ........ ...............
+ + т
т= (K 1)12 М2
Figure 3-38 Frequeney-domain reetangular funetion of width К samples
defined over N samples,
(а)
о i Н(т) ...........
о.............,.......,.... , J 111 J 111 J , .....,......................
о 4 т
(Ь)
о 2 [ h real (n)
015
01
005 · ·
._. .... 8. . 8 .... .
о ..................'.ffi.'тж-Т;-н++. IIIIIIIIIII .4нtТ+++t'.t++i'.' ..
1 · 16 12 .... 4 О 4 .... 12 16 .. 26 28 3 n
-<J05l.
...
. .
0.172
(о)
0.4 1 himag (n)
0.3
0.2
0.1
0111111' "'II!II!I" ",'IIII!I' .,. 'II!II!I .,.,."", .,.",.",..,...., IIII!I''''IJII!I'' ...., I!II!II!I
28 24 20 16 12 о 4 8 12 16 20 24 28 n
(d)
о 2 j lh(n)1
01
01
.' .
00
.. .-. .-. .... ..... . .... .... .-. .-. .
о , , , IIII ' 1111 ' 1111111111" IIII'T 1111 , 't+++t'
I 28 24 20 16 12 о 4 8 12 16 20 24 28 n
...
. .
0172
(е)
[ ''" . ...... :] ..
28 -24 20 16 12 о I I I
о l ... +r+-I-++. ... +++Ж l ... ...'. ...'. ...,.
= \ I \ 4 8 12 16 20 24 28 n
з... ...... ...... 7t
l
Figure 3-39 Inverse OFТ of а rectangular funetion eentered about т = о:
(а) orlglпal Н(т); (Ь) h,eal(п); (е) himag(п); (d) magпitude of h{п);
(е) phase of h{п) in radians,
The OFТ Frequeпcv Respoпse to а Сотр/ех /пput 119
Equation (3 59) te11s us that the inverse DFТ of the symmetrical rectan-
gular function in Figure 3 38 is itself а real function that we can i11ustrate Ьу
example, We'll реrfопn а 64--point inverse DFТ of the sequence in Figure
3 39(a). Our Н(т) is 11 unity valued samples centered about the т = О
index. Here the inverse DFТ results in an h(n) whose real and imaginary
parts are shown in Figure 3 39(b) and Figure 3 39(c), respectively, As Eq,
(3 47) predicted, hreal(n) is nonzero and himag(n) is zero. Тhe magnitude and
phase of h(n) are depicted in Figure 3-39(d) and Figure 3 39(e), (Again,
we've made the functions in Figure 3 39 easy to compare with the forward
DFТ function in Figure 3 29,) It is h(n)'s real part that we're really after here,
Тhose values of hreal(n) are used for the time domain filter coefficients in the
design of FIR low pass filters that we'll discuss in Section 5.3.
3.14 The DFT Frequency Response to а
Complex Input
In this section, we'll determine the frequency response to an N point DFT
when its input is а discrete sequence representing а complex sinusoid
expressed as хс(п). Ву frequency response we mean the DFТ output sam
ples when а complex sinusoidal sequence is app1ied to the DFТ, We begin
Ьу depicting an хс(п) input sequence in Figure 3-40. Тhis time sequence is
of the fопn
Аеаl part 01 хс(п)
(а)
о
1
. ...
\. . ':.
, ,. · i ·
.: i:.
.' '.
...
.I!I
.:' ....
.. i '.. ,.}.., Time (п)
...
t Imа9iП . ary part 01 хс(п)
1 .... ... ...
. .: ; · . .I!I , · . 1
0 + . 1111 j II I!I !! 11 ! IIJI IIIIIII I1J1 III . .lll
. ' : :. .: : .. Т1те (п)
1 ... ...
(Ь)
Flgure 3-40 Complex tlme-domain sequeпce хс(п) = eJ2ппk/N haviпg two
complete cycles (k = 2) over N samples: (а) real part 01 хс(п); (Ь)
imaglnary part 01 хс(п),
120 The Discrete Fourier Traпsforт
хс(п) = ej21tnk/N ,
(3 60)
where k is the number of complete cycles occurring in the N samples.
Figure 3 40 shows xin) if we happen to let k = 2, If we denote our DFТ
output sequence as Хс(т), and apply our хс(п) input to the DFТ expression
in Eq, (3 2) we have
N l N l
Хс(т) = Lxc(n)e j21tnт/N = Lеj2"пk/N'е j21!пт/N
п=О п=О
N l
= Lej21tn(k т)/N,
п=О
(3 61)
If we let N = К, п = р, and q = 21t(k m)/N, Eq. (3--61) becomes
K l
Хс(т) = Le jPq
р=о
(3 62)
Why did we make the substitutions in Eq, (3 61) to get Eq, (3-62)?
Because, happily, we've already solved Eq. (3 62) when it was Eq, (3--39).
Тhat closed form solution was Eq. (3 41) that we repeat here as
Хс(т) = Ie jpq = e jq(K 1)/2. sin(qK /2) ,
р=о sin(q / 2)
(3--63)
Replacing our original variables from Eq, (3 61), we have our answer:
DFf ()f а complex
sinusoid: Х ( ) j[1t(k т) 1t(k т)/NJ sin[1t(k т)]
m e .
с sin[1t(k т) / N]
(3 64)
Like the Diricblet kemel in Eq, (3-43), the Хс(т) in Eq. (3 64) is а сот-
plex expression where а ratio of sine terms is the amplitude of Хс(т) and
the exponential term is the phase angle of Хс(т), At this point, we're inter
ested only in the ratio of sines factor in Eq, (3 64), Its magnitude is shown
in Figure 3--41, Notice that, because хс(п) is complex, there are по negative
frequency components in Хс(т) , Let's think about the shaded curve in
The ОП Frequeпcy Respoпse fo а Сотр/ех /пpuf 121
Figure 3-41 for а moment. That curve is the continuous Fourier transform
of the complex хс(п) and сan Ье thought of as the continuous spectrum of
the х/п) sequence. t Ву continuous spectrum we mеаn а spectrum that's
defined at аН frequencies, not just at the periodic 1./ N analysis frequen-
cies of аn N-point DFТ. The shape of this spectrum with its main lobe and
sidelobes is а direct and unavoidable effect of analyzing anу finite length
time sequence, such as our хс(п) in Figure 3 40,
We сап conceive of obtaining this continuous spectrum analytically Ьу
taking the continuous Fourier transform of our discrete хс(п) sequence, а
process some authors саН the discrete time Fourier transform (DТFr), but
we can't actually calculate the continuous spectrum оп а computer, That's
because the DТFr is defined only for infinitely long time sequences, and
the DТFr's frequency variable is continuous with infinitely fine-grained
resolution. What we сап do, however, is use the DFT to calculate аn
approximation of xc(п)'s continuous spectrum, The DFТ outputs repre-
sented Ьу the dots in Figure 3 41 are а discrete sampled version of xc(п)'s
continuous spectrum, We could have sampled that continuous spectrum
more often, i.e" approximated it more closely, with а larger DFT Ьу
appending additional zeros to the original хс(п) sequence, We actually did
that in Figиre 3-21,
Figure 3-41 shows us why, when аn input sequence's frequency is
exactly at the т = k Ыn center, the DFТ output is zero for аН bins except
where т = k, If our input sequence frequency was k+O,25 cycles in the
sample interval, the DFТ will sample the continuous spectrum shown in
Figure 3 42 where аН of the DFT output Ыns would Ье nonzero, This
effect is а demonstration of DFТ leakage described in Section 3,8,
IХс(т)1 OFТ
, response
This curve is the continuous Fourier
translorm 01 x(n) = ej2xпklN
!!\ /
' ,\
f \
I i \
I , \
'''1 /'",r-,/"v/"V! [ \f'\/"\ ,,,__ ,,,-
i ' i ' i i ' i ' i
О k 5 k-4 k--З 1f-2 k 1 k k+ 1 k+2 k+3 k+4 k+5 m
N
I
k
Figure 3-41 N-point DFТ frequency тagnitude response to а coтplex sinusoid
having integral k cycles in the N-point time sequence хс(п) = eJ2xпk/N
t Just as we used L'Hospita!'s ru!e to find the peak уа!ие of the Dirich1et kerne! in
Eq, (3 44), we cou!d a!so eva!uate Eq, (3-64) to show that the peak of Хс(т) is N when т = k,
122 The Discrete Fourier Traпsforт
IХс(т)1 DFТ
response
k
m
о
Flgure 3-42 N-poiпt DFТ frequeпcy magпltude respoпsв showlпg spectralleakage ot а
complex slпusold havlпg k+O,25 cycles lп the N-poiпt tlme sequeпce хс(п),
Again, just as there are several different expressions for the DFr of а
rectangular function that we listed in ТаЫе 3 2, we сап express the ampli-
tude response of the DFr to а complex input sinusoid in different ways to
arrive at ТаЫе 3-3.
ТаЫе 3-3 Varlous Forms ot the Amplitude Response ot the DFТ to а
Complex lпput Sinusoid Having k Cycles (п the Sample lпteNal
Descriptioп Expressioп
Сотр!ех input OFf amplitude Х (т)= sin[1t(k m)]
response in terms of the integra!
frequency variable т [From Eq, (З 64)] с sin[1t(k m)/N]
Altemate form of the complex input OFf Х (т)= Nsin[1t(k m)]
arnplitude response in terms of the integra! ' 1t(k m)
frequency variable т [based оп Eq, (3-49)]
Amplitude normalized сотр!ех input OFf Х (т)= sin[1t(k m)]
response in terms of the integral frequency с 1t(k m)
variable т
Сотр!ех input OFf response in terms Х mj,)= Nsin[1t(k m)fs]
of the samp!e rate f. in Hz с( s 1t(k m)f,
Amplitude normalized сотр!ех input OFf Х (mj,) = sin[1t(k m)f,]
response in terms of the sample rate f. in Hz с s 1t(k m)fs
Amplitude normalized сотр!ех input OFf Х (тю)= 2sin[(k т)юs/2]
response in terms of the sample rate (о. с s (k т)юs
(3-65)
(3-66)
(3-67)
(3-68)
(3-69)
(3 70)
The OFТ Frequeпcy Respoпse to а Reaf Cosiпe fпput 123
At this point, the thoughtfиl reader тау notice that the DFТ' s response
to а complex input of k cycles per sample interval in Figure 3-41100ks sus-
piciously like the DFТ's response to an а11 ones rectangular function in
Figure 3 32(c). Тhe reason the shapes of those two response curves look
the same is because their shapes are the same. If our DFТ input sequence
was а complex sinusoid of k = О cycles, i.e., а sequence of identical con-
stant values, the ratio of sines term in Eq, (3-64) becomes
sin[1t(O т)]
sin[1t(O т) / N]
=
sin(1tт)
sin( хт / N)
sin( хт)
sin(1tт / N) ,
which is identical to the а11 ones form of the Dirich1et kernel in Eq, (3-48).
Тhe shape of our Хс(т) DFТ response is the sinc function of the Dirich1et
kemel.
3.15 The DFT Frequency Response to а Real
Cosine Input
Now that we know what the DFT frequency response is to а complex
sinusoidal input, it's easy to determine the DFТ frequency response to а
real input sequence. Say we want the DFT's response to а real cosine
sequence, like that shown in Figure 3-40(а), expressed as
Xr(п) = cos(21tпk/N) ,
(3-71)
where k is the integral number of complete cycles occurring in the N sam-
ples, Remembering Euler's relationship COS(I2I) = (e io + e io)/2, we сап show
the desired DFТ as Х,(т) where
N l N l
Xr(т) = LXr(п)e i2Itnт/ N = L cos(21tпk / N). e i2Itnт/ N
п=О п=О
N l
= L(ei2Itnk/N +e i2Itnk/N)/2'e i2"nт/N
п=О
N l N l
= Lei21tn(k т)/N + Le i2"n(k+т)/N ,
п=О п=О
(3 72)
124 The D/screte Four/er Traпsforт
F ortunately, in the previous section we just finished determining the
closed form of а summation expression like those in Eq. (3-72), so we сап
write the closed form for Х/m) as
DFr of а rea!
cosine: --7
Х (т) = ej[1t(k тHt(k т)/ N] ..! sin[1t(k т)]
r 2sin[1t(k m)/N]
J '[1t(k+т) 1t(k+т ) /N] 1 sin[1t(k + т)]
+е .
2 sin[1t(k + т) / N]
(3-73)
We show the magnitude of those мо ratio of sines terms as the sinc
functions in Figure 3 43, Here again, the DFT is sampling the input
cosine sequence's continuous spectrum and, because k = т, оnlу опе DFf
Ьin is nonzero. Because the DFf's input sequence is real, Х/m) has both
positive and negative frequency components. The positive frequency
portion of Figure 3-43 corresponds to the first ratio of sines term in Eq.
N
Т /\
f \
i ,
/\ I ' \ /\
"",, / V \ /'-J""
' i ' i i ' i ,--1:
--4--1< --I< 2--1< ' --1< --1<+,--1<+2--1<+з--1<+4 О
IХ,(т)1 DFТ
response N
"2
Тhis curve, 810ng w h its
-- / ' i\ negative Irequency
I ре"/оп, is the continuous
I \ j / Fourier tr8nslorm 01
( х(п) = cos(21tпk/1l/),
L /'..rJV ! /\/\/'.. ,,)'\........
l\' i ' i i ' i '
k--4 k k k '. k k+' k+2 k+З k+4 m
Flgure 3-43 N-point OFТ frequency magnitude response to о reol cosine havlng
integrol k cycles in the N-point tlme sequence х,(п) = cos(21tпk/N),
IX,(m)1 DFТ
response
N N
Т ' ( \ , 1 ,\ т j 1\
" i , i \
! ,1
11'f i\'i'i'l '" \ T'j"Yi i I Y1 V Y'T
--1<--4 --I< --I<--2--1< ' t-* --1<+,--1<+2 --I<+З --1<+4 О k--4 k k--2 k ' k k+ 1 k+2 k+З k+4 m
k ,25 k+O,25
Flgure 3-44 N-point OFТ frequency magnitude rвsponse showing spectrolleokoge of
о reol cosine hoving k+Q,25 cycles in the N-point tlme sequence х,(п),
Тhв OFТ S/пg/e-Biп Frequeпcy Respoпse to а Rea/ Cos/пe /пput 125
ТаЫе 3-4 Various Forms of the Positive Frequeпcy Amplltude Response of the
OFТ to а Real Coslпe lпput Hav;пg k Cycles iп the Sample lпterval
Descriptioп Expressioп
Real input DFТ amplitude response in Х (т) = sin[1t(k т)]
terms of the integral frequency variable т r 2sin[1t(k m)/N]
[From Eq. (3-73)]
AItemate form of the real input DFТ Х (т) N sin[1t(k т)]
amplitude response in terms of the integral r 21t(k m)
frequency variable т [based оп Eq, (3-49)]
Amplitude normalized real input Xr(m) sin[1t(k m)]
DFТ response in terms of the integral
frequency variable т 21t(k m)
Real input DFТ response in terms of X,(m!s) Nsin[1t(k m)!s]
the sample rate fs in Hz 21t(k m)f,
Amplitude normalized real input Xr(mfs) sin[1t(k m)!sJ
DFТ response in terms of the sample
rate fs in нz 21t(k m)f,
Amplitude normalized real input Х (тш ) sin[(k т)ш, /2]
DFТ response in terms of the sample
rate W s in radians / s " (k m)w,
(3-74)
(3-75)
(3-76)
(3-77)
(3-78)
(3-79)
(3.73) and the second ratio of sines term in Eq. (3-73) produces the nega.
tive frequency components of Х,(т).
DFТ leakage is again demonstrated if our input sequence frequency
were shifted from the center of the kth Ьin to k+O,25 as shown in Figure
3-44. (We used this concept of real input DFT amp1itude response to
introduce the effects of DFТ leakage in Section 3.8.)
In Table 3.4, the various m.athem.atical expressions for the (positive fre.
quency) amplitude response of the DFf to а real cosine input sequence are
simply those expressions in Table 3.3 reduced in amplitude Ьу а factor of 2.
3.16 The DFТ Single-Bin Frequency Response to а
Real Cosine Input
Now that we understand the DFТ's overal1 N.point (or N-bin) frequency
response to а real cosine of k cycles per sample interval, we conclude this
chapter Ьу determining the frequency response of а siпgle DFТ Ьin. We
can think of а single DFТ Ьin as а kind of bandpass filter, and this useful
notion is used, for example, to describe DFТ scalloping 10ss (Section 3.10),
employed in the design of frequency.domain filter banks, and applied in
126 Тhe D/screte Four/er Traпsform
а telephone frequency multiplexing technique known as transmultiplex-
ing[15]. То determine а DFI single bin's frequency response, consider
applying а real х.(п) cosine sequence to а DFI and monitoring the output
magnitude of just the т = k Ыn. Let's say the initial frequency of the input
cosm.e sequence starts at а frequency of k<т cycles and increases up to а
frequency of k>т cycles in the sample interval. If we measure the DFT's
Output (\ с=с>
magnitude 01
the mth Ып
! \
(а) I \
'\, /
о
т--1
Output
magnitude 01
the mth Ып
(Ь)
о
т--1 пн-1
Output
magnitude 01
the mth Ып
(с)
О
Single-bin
magnilude
responSe
(d)
о
k
k
' "
k
This curve is lhe
COnlinuous frequency
magnilude response
01 а OFТ'. m!h Ып
when the OFТ Input is
reai.
Input
Irequency k
Flgure 3-45 Determinlng the output magnitude 01 the тth Ыn 01 оп N-point DF1":
(а) when the real х,(п) has k = т--2,5 cycles In the tlme sequence;
(Ь) when the real Х,(п) has k = т--l,5 cycles in the tlme sequence;
(е) when tt1e real Х,(п) has k = т eyeles In the tlme sequenee;
(d) the DF1" slngle-bin frequeney magn\tude response 01 the т = k Ыn,
Refereпces 127
т = k Ьin during that frequency sweep, we'll see that its output magni-
tude must track the input cosine sequence's continuous spectrum, shown
as the shaded curves in Figure 3-45,
Figure 3-45(а) shows the т = k bin's output when the input xr(п)'s fre
quency is k = m 2.5 cycles per sample interval, Increasing xr(п)'s fre-
quency to k = m 1.5 cycles per sample interval results in the т = k bin's
output, shown in Figure 3-45(Ь), Continuing to sweep xr(п)'s frequency
blgher Figure 3-45(с) shows the т = k bin's output when the input fre-
quency is k = т, Тhroughout our input frequency sweeping exercise, we
сап see that the т = k bin's output magnitude must trace out the cosine
sequence's continuous spectrum, shown Ьу the solid curve in Figure
3-45(d), This means that а DFTs single-bin frequency magnitude
response, to а real input sinusoid, is that solid sinc function curve
defined Ьу Eqs. (3-74) through (3-79).
References
[1] Bracewell, R. "Тhe Fourier Transform," Scientific American, June 1989,
[2] Struik, О. А Concise Нistory о/ Мathematics, Dover Pиblications Inc" New York,
1967, р. 142.
[3] Williams, С. 5, Designing Digital Filters, Section 8,6, Prentice-Hall, Englewood
Cliffs, New Jersey, 1986, р, 122,
[4] Press, W" et аl, Numerical Recipes Тhe Art o/Scientific Computing, Cambridge
University Press, 1989, р, 426,
[5] Geckin1i, N, с., and Yavuz, О, "Some Novel Windows and а Concise Tиtorial
Comparison оЕ Window Families," IЕЕЕ Trans, оп Acoust, Speech, and Sigпal
Proc" Vol, ASSP 26, No, 6, December 1978, (Ву the way, оп page 505 оЕ this
paper, the phrase "sиch that W(f) ;:: о '<1 Е" indicates that W(f) is never nega-
tive, Тhe symbol '<1 means "for all,")
[6] O'Donnell, J, "Looking Тhroиgh the Right Window Improves Spectral
Analysis," EDN, November 1984,
[7] Kaiser, J, F. "Digital Filters," in System Analysis Ьу Digital Computer, Ed, Ьу F,
F. Кио and J, F, Kaiser, John Wiley and Sons, New York, 1966, рр, 218---277'
[8] Rabiner, L, R., and Gold, В, The Theory and Application о/ Digital Signal
Processing, Prentice-Hall, Englewood Cliffs, New Jersey, 1975, р, 88,
[9] Schoenwald, J, "Тhe Sиrface Acoиstic Wave Filter: Window Functions," RF
Design, March 1986,
[10] Harris, F, "On the Use оЕ Windows for Harmonic Analysis with the Discrete
Fourier Transform," Proceedings о/ the IЕЕЕ, Vol, 66, No. 1, Janиary 1978,
128 Тhв D/screte Four/er Traпsform
[11] Nиttall, А. Н, "Some Windows with Very Good Side!obe Behavior," [ЕЕЕ
Traпs, оп Acoust, Speech, aпd Signa/ Proc., Vo!, ASSP-29, No, 1, Febrиary 1981.
[12] Yanagimoto, У. "Receiver Design for а Combined RF Network and Spectrum
Analyzer," Hewlett Packllrd lourпa/, October, 1993,
[13] Gul!emin, Е, А. The Мathematics о! Circuit Aпa/ysis, John Wiley and Sons, New
York, 1949, р. 511.
[14] Lanczos, С. Discourse оп Fourier Series, Chapter 1, Hamer Pиblishing Со" New
York, 1966, р. 7 47,
[15] Freeny, 5, "TDM/FDM Trans!ation As аn Application оЕ Digita! Signal
Processing," [ЕЕЕ Commuпicatioпs Мagaziпe, Janиary 1980.
!CHAPTER FOURI
The Fast Fourier
Transform
Although the DFf is the most straightforward mathematical procedure
for determining the frequency content of а time domain sequence, it's ter
ribly inefficient, As the number of points in the DFf is increased to hun
dreds, or thousands, the amount of necessary number crunching becomes
excessive. In 1965 а paper was published Ьу Cooley and Tukey describing
а very efficient algorithrn to implement the DFf[l]. Тhat algorithrn is now
known as the fast Fourier transform (FFf).t Before the advent of the FFf,
thousand point DFfs took so long to perform that their ше was restricted
to the larger research and university computer centers. Тhanks to Cooley,
Tukey, and the sem.iconductor industry, 102 point DFfs can now Ье per
formed in а few seconds оп home computers.
Volumes have been written about the FFf, and, like по other innоуа
tion, the development of this algorithrn transformed the discipline of dig
ital signal processing Ьу making the power of Fourier analysis affordable,
In this chapter, we'll show why the most popular FFf algorithrn (called
the radix 2 FFf) is superior to the classical DFf algorithrn, present а series
of recommendations to enhance our use of the FFf in practice, and pro
vide а list of sources for FFf routines in various software languages. We
conclude this chapter, for those readers wanting to know the internal
details, with а derivation of the radix 2 FFf and introduce several differ-
ent ways in which this FFf is implemented.
t Actual1y, the FFf has an interesting history. While analyzing X-ray scattering data, а сои-
ple of physicists in the 1940s were taking advantage of the syrnmetries of sines and cosines
using а mathematical method based оп а technique published in the early 1900s,
Remarkably, over 20 years passed before the FFf was (re)discovered, Reference [2] tel1s the
Мl story.
129
130 The Fast Four/er Traпsform
4.1 Relationship о, the FFT to the DFТ
Although тапу different FFТ algorithms have Ьееп developed, in this
section we'll see why the radix 2 FFТ algorithm is so popular and leam
how it's related to the classical DFТ algorithm. Тhe radix 2 FFТ algorithm
is а very efficient process for performing DFTs under the constraint that
the DFI size Ье ап integral power of two, (Тhat is, the number of points
in the transform is N = 2 k , where k is some positive integer,) Let's see just
why the radix-2 FFI is the favorite spectral analysis technique used Ьу
signa1 processing practitioners.
Recall that our DFI Example 1 in Section 3.1 illustrated the number of
redundant arithmetic operations necessary for а simple point DFI. (For
example, we ended ир calculating the product of 1.0607' 0,707 four sepa
rate times.) on the other hand, the radix 2 FFТ eliminates these redundan
cies and great1y reduces the number of necessary arithmetic operations, То
appreciate the FFТ's efficiency, let's consider the number of complex multi
plications necessary for our old friend, the expression for an N point DFI,
N l
Х(т) = Lx(п)e j21tnт/N,
п=О
(4-1)
For an 8 point DFI, Eq. (4 1) tells us that we'd have to perform N'2 or 64
complex multiplications. (Тhat's because, for each of the eight X(m)s, we
have to sum eight complex products as п goes from О to 7.) As we'll ver
Ну in later sections of this chapter, the number of complex multiplications,
for an N point FFI, is approximately
N
.10g2 N.
2
(4-2)
(уУе say approximately because some multiplications tum out to Ье mul-
tiplications Ьу +1 or 1, which amount to mere sign changes,) Well, this
(N /2)10g2N value is а significant reduction from the N'2 complex multipli
cations required Ьу Eq. (4 1), particularly for large N. То show just how
significant, Figure 4 1 compares the number of complex multiplications
required Ьу DFIs and radix 2 FFТs as а function of the number of input
data points N, When N = 512, for example, the DFI requires 200 times
more complex multiplications than those needed Ьу the FFT, When
N = 8192, the DFI must calculate 1000 complex multiplications for each
complex multiplication in the FFТ!
H/пts оп Usiпg FFТs iп Pract/ce 131
109
108
107
106
105
104
103
102
101
101
Number of complex multiplications
у ./
OFТ / f""""
.......
FFТ
,......,
...r--"i-"'"
102
1(/
1()"
1ff' N
Figure 4-1 Number of complex multlplications 'п the OFТ and the radlx-2 FFТ
05 а function of N,
It's appropriate now to make clear that the PFТ is not an approximation
to the DFТ. It's exactly equal to the DFT; it is the DFТ. Moreover, аН of the
performance characteristics of the DFТ described in the previous chapter,
output symmetry, linearity, output magnitиdes, leakage, scaHoping loss,
etc., also describe the behavior of the PFТ.
4.2 Hints оп Using FFTs in Practice
Based оп how usefиl FFТs are, here's а list of practical pointers, or tips, оп
acquiring input data samples and using the radix 2 FFT to analyze real
world signals or data,
4.2.1 Sample Fast Enough and Long Enough
When digitizing continuous signals with an A/D converter, for ехат-
ple, we know, from Chapter 2, that our sampling rate must Ье greater
than twice the bandwidth of the continuous A/D input signal to pre
vent frequency-domain aliasing. Depending оп the application, practi
tioners typicaHy sample at 2,5 to four times the signal bandwidth, If we
know that the bandwidth of the continuous signal is not too large rela
tive to the maximum sample rate of our A/D converter, it's easy to
avoid aliasing, If we don't know the continuous A/D input signal's
132 The Fost Fourier Traпsforт
bandwidth, how do we tell if we're having aliasing problems? Well, we
shou1d mistrust any РРТ results that have significant spectra1 сотро-
nents at frequencies near half the samp1e rate, Ideally, we' d like to work
with signa1s whose spectra1 amplitudes decrease with increasing fre
quency, Ве very suspicious of aliasing if there are any spectra1 сотро-
nents whose frequencies appear to depend оп the samp1e rate, If we
suspect that aliasing is occurring or that the continuous signa1 contains
broadband noise, we'll have to use an ana10g 10w-pass filter prior to
А/О conversion, The cutoff frequency of the 10w-pass filter must, of
course, Ье greater than the frequency band of interest but 1ess than half
the samp1e rate,
Although we know that an N-point radix-2 FFf requires N == 2 k input
samp1es, just how many samp1es must we collect before we perform our
FFf? Тhe answer is that the data collection time interval must Ье 10ng
enough to satisfy Our desired FFf frequency resolution for the given sam-
ple rate f s ' Тhe data collection time interva1 is the reciproca1 of the desired
FFf frequency resolution, and the longer we sample at а fixed fs sample
rate, the finer our frequency reso1ution will Ье; that is, the tota1 data co1-
lection time interva1 is N/fs seconds, and our N-point FFf bin-to-bin fre-
quency resolution is f/N Hz, 50, for examp1e, if we need а spectra1
resolution of 5 Hz, then,fs/N = 5 Hz, and
N = !s = ь.. = О,2!,.
desired reso1ution 5 s
(4-3)
In this case, iffs is, say, 10 kНz, then N must Ье at least 2,000, and we'd
choose N equal to 2048 because this number is а power of 2,
4.2.2 t,,1anipulatlng the Time Data Prior to Transformation
When using the radix-2 РРТ, if we don't have control over the length of
our time domain data sequence, and that sequence length is not an
integral power of two, we have two options, We could discard enough
data samples so that the remaining РРТ input sequence length is some
integra1 power of two, Тhis scheme is not recommended because ignor-
ing data samples degrades our resu1tant frequency-domain resolution,
(The larger N, the better our frequency resolution, right?) А better
approach is to append enough zero valued samples to the end of the
Нте data sequence to match the number of points of the next largest
radix 2 РРТ. For examp1e, if we have 1000 time samples to transform,
rather than analyzing only 512 of them with а 512-point FFf, we should
H/пts оп Us/пg FFТs /п Practice 133
add 24 trailing zero-valued samples to the original sequence and use а
1024 point FFT, (This zero stuffiпg technique is discussed in more detail
in Section 3,11,)
FFТs suffer the same ill effects оЕ spectralleakage that we discussed for
the DFr in Section 3.8. We сап multiply the time data Ьу а window func
tion to al1eviate this leakage problem, Ве prepared, though, for the fre
quency resolution degradation inherent when windows are used. Ву the
way, if appending zeros is necessary to extend а time sequence, we have
to make sure that we append the zeros after mu1tiplying the original time
data sequence Ьу а window function, Applying а window function to the
appended zeros wi11 distort the resu1tant window and worsen our FFr
leakage problems.
Although windowing will reduce leakage problems, it wi11 not elimi
nate them a1together, Even when windowing is employed, high level
spectral components can obscure nearby low level spectral components,
Тhis is especial1y evident when the original time data has а nonzero aver
age, i,e., it's riding оп а ос bias, When the FFr is реrfопnеd in this case,
а large amplitude DC spectral component at О Hz will overshadow its
spectral neighbors, We сап eliminate this problem Ьу calculating the aver
age оЕ the time sequence and subtract that average value Етот each sam
ple in the original sequence, (Тhe averaging and subtraction process must
Ье performed before windowing,) Тhis technique makes the new time
sequence's average (mean) value equal to zero and eliminates any high
level, O НZ component in the FFТ resu1ts,
4.2.3 Enhanclng FFТ Results
If we're using the FFr to detect signal energy in the presence оЕ noise and
enough time domain data is available, we сап improve the sensitivity оЕ
our processing Ьу averaging mu1tiple FFrs. Тhis technique, discussed in
Section 10,7, сап Ье implemented very efficient1y to detect signal energy
that's actual1y below the average noise level; that is, given enough time
domain data, we сап detect signal components that have negative signal-
to-noise ratios,
If our original time domain data is real-valued only, we сап take
advantage оЕ the 2N Point Real FFr technique in Section 10.5 to speed ир
our processing; that is, а 2N-point real sequence сап Ье trапsfопnеd with
а single N-point complex radix 2 FFr, Тhus we сап get the frequency res-
olution оЕ а 2N point FFr for just about the computational price оЕ per
forming а standard N-point FFr, Another FFТ speed enhancement is the
possible use of the frequency domain windowing technique discussed in
134 The Fast Fourier Traпsforт
5ection 10.3. If we need the FFТ of unwindowed time domain data and,
at the same time, we also want the FFТ of that same tirne data with а win
dow function applied, we don't have to perform two separate FFТs. We
сап perform the FFТ of the unwindowed data, and then we can perform
frequency-domain windowing to reduce spectralleakage оп any, or аН, оЕ
the FFТ Ьin outputs.
4.2.4 Interpretlng FFТ Results
Тhe first step in interpreting PFТ results is to compute the absolute fre
quency оЕ the individual FFТ Ьin centers, Like the DFТ, the FFТ Ьin spac
ing is the ratio оЕ the sampling rate (f.) over the number оЕ points in the
FFТ, or f/N. With our PFТ output designated Ьу Х(т), where т = О, 1, 2,
3, ' . ., N l, the absolute frequency оЕ the mth bin center is mfs/N, If the
FFТ's input tirne samples are real, only the Х(т) outputs from т = О to
т = N /2 are independent. 50, in this case, we need determine only the
absolute FFТ Ьin frequencies for т over the range оЕ О S; т S; N /2. If the
PFТ input samples are complex, аН N оЕ the FFТ outputs are independent,
and we should compute the absolute FFТ Ьin frequencies for т over the
full range of О S; т S; N l.
If necessary, we can determine the true amplitude оЕ tirne domain sig
nals from their FFТ spectral results. То do so, we have to keep in mind that
radix 2 PFТ outputs are complex and оЕ the form
Х(т) = Xrea1(m) + jXimag(m) ,
(4 4)
AIso, the FFТ output magnitude samples,
Xmag(m) =1 Х(т) 1= XreaJ(m)2 + X imag (m)2 ,
(4 5)
are all inherent1y multiplied Ьу the factor N /2, as described in Section 3.4,
when the input samples are real. If the FFТ input samples are complex, the
scaling factor is N. 50 to determine the correct amplitudes оЕ the time
domain sinusoidal components, we' d have to divide the FFТ magnitudes Ьу
the appropriate scale factor, N /2 for real inputs and N for complex inputs.
If а window function was used оп the original tirne-domain data, some
оЕ the FFТ input samples will Ье attenuated. Тhis reduces the resultant
FFТ output magnitudes from their true unwindowed values. То calculate
the correct amplitudes оЕ various time domain sinusoidal components,
then, we'd have to further divide the FFТ magnitudes Ьу the appropriate
H/пts оп Us/пg FFТs /п Pract/ce 135
processing 10ss factor associated with the window function used,
Processing 10ss factors for the most popular window fиnctions are listed
in Reference [3].
Should we want to determine the power spectrum Xps(m) of an РП
result, we'd calculate the magnitude squared va1ues using
Xps(m) = 1 Х(т) 12 = X real (m)2 + X imag (m)2,
(4-6)
Doing So wou1d allow us to compute the power spectrum in decibe1s with
XdB(m) = 10 .10g10 ( I Х(т) 12) dB.
(4-7)
Тhe normalized power spectrum in decibe1s сап Ье calculated using
, ( 1 Х(т) 12 )
norma1lzed XdB(m) = 10 . log 10 2 '
(1 Х(т) Imax)
(4-8)
or
, ( I Х(т) I )
normal1Zed XdB(m) = 20 .10glO '
1 Х(т) 'тах
(4-9)
In Eqs, (4-8) and (4-9), the term I Х(т) 1 тах is the 1argest PFТ output mag-
nitude samp1e. In practice, we find that p10tting XdB(m) is very informa-
tive because of the enhanced 10w magnitude reso1ution afforded Ьу the
10garithmic decibe1 sca1e, as described in Appendix Е. If either Eq, (4-8)
or Eq, (4 9) is used, по compensation need Ье performed for the above
mentioned N or N /2 РП sca1e or window processing 10ss factors,
Normalization through division Ьу (1 Х(т) 1 таУ or 1 Х(т) 1 тах elimi
nates the effect of any abso1ute PFТ or window sca1e factors.
Кnowing that the phase ang1es Х,,(т) of the individua1 FFТ oиtputs are
given Ьу
Х",(т) = tan l ( Ximag(m) ) ,
X real (т)
(4 10)
it's important to watch out for Xreal(m) values that are equal to zero, Тhat
would invalidate our phase ang1e calculations in Eq, (4-10) due to division
136 The Fost Four/er Troпsforт
Ьу а zero condition. In practice, we want to make sure that our calculations
(or software compiler) detect осcuпеnсеs of Xreal(m) = О and set the corre--
sponding Хе(т) to 900 if Ximaim) is positive, set Хе(т) to 00 if Ximaim) is
zero, and set Хе(т) to 900 if Ximag(m) is negative. While we're оп the subject
of FFf output phase angles, Ье aware that FFf outputs containing signifi
cant noise components can cause large f1uctuations in the computed Хе(т)
phase angles, This mеans that the Х/т) samples are only meaningful when
the corresponding I Х(т) I is well above the average FFf output noise level,
4.3 FFТ Software Programs
For readers seeking actual FFТ software routines without having to Ьиу
those high priced signal processing software packages, риЬНс domain
radix 2 FFТ routines are readily available, References [4---7] provide stan-
dard FFТ program listings using the FORТRAN language. Reference [8]
presents an efficient FFT program written in FORTRAN for real only
input data sequences. Reference [9] provides а standard FFТ program
written in HP BASICTM, and reference [10] presents an FFТ routine writ
ten in Applesoft BASICTM. Readers interested in the Ada language сап
find FFТ related subroutines in reference [11].
4.4 Derivation о, the Radix 2 FFТ Algorithm
тhis section and those that follow provide а detailed description of the
internal data structures and operations of the radix 2 FFТ for those read-
ers interested in developing software FFТ routines or designing FFf hard-
ware. То see just exact1y how the FFТ evolved from the DFТ, we retum to
the equation for an N point DFТ,
N l
Х(т) = x(n)e j211пт!N,
п=О
( 1l)
А straightforward derivation of the FFТ proceeds with the separation of
the input data sequence х(п) into two parts, When х(п) is segmented into
its even and odd indexed elements, we сап, then, break Eq, (4 11) into
two parts as
(N /2Н (N /2Н
Х(т)= x(2n)e j211(2п)т!N + x(2n+1)e j211(2п+l)т!N, (4 12)
п=О п=О
Der/vat/oп of the Radix-2 FFТ A/gor/thm 137
Pulling the constant phase angle outside the second summation,
(N/2H (N l
Х(т) = Lx(2n)e j27t(2n)т/N + e j27tm/N LX(2n + 1)e j27t(2n)m/N. (4-13)
п=О п=О
Well, here the equations get so long and drawn out that we'll use the
standard notation to simplify things. We'll define W N = e j27t/N to repre
sent the complex phase angle factor that is constant with N. 50 Eq. (4-13)
becomes
(N/2H (N/2H
Х(т) = L-x(2n)w nm + W!J L-x(2n + l)W nm .
п=О п=О
(4-14)
Because W = e j27t2/(N) = e j27t/(N/2), we сап substitute W N /2 for in
Eq. (4-14), as
(N/2) 1 (N/2) 1
Х(т) = L-х(2n)W i2 + W!'l L-x(2n + 1)W i2 .
п=О п=О
( 15)
50 we now have two N /2 summations whose results can Ье combined
to give us the N-point DFТ. We've reduced some of the necessary number
crnnching in Eq, ( 15) relative to Eq. (4-11) because the W terms in the
two summations of Eq. (4-15) are identical. Тhere's а further benefit in
breaking the N-point DFТ into two parts because the upper half of the
DFТ outputs are easy to calculate. Consider the X(m+N/2) output. If we
plug m+N/2 in for т in Eq. (4-15), then
(N/2H
Х(т + N / 2) = '" X(2n)W n (m+N /2)
N/2
п=О
(m+N /2) (N /2) 1
+ W '" Х ( 2п + l ) W n (m+N /2)
N N /2
п=О
(4-16)
It looks like we're complicating things, right? Well, just hang in there for
а moment. We сап now simplify the phase angle terms inside the sum-
mations because
138 The Fost Foиrier Troпsforт
W п (m+N/2) =W пm W пN / 2 =Wпm ( e j2"п2N/2N )
N/2 N/2 N/2 N/2
= W;J/2(1) = W;J/2'
(4-17)
for anу integer n, Looking at the so-called twiddle factor in front оЕ the sec
ond summation in Eq. (4 16), we сап simplify it as
W m+N /2) = WNW/J /2 = WN(e j27tN /2N) = WNH) = WN' (4--18)
ОК, using Eqs, (4--17) and (4--18), we represent Eq. (4 16)'s X(m+N/2) as
(N/2H (N/2H
X(m+N /2)= LX(2n)W;J/2 WN LX(2n+1)W;J/2' (4-19)
п=О п=О
Now, let's repeat Eqs, (4--15) and (4-19) to see the similarity;
(N l (N/2H
Х(т)= Lx(2п)W;J/2 +W N LX(2n+1)W;J/2'
п=О п=О
(4--20)
and
(N l (N l
X(m+N /2)= Lx(2n)W /2 W;'J LX(2n+1)W N / 2 .
п=О п=О
(4-20')
50 here we are, We need not perform аnу sine or cosine multiplications to
get X(m+N/2). We just change the sign оЕ the twiddle factor W N and use
the results of the two summations from Х(т) to get X(m+N /2), ОЕ course,
т goes from О to (N /2) 1 in Eq. (4-20) which means, for аn N-point ОРТ,
we perform an N /2-point ОРТ to get the first N /2 outputs and use those
to get the last N/2 outputs, For N = 8, Eqs. (4 20) and (4--20') are imple
mented as shown in Figure 4 2.
If we simplify Eqs. (4-20) and (4-20') to the form
Х(т) = А(т) + W N В(т) ,
(4-21)
and
DerivQtioп о' the RQdix 2 FFТ A/gorithт 13«;
х(6)
Х(З)
х(О) Х(О)
4-point
OFТ
х(2) Х(1)
з
L nт
х(2n) W 4
х(4) О Х(2)
х(1)
Х(4)
Х(З)
Х(5)
х(5)
Х(6)
х(7)
Х(7)
Flgure 4-2 FFТ Implementotlon of оп 8-polnt DFТ uslng two 4-point DFТs,
X(m+N/2) = А(т) W;:; В(т) ,
(4 21')
we сап go further and think about breaking the two 4 point DFTs into
four 2-point DFТs. Let's see how we сап subdivide the upper 4 point DFT
in Figure 4-2 whose four outputs are А(т) in Eqs, (4 21) and (4 21'), We
segment the inputs to the upper 4-point OFТ into their odd and even
components
140 The Fost Four/er Traпsforт
(N/2H (N H (N H
А(т) = Lx(2п)W i2 = Lx(4п)WJ'i + LX(4п + 1)wt 4 i;1)т ,
п=О 2п=О 2п=О
or
(N l (N -l
А(т)= Lx(4п)WJ'i +W/J/2 Lx(4п+1)WJ'i
2п=О 2п=О
(4 22)
If we let р = 2п, the index of the sumrnations in Eq. (4 22) can Ье simpli
fied to give А(т) the more familiar form of
(N l (N l
А(т)= Lx(2p)W 4 +W/J/2 Lx(2p+1)W 4'
р=О р=О
(4-23)
Notice the similarity between Eq. (4 23) and Eq. (+20). This capability to
subdivide ап N /2 point DFТ into two N / +point DFТs gives the FFТ its
capacity to greatly reduce the number of necessary mu1tiplications to imple
ment DFТs. (We're going to prove this shortly.) Following the same steps that
we used to obtain А(т), we can show that Eq. (+21)'s В(т) is given Ьу
(N l (N l
В(т)= Lx(2p)W 4 W/J/2 Lx(2p+1)W 4'
р=О р=О
(+24)
For our N = 8 example, Eqs. (+23) and (4 24) are implemented as shown
in Figure 4-3. The FFTs well known butterfly pattem of signal flows is cer
tain1yevident, and we see the fиrther shuffling of the input data in Figure
4 3. The twiddle factor Wfj/2 in Eqs. (4 23) and (4 24), for our N = 8 exaт
ple, ranges from W to Wl because the т index, for А(т) and В(т), goes
from О to 3. For апу N point DFТ, we can break each of the N /2-point
DFТs into two N/4 point DFТs to fиrther reduce the number of sine and
cosine multiplications, Eventually, we wou1d arrive at ап array of 2 point
DFТs where по fиrther computational savings could Ье realized. Тhis is
why the number of points in our FFТs are constrained to Ье some power
of 2 and why this FFТ algorithm is referred to as the radix 2 FFТ.
Moving right along, let's go опе step further, and then we'll Ье fin
ished with our N = 8 point FFТ derivation. The 2 point DFТ fиnctions in
х(О)..........
2-polnt
DFT
х(4) .......
Derlvatfoп of the Radlx-2 FFТ Algorlthm 141
А(О)
Х(О)
Х(1)
х(2)....... Х(2)
2-polnt
DFТ
х(б)....... Х(З)
х( 1) ........ )«4)
2-polnt
DFТ
х(5)....... X(S)
Х(З) .......
2-point
DFТ
х(7) ..........
В(З)
Х(6)
Х(7)
Flgure 4-3 FFТ implementotlon of оп 8-polnt DFТ 05 two 4-polnt DFТs ond four
2-polnt DFТs,
Figure 4 3 cannot Ье partitioned into smaller parts we've reached the
end of our DFT reduction process arriving at the butterfly of а single
2 point DFT as shown in Figure 4 4. From the definition of W N ,
W = e j21t01N = 1 and WiJ/2 = e j21tN 12N = j1t = 1, 50 the 2 point DFТ blocks
in Figure 3 сап Ье replaced Ьу the butterfly in Figure 4 to give us а
full 8 point PFТ implementation of the DFТ as shown in Figure 4 5.
142 The Fost Four/er Troпsforт
x(k)
x(k+N/2)
Figure 4-4 Single 2-point DFТ butterf\y.
ОК, we've gone through а fair amount of algebraic foot shuff1ing
here. То verify that the derivation of the РРТ is valid, we сап apply the
8-point data sequence of Chapter 3's DFT Example 1 to the 8 point РРТ
represented Ьу Figure 4-5. The data sequence representing
х(п) = sin(21t1000nt s ) + O,5sin(21t2000nt s +3п/ 4) is
х(О) = 0,3535,
х(2) = 0.6464,
х(4) = 0,3535,
х(6) = 1.3535,
х(l) = 0.3535,
х(3) = 1,0607,
х(5) = l,0607,
х(7) = O,3535.
(4 25)
We begin grinding through this example Ьу applying the input values
from Eq. (4 25) to Figure 4-5, giving the data values shown оп left side of
Figure 4-6, The outputs of the second stage of the FFf are
А(О) = 0.707 + ( 0.707) = 0.707 + (1 + jO)( 0.707) = О + jO,
А(l) = 0,0 + wl (1.999) = 0,0 + (О j1)(1.999) = О j1,999,
А(2) = 0,707 + vq ( 0.707) = 0.707 + Н + jO)( O,707) = 1,414 + jO,
А(3) = 0,0 + wl (1.999) = 0,0 + (О + j1)(1.999) = О + j1.999,
8(0) = O,707 + (0.707) = 0.707 + (1 + jO)(O,707) = О + jO,
8(1) = 1.414 + wl (1.414) = 1.414 + (О j1)(1.414) = 1.414 j1,414,
8(2) = 0.707 + vq (0,707) = O,707 + Н + jO)(0.707) = 1.414 + jO, and
В(3) = 1.414 + wl (1,414) = 1.414 + (О + j1)(1.414) = 1.414 + j1.414 ,
Ca1culating the outputs of the third stage of the FFf to arrive at our final
answer
Der/vatloп of the Radix-2 FП A/gorithm 143
Х(О)
А(О)
Х(О)
х(4)
Х(1)
х(2)
Х(2)
х(6)
Х(З)
х(1)
Х(4)
х(5)
Х(5}
х(З)
Х(6}
х(7)
Х(7)
Flgure 4-5 Full decimotlon-in-tlme FFТ Implementotlon of оп 8-polnt DFТ.
Х(О) = А(О) + В(О) = О + jO + (1 + jO)( О + jO) = О + jO + О + jO = О LO°,
Х(l) = А(l) + В(l) = О j1.999 + (0,707 jO,707)(l,414 j1,414)
= О j1.999 + О j1.999 = О j4 = 4 L 90°
Х(2) = А(2) + В(2) = 1.414 + jO + (О j1)( 1.414 + jO)
= 1,414 + jO + О + j1,4242 = 1.414 + j1,414 = 2 З50,
Х(З) = А(З) + В(З) = О + j1.999 + ( O,707 jO.707)(l,414 + j1,414 )
= О + j1.999 + О j1,999 = О LO°,
144 The Fast Fourier Traпsform
х(2) Z O'6464
"'{),707
1
х(6) 1,3535 1 1.999
Х(О)
Х(1)
Х(2)
Х(З)
Х(4)
Х(5)
Х(6)
х(7)
"'{),3535
I
1
1 51 Stage
1.414
11
I
2nd Stage
В(З)
11
Х(7)
1
зrd Stage
Figure 4-6 8-polnt FFТ of Example 1 f10m Sectlon 3,1,
Х(4) = А(О) + W В(О) = О + jO + Н + jO)(O + jO)
= О + jO + О + jO = О LO°,
Х(5) = А(l) + wg В(l) = О j1.999 + ( 0.707 + jO.707)(l,414 j1.414)
= О j1.999 + О + j1.999 = О LO°,
Х(6) = А(2) + wg В(2) = 1,414 + jO + (О + тН,414 + jO)
= 1,414 + jO + О j1,414 = 1.414 j1.414 = 2 L 45°, and
Х(7) = А(3) + В(3) = О + j1.999 + (0.707 + jO.707)(1.414 + j1.414)
= О + j1.999 + О + j1.999 = О + j4 = 4 L90°,
FFТ lпput/Output Data lпdex Bit Reversal 145
6 3
W 8 =W 4
4 2
W 8 = W 4
7
W 8
о о
W =W
8 4
з
W 8
1
W 8
2 1
W 8 = W 4
Flgure 4-7 Cycllc redundancles In the twlddle factors of an 8-polnt FFТ,
50, happily, the FFТ gives us the correct results, and again we remind
.the reader that the FFТ is not an approximation to а DFТ; it is а DFТ with
а reduced number о! necessary arithmetic operations, You've seen from
the above example that the 8 point FFТ example required less effort than
the 8 point DFТ Example 1 in Section 3.1. Some authors like to explain this
arithmetic reduction Ьу the redundancies inherent in the twiddle factors
W N . Тhey illustrate this with the starburst pattern in Figure 4--7 showing
the equivalencies о! some о! the twiddle factors in an 8 point DFТ.
4.5 FFТ Input/Output Data Index Bit Reversal
ок, let's look into some оЕ the special properties оЕ the FFT that are
important to FFT software developers and FFT hardware designers,
Notice that Figure 4 5 was titled "Ful1 decimation in time FFТ imple
mentation о! an 8 point DFT." The deciтatioп iп tiтe phrase refers to
how we broke the DFТ input samples into odd and even parts in the
derivation о! Eqs. (4--20), (4 23), and (4 24). Тhis time decimation leads to
the scrambled order оЕ the input data's index п in Figure 4 5. Тhe pattern
оЕ this shuffled order сап Ье understood with the help о! ТаЫе 4 1. Тhe
shuffling о! the input data is known as bit reversal because the scrambled
order о! the input data index сап Ье obtained Ьу reversing the bits оЕ the
binary representation о! the normal input data index order, 50unds con
fusing, but it's real1y not Table 4-1 illustrates the input index bit rever-
sal for our 8 point FFТ example. Notice the nопnаl index order in the left
146 The Fast Four/er Traпsforт
ТаЫе 4-1 Input Index Bit Reversal for оп 8 point FFТ
Normal order Biпary bits Reversed bits Bit reversed
о/ iпdex n о/ iпdex n о/ iпdex n order о/ iпdex n
о 000 000 о
1 001 100 4
2 010 010 2
3 011 110 6
4 100 001 1
5 101 101 5
6 110 011 3
7 111 111 7
colurnn of Table 4-1 and the scrambled order in the right colurnn that cor-
responds to the final decimated input index order in Figure 4-5. We've
transposed the original binary bits representing the normal index order
Ьу reversing their positions. Тhe most significant bit becomes the least
significant bit and the least significant bit becomes the most significant
bit, the next to the most significant bit becomes the next to the least sig-
nificant bit, and the next to the least significant bit becomes the next to
the most significant bit, and so on,+
4.6 Radix-2 FFТ Butterfly Structures
Let' s explore the butterf1y signal f10ws of the decimation-in-time PFТ а bit
further. То simplify the signal f1ows, let's replace the twiddle factors in
Figure 4-5 with their equivalent values referenced to W N , where N = 8. We
сап show just the exponents m of W N , to get the FFТ structure shown in
Figure 4-8, Тhat is, Wl from Figure 4-5 is equal to Wl and is shown as а 2
in Figure 4-8, WJ from Figure 4-5 is equal to W: and is shown as а 4 in
Figure 4-8, etc. Тhe ls and ls in the first stage of Figure 4-5 are replaced
in Figure 4-8 Ьу Os and 4s, respectively. Other than the twiddle factor nota-
tiOIl, Figure 4-8 is identical to Figure 4-5. We сап shift around the signal
nodes in Figure 4-5 and апivе at an 8-point decimation-in-time PFТ as
shown in Figure 4-9, Notice that the input data in Figure 4-9 is in its nor-
mal order and the output data indices are bit-reversed. In this case, а bit-
reversal operation needs to Ье performed at the output of the РРТ to
unscramble the frequency-domain results,
t Маnу that are first shall Ье last; and the last first, [Mark 10:31]
Rad/x-2 FFТ Butterffy structures 147
Х(О)
Х(О)
Х(1)
Х(2)
Х(З)
Х(4)
Х(5)
Х(6)
Х(7)
Х(О)
Х(4)
Х(2)
,...... Х(6)
Х(1)
Х(5)
Х(З)
Х(7)
Figure 4-10 shows аn FFТ signal flow structure that avoids the bit-
reversal problem altogether, and the graceful weave of the traditional FFf
butterflies is replaced with а tangled, but effective, configuration,
Not too long ago, hardware implementations of the FFТ spent most of
their time (clock cycles) performing multiplications, and the bit reversal
process necessary to access data in memory wasn't а significant portion of
х(4)
х(2)
х(6)
х(1)
х(5)
х(З)
х(7)
Flgure 4-8 8-point decimation-In-time FFТ with biHeversed inputs,
Х(О)
х(1)
х(2)
х(З)
х(4)
х(5)
х(6)
х(7)
Figure 4.9 8-point decimatlon-in-tlme FFТ with biHeversed outputs,
148 The Fost Four/er Troпsforт
Х(О) Х(О)
х(1) Х(1)
х(2) Х(2)
Х(З) Х(З)
х(4) Х(4)
х(5) Х(5)
х(6) Х(6)
х(7) Х(7)
Flgure 4-10 8-point decimatlon-in-tlme FFТ wlth inputs and outputs in normal
order.
the overall РРТ computational problem, Now that high speed multi
plier/accumulator integrated circuits can multiply two numbers in а Бin
gle clock cycle, PFТ data multiplexing and memory addressing have
Ьесоте тиш more important, This has led to the development of еШ
cient algorithms to perform bit reversal[12 lS].
There's another derivation for the PFТ that leads to butterfly struc
tures looking like those we've already covered, but the twiddle factors in
the butterflies are different, This altemate FFТ technique is known аБ the
decimation-in frequency algorithm. Where the decimation-in-time PFТ
algorithm is based оп subdividing the input data into its odd and even
components, the decimation-in frequency РРТ algorithm is founded
upon calculating the odd and even output frequency samples separately.
The derivation of the decimation in-frequency algorithm is straightfor
ward and included in тапу tutoriaI papers and textbooks, БО we won't
go through the derivation here[4,S,lS,16]. We will, however, illustrate
decimation in frequency butterfly structures (analogous to the structures
in Figures 4-8 through 4-10) in Figures 4--11 though 4-13.
50 an equivalent decimation in-frequency PFТ structure exists for each
decimation-in time PFТ structure. It's important to note that the number
of necessary multiplications to implement the decimation in-frequency
PFТ algorithms is the Бате аБ the number necessary for the decimation
in time PFТ algorithms, There are БО тапу different PFТ butterfly struc
tures described in the literature, that it' s еаву to Ьесоте confused about
Rad/x-2 FFТ 8utterfly Structures 149
х(О) Х(О)
х(4) Х(1)
х(2) Х(2)
х(6) Х(3)
О
х(1) 4 X(4)
О
х(5) 4 X(5)
О
х(3) 4 X(6)
О
х(7) 4 X(7)
Flgure 4.11 8-polnt declmatlon-in-frequency FFТ with bIt-rеvеrsеd inputs
х(О) Х(О)
х(1) Х(4)
х(2) Х(2)
х(з) Х(6)
х(4) Х(1)
х(5) Х(5)
х(6) Х(3)
х(7) Х(7)
Flgure 4.12 8-polnt declmatlon-In-frequency FFТ with bit-reversed outputs,
wblch structures are decimation-in-time and which are decimation in
frequency, Depending оп how the material is presented, it's easy for а
beginner to fall into the trap of believing that decimation in time FFТs
always have their inputs bit reversed and decimation-in frequency FFТs
always have their outputs bit reversed. This is not true, as the аЬоуе fig
ures show, Decimation-in time or frequency is determined Ьу whether
150 The Fast Fourier Traпsforт
х(О) Х(О)
х(1 ) Х(1)
х(2) Х(2)
х(З) Х(З)
О
х(4) 4 X(4)
О
х(5) 4 X(5)
О
х(6) 4 X(6)
О
х(7) 4 X(7)
Flgure 4-13 8-point decimation-In-frequency FFТ with inputs and outputs in
normai order,
the DFT inputs or outputs are partitioned when deriving а particular PFТ
butterfly structure from the DFТ equations,
Let's take one more look at а single butterfly, Тhe FFТ butterfly struc
tures in Figures 4 8, 4 9, 4 11, and 4 12 are the direct result of the deriva
tions of the decimation in time and decimation-in frequency algorithms,
Although it's not very obvious at first, the twiddle factor exponents
shown in these structures do have а consistent pattern, Notice how they
always take the general forms shown in Figure 4 14(a),t То implement
the decimation in time butterfly of Figure 4 14(a), we'd have to perform
two complex mu1tiplications and two complex additions, Well, there's а
better way. Consider the decimation-in time butterfly in Figure 4 14(а), If
the top input is х and the bottom input is у, the top butterfly output
would Ье
х' = х + W у ,
(4-26)
and the bottom butterfly output would Ье
у' = х + W +N/2 у.
(4-27)
t Remember, for simplicity the butterfly structures in Figures 4-8 through 4 13 show only the
twiddle factor exponents, k and k+N /2, and nо! the еnШе complex twiddle factors,
Radix-2 FFТ 8utterfly Struetures 151
Decimation in time Decimation in frequency
х х' х х"
(а) W k
у у' У W k +N/2 "
N У
х х' х х"
(Ь)
У у' У
х х' х х"
(с)
k k
y........W N у' У W y"
Flgure 4-14 Deelmation-in-time and deeimation-in-frequeney butterfly structures:
(а) original form; (Ь) simplifted form; (е) optimized form,
Fortunately, the operations in Eqs, (4 26) and (4-27) сап Ье simplified
because the two twiddle factors are related Ьу
wt+ N /2 = wtwй /2 = wt(е j2пN /2N) = wtH) = wt,
(4-28)
So we сап replace the W +N/2 twiddle factors in Figure 4-14(а) with
wt to give us the simplified butterflies shown in Figure 4-14(Ь).
Rea1izing that the twiddle factors in Figure 4-14(Ь) differ only Ьу their
signs, the optimized butterflies in Figure 4-14(с) сап Ье used, Notice
that these optiтized butterflies require two complex additions but only
152 Тhe Fast Four/er Traпsforт
one complex multiplication, thus reducing our computational work-
load,+
We'll often see the optimized butterfly structures of Figure 4 14(c) in
the literature instead of those in Figure 4--14(а), Тhese optimized butter-
flies give us an easy way to recognize decimation in time and decimation-
in-frequency algorithms, When we do соте across the optimized
butterflies from Figure 4-14(с), we'll know that the algorithm is decima
tion in-time if the twiddle factor precedes the 1, or else the algorithm is
decimation-in-frequency if the twiddle factor follows the 1,
50metimes we'll encounter FFТ structures in the literature that use the
notation shown in Figure 4 15 [5, 17], Тhese wingless butterflies are equiv-
alent to those shown in Figure 4 14(c), The signal flow convention in
Figure 4-15 is such that the plus output of а circle is the sum of the two
samples that enter the circle from the left, and the minus output of а cir-
cle is the difference of the samples that enter the circle, 50 the outputs of
the decimation-in time butterflies in Figure 4-14(с) and Figure 4-15(а) are
given Ьу
х' = х + wt у' and у' = х у .
(4-29)
The outputs of the decimation-in-frequency butterflies in Figure 4 14(c)
and Figure 4 15(Ь) are
" d " W k ( ) W k W k
Х =х+у,an у = N x y = NX NY'
(4--30)
х
х'
х
у"
у
у'
у
(а)
(Ь)
Flgure 4-15 Alternate FFТ butterfly nototlon: (о) declтatlon-In-tlтe;
(Ь) declтotlon-In-frequency,
+ It's because there are (N/2)log2N butterf1ies in an N-point FFТ that we said the number of
complex multiplications perform.ed Ьу an FFТ is (N /2)log2N in Eq, (4 2),
Radix-2 FFТ Butterfly Structures 153
So which FFf structure is the best one to use? It depends оп the appli
cation, the hardware implementation, and convenience. If we're using а
software routine to perform FFТs оп а general purpose computer, we usu
аllу don't have а lot оЕ choices, Most folks just use whatever existing FFf
routines happen to Ье included in their commercial software package.
Тheir code mау Ье optimized for speed, but you never know, Examination
оЕ the software code mау Ье necessary to see just how the FFf is imple
mented. If we feel the пeed for speed, we should check to see if the software
calculates the sines and cosines each time it needs а twiddle factor,
Trigonometric calculations normally take many machine cycles, It mау Ье
possible to speed up the algorithm Ьу calculating the twiddle factors
ahead оЕ time and storing them in а table. Тhat way, they сап Ье looked ир,
instead ofbeing calculated each time they're needed in а butterf1y. Ifwe're
writing our own software routine, checking for butterf1y output data
overf1ow and careful magnitude scaling mау allow our FFf to Ье per-
formed using integer arithmetic that сап Ье faster оп some machines.+
Care must Ье taken, however, when using integer arithmetic; some
Reduced Instruction Set Computer (RISC) processors actually take longer
to perform integer calculations because they're specifically designed to
operate оп f1oating point numbers.
If we're using commercial array processor hardware for our calcula
tions, the code in these processors is always optimized because their pur
pose in life is high speed. Array processor manufacturers typicaHy
publicize their products Ьу specifying the speed at which their machines
perform а 1024 point FFf, Let's look at some оЕ our options in selecting а
particular FFf structure in case we're designing special purpose hard
ware to implement an FFf.
The FFT butterf1y structures previously discussed typically ЕаН into
one оЕ two categories: in place FFT algorithms and double memory
FFf algorithms. An in place algorithm is depicted in Figure 4 5, The
output оЕ а butterf1y operation сап Ье stored in the same hardware
memory locations that previously he1d the butterf1y's input data, No
intermediate storage is necessary, This way, for an N-point FFT, only 2N
memory locations are needed, (Тhe 2 comes from the fact that each but-
terf1y node represents а data value that has both а real and an imagi-
nary part.) Тhe rub with the in place algorithms is that data routing and
memory addressing is rather complicated, А double memory FFT
t Overflow is what happens when the result о! an arithmetic operation has too тапу bits, or
digits, to Ье represented in the hardware registers designed to contain that result, FFТ data
overflow is described in Section 9,3,
154 Тhe Fast Four/er Traпsforт
structure is that depicted in Figure 4 10. With this structure, intermedi
ate s torage is necessary because we по longer have the standard but
terflies, and 4N memory locations are needed. However, data routing
and memory address control is much simpler in double memory FFT
structures than the in place technique. The use of high speed, floating-
point integrated circuits to implement pipelined FFT architectures
takes better advantage of their pipelined structure when the double
memory algorithm is used[18],
There' s another class of FFТ structures, known as constant geometry
algorithms, that make the addressing of memory both simple and con
stant for each stage of the FFr. Тhese structures are of interest to those
fo1ks who build special purpose FFf hardware devices[4,19]. From the
standpoint of general hardware the decimation in time algorithms are
optimum for real input data sequences, and decimation in frequency is
appropriate when the input is complex[8], When the FFf input data is
symmetrical in time, special FFI structures exist to eliminate unnecessary
calcиlations. Тhese special butterfly structures based оп input data sym
metry are described in the literature[20],
For two dimensional FFT applications, such as processing photo
graphic images, the decimation in frequency algorithms appear to Ье the
optimum choice[21]. Your application тау Ье such that FFf input and
output bit reversal is not an important factor. Some FFT applications
aHow manipulating а bit reversed FFf output sequence in the frequency
domain without having to unscramble the FFf's output data, Тhen an
inverse transform that' s expecting bit-reversed inputs will give а time
domain output whose data sequence is correct, Тhis situation avoids the
need to perform any bit reversals at аН, Multiplying two FFf outputs to
implement convolution or correlation are examples of this possibility,t As
we сan see, finding the optimum FFf algorithm and hardware architec
ture for an FFf is а fairly complex problem to solve, but the literature pro
vides guidance[4,22,23],
References
[1] Cooley, J, and Tukey, J, "An Algorithm for the Machine Calculation оЕ Complex
Fourier Series," Мath, Coтput., Vol, 19, No, 90, Apr, 1965, рр, 297 301,
[2] Cooley, J" Lewis, р" and We1ch, р, "Historical Notes оп the Fast Fourier
Transform," IEEE Traпs, оп Audio aпd E/ectroacoustics, Vol, AU-15, No, 2, June
1967,
t See Section 10,10 for an example of using the PFТ to perform convolution,
Refereпces 155
[3] Harris, F. J, "On the Use оЕ Windows for Harmonic Analysis with the Discrete
Fourier Transform," Proceediпgs о/ the IEEE, Vol, 66, No, 1, рр. 54, Janиary 1978,
[4] Oppenheim, А, V., and Schafer, R. W, Discrete-Tiтe Sigпa/ Processiпg,
Prentice-Hall, Englewood Cliffs, New Jersey, 1989, рр, 608.
[5] Rabiner, L, R, and Gold, В, тпеоту aпd AppIicatioп о/ Digita/ Signa/ Processiпg,
Prentice-Hall, Englewood Cliffs, New Jersey, 1975, рр, 367,
[6] Steams, S. Digita/ Sigпa/ Aпa/ysis, Hayden Book Со" Rochelle Park, New
Jersey, 1975, рр, 265,
[7] Prograтs /оу Digita/ Sigпa/ Processiпg, Chapter 1, IEEE Press, New York, 1979,
[8] Sorenson, Н. V" Jones, D, L" Heideman, М. т., and Burrиs, С. 5, "Real-Valиed
Fast Fourier Transform Algorithms," IEEE Traпs, оп Acoust. Speech, aпd Signa/
РУос" Vol. ASSP 35, No, 6, June 1987.
[9] Bracewell, R. тпе Fourier Traпsforт aпd [t's AppIicatioпs, 2nd Edition, Revised,
McGraw-Нill, New York, 1986, рр, 405.
[10] СоЬЬ, F. "Use Fast Fourier Transform Programs to Simplify, Enhance Filter
Analysis," EDN, 8 March 1984,
[11] Carlin, F. "Ada and Generic FFТ Generate Routines Tailored to Yoиr Needs,"
EDN, 23 April1992,
[12] Evans, D, "An Improved Digit-Reversal Permиtation Algorithm for the Fast
Foиrier and Hartley Transforms," IEEE Traпs, оп Acoust, Speech, aпd Sigпa/
РУос" Vol. ASSP-35, No, 8, Aиgust 1987.
[13] Bиrrus, С. 5, "Unscrambling for Fast DFТ Algorithms," IEEE Traпs, оп Acoust,
Speech, aпd Signa/ Рroс" Vol, 36, No, 7, July 1988,
[14] Rodriguez, J. J, "An Improved FFТ Digit-Reversal Algorithm," IEEE Traпs, оп
Acoust, Speech, aпd Signal Рroс" Vol, ASSP 37, No, 8, August 1989,
[15] Land, А. "Bit Reverser Scrambles Data for FFТ," EDN, March 2, 1995,
[16] JG-AE Sиbcommittee оп Measиrement Concepts, "What Is the Fast Foиrier
Transform?," IEEE Traпs, оп Audio aпd E/ectroacoustics, Vol. AU-15, No, 2,
Jиne 1967,
[17] Cohen, R" and Perlman, R, "500 kНz Single Board FFТ System Incorporates
DSP-Optimized Chips," EDN, 31 October 1984,
[18] Eldon, J" and Winter, G, Е. "Floating point Chips Carve Oиt FFТ Systems,"
E/ectroпic Design, 4 Aиgust 1983.
[19] Lamb, К. "CMOS Bиilding Blocks Shrink and Speed Up FFT Systems,"
E/ectroпic Design, 6 Aиgust 1987,
[20] Markel, J, О, "FFТ Pruning," IEEE Tyaпs, оп Audio aпd E/ectroacoustics, Vol,
AU-19, No, 4, December 1971,
156 Тhe Fast Foиrier Traпsforт
[21] Wu, Н, R., and Раоl0ni, F. J, "The Structure оЕ Vector Radix Fast Fourier
Transforms," IЕЕЕ Trans, оп Acoиst, Speech, and Sigпa/ Proc" Vol, ASSP-37, No,
8, August 1989,
[22] АН, Z, М, "Нigh Speed FFТ Processor," IЕЕЕ Trans, оп Coттиnications, Vol.
СОМ-26, No, 5, Мау 1978,
[23] Беrglаnd, G, "Fast Fourier Transform Hardware Implementations An
Overview," IЕЕЕ Trans, оп Aиdio and E/ectroacoиstics, Vol, AU-17, June 1969,
I CHAPTER FIVE I
Finite Impulse
Response Filters
Тhe filtering of digitized data, if not the most fundamental, is certainly the
oldest discipline in the field of digital signal processing. Digital filtering's
origins go back forty years. Тhe growing availability of digital computers
in the early 1950s led to efforts in the smoothing of discrete sampled data
and the analysis of discrete data control systems. However it wasn't until
the early to mid 1960s, around the time the Beatles сате to Arnerica, that
the analysis and development of digital equivalents of analog filters
began in earnest. Тhat's when digital signal processing experts realized
that computers could go beyond the mere analysis of digitized signals
into the domain of actually changing signal characteristics through filter
ing. Today, digital filtering is so widespread that the quantity of literature
pertaining to it exceeds that of any other topic in digital signal processing.
In this chapter, we introduce the fundamental attributes of digital filters,
learn how to quantify their performance, and review the principles asso
ciated with the design of finite impulse response digital filters,
50 let's get started Ьу illustrating the concept of filtering а time--domain
signal as shown in Figure 5 1,
In general, filtering is the processing of а time--domain signal resulting
in some change in that signal's original spectral content, Тhe change is
usually the reduction, or filtering out, of some unwanted input spectral
components; that is, filters allow certain frequencies to pass while attenu
ating other frequencies. Figure 5 1 shows both analog and digital versions
of а filtering process. Where an analog filter operates оп а continuous sig
nal, а digital filter processes а sequence of discrete sample values. Тhe
digital filter in Figure 5 1(b), of course, сап Ье а software program in а
computer, а programmable hardware processor, or а dedicated integrated
circuit. Traditionallinear digital filters typically соте in two flavors: finite
157
158 Fiп/te /mpи/se Respoпse Fllters
/\/\.
V V Time
1\ 1\
V V Time
Input signal
Ou!put signal
(а)
, .
.. : :! " .,.
. .
. t i .
r т .
, . : .
... 1.
. .
.. ' '
I i I
. .. .
...
Тiтa
: 11
. .
t t .
,. i ' .
.
.' .. Тiтe
. .
Inpu! sequence
Outpu! sequence
(Ь)
Flgure 5.1 Fllters: (о) оп onolog filter wlth о nolsy tone input ond о reduced-nolse
tone output; (Ь) the dlgitol equlvolent of the onolog filter,
impulse response (FIR) filters and infinite impulse response (IIR) fi1ters.
Бесаusе FIR fi1ters are the simplest type of digital filter to analyze, we'll
ехаminе them in this chapter and cover IIR fi1ters in Chapter 6,
5.1 An Introduction to Finite Impulse Response
FIR Filters
First of аН, FIR digital filters use only current and past input samples, and
попе of the fi1ter's previous output samples, to obtain а current output
sample value, (Тhat's also why FIR fi1ters are sometimes caHed nonrecur
sive filters.) Given а finite duration of nonzero input values, the effect is
that an FIR filter will always have а finite duration of nonzero output val-
ues and that's how FIR fi1ters got their name. 50, if the FIR filter's input
suddenly becomes а sequence of all zeros, the filter's output will eventu
аНу Ье аН zeros. While not sounding аН that unusual, this characteristic is
however, very important, and we'll soon find out why, as we learn more
about digital filters.
FIR fi1ters use addition to ca1culate their outputs in а manner much
the same as the process of averaging uses addition, In fact, averaging is
а kind of FIR fi1ter that we сап illustrate with an example, Let's say
we're counting the number of cars that pass over а bridge every minute,
and we need to know the average number of cars per minute over five-
minute intervals; that is, every minute we'H calculate the average num-
Ап /пtroductioп to Fiпite /mpu/se Respoпse F/R Filters 159
ТаЫе 5-1 Values for the Averaging Example
Numьer о/ Numьer о/ cars/minute
Minute cars/minute over averaged over the
index the /ast minute /ast five minutes
1 10
2 22
3 24
4 42
5 37 27
6 77 40.4
7 89 53,8
8 22 53,4
9 63 57,6
10 9 52
ber of cars/minute over the last Нуе minutes, If the results of оur car
counting for the first ten minutes are those values shown in the center
column of Table 5 1, then the average number of cars/minute over the
previous Нуе one-minute intervals is listed in the right column of the
table, We've added the number of cars for the first Нуе one-minute inter-
vals and divided Ьу Нуе to get our first five minute average output
value, (10+22+24+42+37)/5 = 27, Next we've averaged the number of
cars/minute for the second to the sixth one-minute intervals to get our
second five minute average output of 40.4, Continuing, we average the
number of cars/minute for the third to the seventh one-minute intervals
to get our third average output of 53,8, and so оп, With the number of
cars/minute for the one-minute intervals represented Ьу the dashed Нnе
in Figure 5-2, we show our five-minute average output as the solid line,
(Figure 5-2 shows cars/minute input values beyond the first ten min-
utes listed in Table 5-1 to Шustrаtе а couple of important ideas to Ье dis
cussed shortly,)
There's much to learn from this simple averaging example. 1n Figure
5 2, notice that the sudden changes in our input sequence of cars/minute
are flattened out Ьу our averager, The averager output sequence is соn-
siderably smoother than the input sequence. Кnowing that sudden tran
sitions in а time sequence represent high frequency components, we сап
say that our averager is behaving like а low-pass filter and smoothing
sudden changes in the input, Is our averager аn FIR filter? It sure is no
previous averager output value is used to determine а current output
160 Fiпite /mpu/se Respoпse F//ters
Number 01 cars
:: : : : : : : : : : : : l\/ ? i : 1 :- : :e : : : : : : : ; - I
70 - - .. . - .. - . t - . -1- - - . - - -. carslm.nule over lasl five - -9. - - - - - . - . - . - . - .
60 - - - - - - . - - -, . - -.i :'-f -j/'- п !е _ - - - - . - -. - . - . f\- - - . - - - - - . - - - -
: ::iIч-r ;\:д<j;,!с: ::::::::::
: :-A:::?::i:'j: ::\j:::_ :::
" ----------1--------------- ---\
о . .............
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1920 21 22 2324 25 Minules
Figure 5-2 Averaging the number of cars/minute, The dashed line shows the
indivlduai cars/minute, and the soiid Ilne is the number of cars/minute
averaged over the iast five minutes,
value; only input values are used to calculate output values, In addition,
we see that, if the bridge were suddenly closed at the end оЕ the 19th
minute, the dashed line immediately goes to zero cars/minute at the end
оЕ the 20th minute, and the averager's output in Figure 5 2 approaches
and settles to а value оЕ zero Ьу the end of the 24th minute,
Figure 5 2 shows the first averager output sample occurring at the end
оЕ the 5th minute because that's when we first have five input samples to
calculate а valid average. Тhe 5th output of our averager can Ье denoted
as у ave(5) where
1
Уауе(5) = [x(l) + х(2) + х(З) + х(4) + х(5)]
5
(5 1)
In the general case, if the kth input sample is x(k), then the nth output is
1 1 п
Уаvе(n)= [х(n 4)+х(n з)+х(n 2)+х(п 1)+х(n)]= Lx(k) . (5 2)
5 5 k=п 4
Look at Eq. (5 2) carefully now, It states that the nth output is the average
of the nth input sample and the four previous input samples,
We can formalize the digital filter nature оЕ our averager Ьу creating
the block diagram in Figure 5 З showing how the averager calculates its
output samples.
Ап /пtroductioп to F/п/te /mpu/se Respoпse F/R F/lters 161
fifth output = 27
1/5
Flgur. 5-3 Averaglng fllter bIock dlagram when the flfth Input 5ample value,
37,15 applied,
This block diagram, referred to as the filter structure, is а physical
depiction оЕ how we might calculate our averaging filter outputs with
the input sequence оЕ values shifted, in order, from left to right along the
top оЕ the filter as new output calculations are performed, Тhis structure,
implementing Eqs. (5 1) and (5 2), shows those values used when the
first five input sample values are available. Тhe delay elements in Figure
5 3, called ипа delays, merely indicate а shift register arrangement where
input sample values are temporarily stored during an output calculation.
In averaging, we add five numbers and divide the sum Ьу five to get
our answer. In а conventional FIR filter implementation, we сап just as
well multiply each оЕ the five input samples Ьу the coefficient 1/5 and
then perform the summation as shown in Figure 5 4(a). ОЕ course, the two
methods in Figures 5-3 and 5-4(а) are equivalent ЬесаuБe Eq, (5 2) describing
the structure shown in Figure 5 3 is equivalent to
1 1 1 1 1
Yave(n) = sх(п 4) + sх(п 3) + sх(п 2) +sх(п 1) + sх(п)
n 1
= L Sx(k)
k=n 4
(5 3)
that describes the strиcture in Figure 5 4(a).t
Let's make sure we understand what's happening in Figure 5 4(a).
Each оЕ the first five input values are multiplied Ьу 1/5, and the five
products are summed to give the 5th filter output value, Тhe left to right
t We've used the venerable distributive law for multiplication and addition of scalars,
a(b+c+d) = ab+ac+ad, in moving Eq, (5-2)'s factor of 1/5 inside the summation in Eq. (5 3),
162 Fiп/te /mpu/se Respoпse Filters
37
(а)
1/5
5th output = 27
77
6th output = 40.4
(Ь)
1/5
89
7th output = 53,8
(с)
1/5
Figure 5-4 Alternote overaging fllter strueture: (а) Input volues used for the 5th
output volue; (Ь) input volues used for the 6th output volue; (е) input
volues used for the 7th output volue,
sample shifting is i1lustrated in Figures 5 4(b) and 5 4(c). То calculate
the filter's 6th output value, the input sequence is right shifted discard
ing the 1st input value of 10, and the 6th input value 77 is accepted оп
the left, Likewise, to calculate the filter's 7th output value, the input
sequence is right shifted discarding the 2nd value of 22, and the 7th
input value 89 arrives оп the left. 50, when а new input sample value is
applied, the filter discards the oldest sample value, multiplies the sam-
ples Ьу the coefficients of 1/5, and sums the products to get а single new
output value, The filter's structure using this bucket brigade shifting
process is often са1lед а transversal filter дие to the cross directional
flow of the input samples. Because we tap off five separate input sample
Coпvo/utioп iп F/R Fi!ters 163
values to ca1culate an output value, the structure in Figure 5 4 is called
а 5 tap FIR fi1ter in digital fi1ter vemacular,
One important and, perhaps, most interesting aspect of understanding
FIR fi1ters is leaming how to predict their behavior when sinusoidal sam
ples of various frequencies are applied to the input, i,e., how to estimate
their frequency domain response, Two factors affect an FIR fi1ter's fre-
quency response: the number of taps and the specific values used for the
multiplication coefficients, We'll explore these two factors using our aver-
aging example and, then, see how we сап ше them to design FIR filters,
Тhis brings us to the point where we have to introduce the С word: con
volution, (Actually, we already slipped а convolution equation in оп the
reader without saying so, It was Eq. (5 3), and we'll ехаminе it in more
detaillater, )
5.2 Convolution in FIR Filters
ок, here's where we get serious about understanding the mathematics
behind FIR fi1ters. We can graphically depict Eq. (5 3)'s and Figure 5 4's
ca1culations as shown in Figure 5 5. Also, let's Ье formal and use the stan
dard notation of digital fi1ters for indexing the input samples and the ш
ter coefficients Ьу starting with an initial index value of zero; that is, we'll
саН the initial input value the Oth sample х(О), Тhe next input sample is
represented Ьу the tепn х(l), the following input sample is called х(2),
and so оп, Likewise, our five coefficient values will Ье indexed from zero
to fощ h(O) through h(4), (Тhis indexing scheme makes the equations
describing our example consistent with conventional filter notation found
in the literature,)
In Eq. (5 3) we used the factor of 1/5 as the fi1ter coefficients mиlti
plied Ьу our averaging fi1ter's input samples, The left side of Figure 5-5
shows the alignment of those coefficients, black squares, with the fi1ter
input sample values represented Ьу the white squares, Notice in Figure
5-5(а) through 5 5(e) that we're marching the input samples to the right,
and, at each step, we ca1culate the fi1ter output sample value using Eq.
(5 3), The output samples оп the right side of Figure 5-5 match the first
five values represented Ьу the black squares in Figure 5-2, The input
samples in Figure 5-5 are those values represented Ьу the white squares
in Figure 5 2, Notice the time order of the inputs in Figure 5 5 has been
reversed from the input sequence order in Figure 5 2! That is, the input
sequence has been f1ipped in the time domain in Figure 5-5, This time
order reversal is what happens to the input data using the filter struc-
ture in Figure 5-4,
164 F/п/te /mpu/se Respoпse FlIters
1;1 I;J Flrst Input ssmpl., >(0) Зuт 01 the first
1;1 I tive products
9 1;1 /
(а) 1;1 1;1 о
I !1J Compute )1(4) 11
I c::t> I .
Statlonary , 11 11 11 '!' 4 п
coofficlonts О 2 4 Output, )l(п)
1;1 I;J Зиm 01 the
88COnd
1;1 1;1 о 1;1 [] '!' flvo products
(Ь) О Compute )1(5) 11
=с> I .
11 11 11 11 11 4 5 п
i Output, )l(п)
О 2 4
[] [] ,
[] i
[] 1;1 11 I
1;1 а [] 1;1 Compute )1(6)
(е) 11 i
=с> .
11 11 11 11 11 4 5 6 п
О 2 4 Output, )l(п)
[] !1J 11 11
1;1
[] [] 11
[] а о 1;1 Compute )1(7)
(d) =с> 11
11 11 11 11 11 4 5 6 7 .
п
О 2 4 Output, )l(п)
О Flfth Input ssmpl., >(4) 11
1;1 []/I;I '!' 11
1;1 I
[] а 11 i
I;J 1;1 Compute )1(8)
(е)
=с> 11 .
11 11 11 11 11 4 5 6 7 8 п
О 2 4 Output, )l(п)
Flgure 5-5 Averaging fllter eonvolutlon: (а) flrst flve input samples aligned with the
statlonary fllter eoeffieients. index п = 4; (ь) input samples shift to the
right and index п = 5; (е) Index n = 6; (d) Index п = 7; (е) Index n = 8,
Repeating the first part of Eq. (5-3) and omitting the subscript оп the
output term, our original FIR filter's y(п)th output is given Ьу
1 1 111
у(п) = sx(n 4) +sx(n 3) +sx(n 2) + sx(n 1) + sx(n) (5-4)
Coпvo/ut/oп /п F/R F//ters 165
Because we'll explore filters whose coefficients are not all the same value,
we need to represent the individual filter coefficients Ьу а variable, such
as the terrn h(k), for example, Тhus we сап rewrite the averaging filter's
output from Eq. (5 4) in а more general way as
у(п) = h( 4)х(п 4) + h(3)x(n 3) + h(2)x(n 2) + h(1)x(n 1) + h(O)x(n)
4
= )(k)x(n k) ,
k=O
(5-5)
where h(O) through h(4) all equal1/5. Equation (5-5) is а concise way of
describing the filter structure in Figure 5-4 and the process illustrated in
Figиre 5 5.
Let's take Eq. (5 5) one step further and say, for а general M tap FIR fil
ter, the nth output is
M l
у(п)= Lh(k)x(n k)
k=O
(5 6)
Well, there it is. Eq. (5 6) is the infamous convolution equation as it
applies to digital FIR filters. Beginners in the field of digital signal pro-
cessing often have trouble understanding the concept of convolution. It
need not Ье that way. Eq. (5 6) is merely а series of multiplications fol
lowed Ьу the addition of the products. The process is actually rather sim
ple. We just flip the time order of an input sample sequence and start step
ping the flipped sequence across the filter's coefficients as shown in
Figиre 5 5. For each new filter input sample, we sum а series of products
to compute а single filter output value.
Let's pause for а moment and introduce а new terrn that's important to
keep in mind, the iтpulse response. Тhe impulse response of а filter is exact
ly what its name implie it's the filter's output time domain sequence
when the input is а single unity valued sample (impulse) preceded and
followed Ьу zero valued samples. Figиre 5 6 il1ustrates this idea in the
same way we deterrnined the filter's output sequence in Figиre 5 5. Тhe
left side of Figure 5 6 shows the alignrnent of the filter coefficients, black
squares, with the filter input impulse sample values represented Ьу the
white squares. Again, in Figure 5-6(а) through 5 6(e) we're shifting the
input samples to the right, and, at each step, we calculate the filter output
sample value using Eq. (5 4), Тhe output samples оп the right side of
166 Fiпite /mpu/se Respoпse F//ters
lтpul,e 'nput $uт о( thв flrst
1[] Compute у( 4) . flve products
(а) =с>
Statlonary .....> · . . . . 4 n
coefflcients О 2 4 Impulse responsв, у(п)
1[] $um о( the '8Cond
Compute у(5) . . ffve producta
(Ь) =с> .
. . . . 4 5 n
О 2 4 Impulse response, у(п)
(с)
19
. . .
Compute у(6)
=с>
. I!I
4 5 6 .n
Impulse response, у(п)
о
2
4
(d)
1 []
. . . . .
Compute у(7)
=с>
. . . .
i
4 5 6 7 .n
Impulse responsв, у(п)
о
2
4
(е)
Compute у(8)
.....J"-o....
. ..
1 О
.....
о
2
4
.
45678 n
Impulse response, у(п)
Figure 5-6 Convolution of fllter eoeffielents and оп Input Impulse to obtain the
fllter's output Impulse response: (а) impulse sample aligned with the flrst
fllter eoeffieient, index п = 4; (Ь) impulse sample shlfts to the rlght and
;пdex п = 5; (е) ;пdex п = 6; (d) ;пdex п = 7; (е) ;ndex п = 8,
Figure 5-6 are the filter's impulse response, Notice the key point here: the
FIR filter's impulse response is identical to the five fi1ter coefficient values.
For this reason, the tепns FIR filter coefficieпts and impulse response are syn-
onyrnous, Тhus, when someone refers to the impu1se response of an FIR
filter, they're also talking about the coefficients,
Returning to our averaging fi1ter, recall that coefficients (or impulse
response) h(O) through h(4) were all equal to 1/5, As it tums out, our fil-
ter's performance сап Ье improved Ьу using coefficients whose values are
not аН the same, Ву реrfопnаnсе we mean how well the fi1ter passes
desired signals and attenuates unwanted signals, We judge that perfor-
mance Ьу determining the shape of the filter's frequency-domain response
that we obtain Ьу the convolution property of linear systems, То describe
this concept, let's repeat Eq. (5-6) using the abbreviated notation of
Coпvo/utioп iп F/R Filters 167
у(п) = h(k) * х(п)
(5 7)
where the * symbol means convolution. (Equation 5-7 is read as "у of п
equals the convolution of h of k and х of п,") Тhe process of convolution,
as it applies to FIR filters is as follows: Тhe discrete Fourier transform
(DFT) of the convolution of а filter's impulse response (coefficients), and
an input sequence is equal to the product of the spectrum of the input
sequence and the DFТ of the impulse response. Тhe idea we're trying to
convey here is that if two time domain sequences h(k) and х(п) Ьауе DFTs
of Н(т) and Х(т), respectively, then the DFT of у(п) = h(k) * х(п) is
Н(т) 'Х(т). Making this point in а more compact way, we state this rela
tionship with the expression
OFТ
у(п) = h(k) * х(п) Н(т). Х(т),
IDFТ
(5 8)
With ЮРТ indicating the inverse DPТ, Eq, ( 8) indicates that two sequences
resulting from h(k) *х(п) and Н(т).Х(т) ате Fourier transform pairs. 50 tak
ing the DPТ of h(k) *х(п) gives us the product Н(т).Х(т) that is the Бресtrшn
of our ffiter output У(т). Likewise, we can determine h(k) *х(п) Ьу taking the
inverse DPТ of Н(т)'Х(т). тhe very important conclusion to learn from Eq,
(5-8) is that convolution in the time--domain is equivalent to multiplication
in the frequency-domain, То help us appreciate this principle, Figure 5-7
sketches the relationship between convolution in the time domain and mul
tiplication in the frequency domain. Тhe process of convolution with regard
to linear systems is discussed in more detail in Section 5.9, Тhe beginner is
encouraged to review that material to get а general idea of why and when
the convolution process can Ье used to analyze digital ffiters.
Equation (5 8) and the relationships in Figure 5 7 tell иБ what we need
to do to determine the frequency response of an FIR filter. Тhe product
Х(т). Н(т) is the DPТ of the ffiter output. Because Х(т) is the DFТ of the
filter's input sequence, the frequency response of the filter is then defined
аБ Н(т), the DFТ of filter's impulse response h(k),t Getting back to our
original problem, we сап determine our averaging filter's frequency-
domain response Ьу taking the DFТ of the individual filter coefficients
(impulse response) in Eq, (5 4). If we take the five h(k) coefficient values
of 1/5 and append 59 zeros, we have the sequence depicted in Figure
5-8(а). Performing а 64 point DPТ оп that sequence, and normalizing the
t We use the terт iтpиlse respoпse here, instead of coefficieпts, because this concept a!so
applies to ПR fi!ters, ПR filter frequency responses are a!so equa! to the DFf of their impulse
responses,
168 F/п/te /тpu/Se Respoпse Fllters
Input х(п)
Coefficients h(k)
Time
domain
t
t
Х(т)
Frequency
domain
Filter output sequence in the
Irequency domain, У(т) = Н(т) . Х(т)
Flgure 5-7 Relationshlps 01 convolutioп 05 applied to FIR dlgltal fllters,
DFl' magnitudes, gives us the filter's frequency magnitude response
I Н(т) I in Figure 5 8(b) and phase response shown in Figure 5 8(c).t Н(т)
is our old friend, the sin(x) / х function from Section 3.13.
Let's relate the discrete frequency response samples in Figures 5-8(Ь)
and 8(c) to the physical dimension of the sample frequency fs. We know,
frorn Section 3.5 and our experience with the DFТ, that the т = N /2 dis-
crete frequency sample, т = 32 in this case, is equal to the folding fre-
quency, or half the sample rate, fs/2. Keeping this in mind, we сan convert
the discrete frequency axis in Figure 5-8 to that shown in Figure 5-9, In
Figure 5-9(а), notice that the filter's magnitude response is, of course, peri
odic in the frequency domain with а period of the equivalent sample rate
fs, Because we're primarily interested in the filter's response between О and
half the sample rate, Figure 9(c) shows that frequency band in greater
t Тhere's nothing sacred about using а 64-point ОРТ here, We could just as well have
appended on1y enough zeros to take а l or 32-point FFТ, We chose 64 points to get а fre-
quency resolution that would make the shape of the response in Figure 5-8(Ь) reasonably
smooth, Remember, the more points in the FFТ, the finer the frequency resolution right?
Coпvo/иt/oп iп F/R Filters 169
(а)
. h(k)
:,:liI11
о . . . . . . . . . . . . . . . .
о 1 3 5 7 9 11 13 15 17 59 61 63 k
(Ь)
. 'Н(т)'
1 .
0 0,: i i 'r. . . . , i . ,. : , ' . ; . ' ,. ; , ;
, i i ,!.
, \1 j 1 \ 1
i [:; i 11'r . . П1П . . .Пi ... . iiШ .. i.i' ': i: i i
О . 1111 JI J 111 J , ,, 11111 IIIIIJ '" 1 i 11 i 111111 " 1 IIJ 11111111
о 3 6 912151821 М V м a м M т
0,4
0,2
(с)
: ! (т) ..
90 ".. -.. ...
45 3" .... .. ;. T!"... '" ...--{ ..... "! ....
о . , IIIIIIIIIII! 1 Т 1 ' 1 1111 J 11 , 1 1"1 , , 11111 1111"11 , . , 1 11111111'111 i .
-45 t !..IL. 9." .15-j....24i2Z .3Q. ..39..А2.4!i,шм. "..54....5Z 60ш63 т
-..: 8. :: 8.
=: =.:: .. l:. - ...:':::::.::.:::: ...ш.. ...... ........... ' '"
Figure 5-8 Averaging FIR fllter: (а) fllter eoeffieient sequenee h(k) wlth
appended zeros; (Ь) norma//zed d/serete freqиeney magпitиde
response I Н(т) I of the h(К) fllter eoeffielents; (е) phase-angle
response of Н(т) In degrees,
detai1 affirming the notion that averaging behaves Шее а low pass filter. It' s
а relatively poor low pass filter compared to an arbitrary, ideal low-pass
filter indicated Ьу the dashed lines in Figure 5-9(с), but our averaging fil
ter will attenuate higher frequency inрии relative to its response to low-
frequency input signals,
We сап demonstrate this Ьу way оЕ example, Suppose we applied а
low frequency sinewave to the averaging FIR fШеr, as shown in Figure
5 10(a), and that the input sinewave's frequency is 1/32 and its peak
amplitude is unity, Тhe filter output in this case would Ье а sinewave of
frequency 1/32, but it's peak amplitude would Ье reduced to а value of
0.96, and the output sinewave's first sample value is delayed Ьу а phase
170 Fiпite /mpu/se Respoпse F//ters
lН(m)1
0,8
0,6
(а) 0,4
0,2
О
'5/8 /4 /2
13S
90
4S
(Ь) О
--4S '5 Freq
O
13S
(с)
0,8
0,6
0,4
0,2
О
'-/8
'5/4
f. 12 Freq
5
Flgure 5-9 Averogiпg FIR fllter frequency response shown os continuous curves: (о)
normollzed frequency mognltude response, I Н(т) 1; (Ь) phose ongle
response of Н(т) in degrees; (с) the fllter's mognitude response
between zero Hz ond holf the somple rote, ',/2 Hz,
angle of 22.50, Next, if we applied а higher frequency sinewave of 3//32
to the FIR filter as shown in Figure 5-10(Ь), the filter output would, then,
Ье а sinewave of frequency 31.132, but it's peak amplitude is еуеп fur-
ther reduced to а value of 0,69. In addition, the Figure 5-10(Ь) output
sinewave's first sample value has ап even larger phase angle delay of
67.50, Although the output amplitudes and phase delays in Figure 5-10
were measured values from actually performing а 5-tap FIR filter
process оп the input sinewave samples, we could have obtained those
amplitude and phase delay values directly from Figures 5-8(а) and 5-
8(Ь), The emphasis here is that we don't have to implement ап FIR filter
and apply various sinewave inputs to discover what its frequency
Coпvo/ufioп 'п F/R Fiiters 171
1 ,1,1 1 ".
;:;.il,1I!I Fr8quencyof
3iiiiililll;\ I;il 7
IIt.!..8
''',f' li: \'J 1!1 !II щi Т,.;
delay = 22.5 ... ! j j "
1..
(а)
.. "' .8 r. . .
.i ; : :. Frequency of
',,, .::. m=6,оrЗI.1З2
. "и., I i IT' " . Т i lи ii , н. ---т:.
.:: 8
Н'.
r. 8 .Т
,ji. 8;; ..' 1
.НН. . rиi . .LL
i iji 8 8! Time
.ij . :.
'. .8
0.69" ..
! ".
(Ь)
'ii
i
.i.
.
ili,;
;'.
'1'.
t il:
Phase ' "
delay . .
= 67,50
r.
.'
L 8
Flgure 5-10 Averaging FIR fllter Input and output responses: (а) with an input sinewave
of frequency '. /32; (ь) with an input sinewave of frequency 3'. /32,
response will Ье. We need merely take the DFТ of the FIR fi1ter's соеЮ
cients (impulse response) to determine the filter's frequency response as
we did for Figure 5-8.
Figure 5 11 is another depiction of how weH our 5 tap averaging FIR
filter performs, where the dashed line is the fi1ter's magnitude response
I Н(т) 1, and the shaded line is the IX(m) I magnitude spectrum of the ш
ter's input values (the white squares in Figure 5 2), The solid line is the
magnitude spectrum of the filter's output sequence which is shown Ьу
the black squares in Figure 5 2, 50 in Figure 5--11, the solid output spec
trum is the product of the dashed filter response curve and the shaded
input spectrum, or IX(m). Н(т) 1. Again, we see that our averager does
indeed attenuate the higher frequency portion of the input spectrum.
Let's pause for а moment to let аН of this soak in а litt1e, 50 far we've
gone through the averaging filter example to establish that
· FIR fi1ters perform time domain convolution Ьу summing the prod
ucts of the shifted input samples and а sequence of filter coefficients,
· an FIR filter's output sequence is equal to the convolution of the input
sequence and а fi1ter's impulse response (coefficients),
172 Fiпite /mpu/se Respoпse Fi/ters
0.2
Filter Input, lX(m)1
1,0
0,8
0,6
0,4
о
fJ8
fJ4
Freq
fJ2
Flgure 5-11 Averaging FIR filter Input magnitude spectrum, frequency
magnltude response, and output magnitude spectrum,
· ап FIR filter's frequency response is the DFf of the filter's impulse
response,t
· ап FIR fi1ter's output spectrum is the product of the input spectrum
and the fi1ter's frequency response, and
· convolution in the time domain and multiplication in the frequency
domain are Fourier transform pairs.
ОК, here's where FIR fi1ters start to get really interesting. Let's change
the va1ues of the five fi1ter coefficients to modify the frequency response
of our 5-tap, low pass filter. In fact, Figure 5-12(а) shows our original five
filter coefficients and two other arbitrary sets of 5 tap coefficients, Figure
5 12(b) compares the frequency magnitude responses of those tЩее sets
оЕ coefficients, Again, the frequency responses are obtained Ьу taking the
DFf of the tЩее individual sets of coefficients and plotting the magni-
tude of the transforms, as we did for Figure 5-9(с). 50 we see tЩее impor
tant characteristics in Figure 5 12, First, as we expected, different sets of
coefficients give us different frequency magnitude responses. Second, а
t In Section 6,3, while treating an FIR filter as а special сзse of an ПR filter, we'n arrive at а math-
ematica! expression for an FIR filter's frequency response in terms of its coefficient values,
.
Coпvo/utioп /п F/R Filters 173
0,2
0,2 -
!
0,2 -
(а)
0,1
0,1 .
0,1 .
I!I :
о
о 2
о .........
о 2 4
о ...........
024
Filter magnitude response, 'Н(m)'
0,8
0,2, 0,2, 0,2, 0,2, 0,2
0,1,0,2,0,2,0,2,0,1
0,6
(Ь)
0,4
0,2
о
'в/д
'в/4
fsl2 Freq
Flgure 5-12 Three sets of 5-top low-poss fllter coefficients: (о) sets of coefficients;
0,2,0,2,0,2,0,2,0,2; 0,],0,2,0,2,0,2, О,]; oпd 0,04, 0,]2, 0,2, 0,]2, 0,04; (Ь)
frequeпcy mognitude response of three low-poss FIR fllters using those
sets of coefficlents,
sudden change in the values of the coefficient sequence, such as the 0.2 to
О transition in the first coefficient set, causes ripples, or sidelobes, in the
frequency response. Тhид, if we minimize the suddenness of the changes
in the coefficient values, such as the third set of coefficients in Figure 5-
12(а), we reduce the sidelobe ripples in the frequency response. However,
redIJcing the sidelobes results in increasing the таin lobe width of our
low-pass fi1ter, (As we'H see, this is exactly the saтe effect encountered in
the discussion of window fиnctions used with the DFТ in Section 3.9.)
То reiterate the fиnction of the fi1ter coefficients, Figиre 5-13 shows the
5-tap FIR fi1ter structure using the third set of coefficients from Figиre 5-12,
Тhe implementation of constant-coefficient transversal FIR filters does
not get any more complicated than that shown in Figиre 5 13, It's that
simple, We сап have а filter with more than five taps, but the input sig
nal sample shifting, the multiplications Ьу the constant coefficients, and
the summation are аН there is to it, (Ву constant coefficients, we don't
mean coefficients whose values are аН the same; we mean coefficients
whose values remain unchanged, or time invariant. Тhere is а class of
174 Fiпite /тpu/se Respoпse Filters
Х(п)
h(O) =
0,04
Flgure 5-13 Five-tap, low-pass FIR fi/ter Implementatlon usiпg the coefficlents 0,04,
0.12,0,2,0,12, and 0,04.
digital filters, called adaptive filters, whose coefficient values are period
ical1y changed to adapt to changing input signal parameters, While we
won't discuss these adaptive filters in this introductory text, their
descriptions are available in the literature[1 5],)
So far, our description of an FIR filter implementation has been pre
sented from а hardware perspective, In Figure 5-13, to calculate а sin-
gle filter output sample, Нуе multiplications and Нуе additions must
take place before the arrival of the next input sample value, In а soft-
ware implementation of а 5 tap FIR filter, however, аН of the input data
samples would Ье previously stored in memory, Тhe software filter rou-
tine's job, then, is to access different five sample segments of the х(п)
input data space, perform the calculations shown in Figure 5-13, and
store the resulting filter у(п) output sequence in an array of memory
loca tions, t
Now that we have а basic understanding of what а digital FIR filter is,
let's see what effect is had Ьу using more than five filter taps Ьу leaming
to design FIR filters,
5.3 Low-Pass FIR Filter Design
ОК, instead of just accepting а given set of FIR filter coefficients and ana
lyziIlg their frequency response, let's reverse the process and design our
t 'N reviewing the literature of FIR filters, the reader will often find the terrn z l replacing
the delay fиnфоп in Figure 5-13, Тhis equivalence is explained in the nех! chapter when we
study IIR filters.
Low-Poss F/R FiJter Desigп 175
own low-pass FlR filter, Тhe design procedure starts with the determina
tion of а desired frequency response followed Ьу calculating the filter соеЕ-
ficients that will give us that response. Тhere are two predominant tech-
niques used to design FIR filters, the window method, and the so-called
optimum method. Let's discuss them in that order.
5.3.1 Window Design Method
Тhe window method оЕ FIR filter design (also called the Fourier series
method) begins with our deciding what frequency response we want for
our low pass filter. We сап start Ьу considering а continuous low-pass ы-
ter, and simu1ating that filter with а digital filter. We'll define the соntin-
uous frequency response Н(/) to Ье ideal, i.e., а low pass filter with unity
gain at low frequencies and zero gain (infinite attenuation) beyond some
cutoff frequeпcy, as shown in Figure 5 14(a). Representing this не/)
response Ьу а discrete frequency response is straightforward enough
because the idea оЕ а discrete frequency response is essentially the same
as а continuous frequency respons with оnе important difference. As
described in Sections 2.2 and 3,13, discrete frequency domain representa
tions are always periodic with the period being the sample rate 1.. Тhe dis-
crete representation оЕ our ideal, continuous low-pass filter Н(/) is the
periodic response Н(т) depicted Ьу the frequency domain samples in
Figure 5-14(Ь),
Fitter frequency
response
CUloff
frequ"ncy
(а)
'.
'';2
О
'./2
'.
Freq
(f)
H(т)
..... . .......1 .II!I.......
(Ь) - .i:":: f ........,...,.T.,..........,.Цll..........'..'r..'.......... : i ' i : , ' .....
Freq
m= N т= N12 т=О т=Nl2 m=N (т)
Н.)
Н,/2)
(О)
('5/2)
('5)
'sl2 10 '512 inlerval
010 '5 inlerval r- :
Flgure 5-14 Low pass filter frequency responses: (а) contlnuous frequency
response Н(f); (Ь) periodlc, dlscrete frequency response Н(т),
176 Fiпlte /тpu/se Response Ff/ters
"'<
1......
! \ ! : I i
............. ._. ....._....."*... I I
+
+1 т= K 1)12
. . . Н(т)
, I I I I
i I : I ) I ."*....._........ ............
+ m
т= (K 1)12 М2
Flgure 5-15 ArbItrary, d/screte /ow-pass F/R ft/ter frequeпcy response deftпed
over N trequeпcy-domain samples coverlпg the trequency
raпge of '. Hz,
We have two ways to determine our low pass filter's time domain
coefficients. Тhe first way is algebraic:
1, Develop an expression for the discrete frequency response Н(т).
2. Apply that expression to the inverse DFТ equation to get the time
domain h(k),
3. Evaluate that h(k) expression as а fиnction of time index k.
Тhe second method is to define the individual frequency domain samples
representing Н(т) and then have а software routine perform the inverse
DFТ оЕ those samples, giving us the FIR filter coefficients, In either
method, we need only define the periodic Н(т) over а single period of!.
Нz. As it turns out, defining Н(т) in Figure 5 14(b) over the frequency
span fs/2 to 1./2 is the easiest form to analyze algebraical1y, and defining
Н(т) over the frequency span О to 1. is the best representation if we use the
inverse DFТ to obtain our filter's coefficients, Let's try both methods to
determine the fi1ter's time domain coefficients.
In the algebraic method, we сап define an arbitrary discrete frequency
response Н(т) using N samples to cover the fs/2 to !.12 frequency range
and establish К unity valued samples for the passband of our low pass fil
ter as shown in Figure 5 15, То determine h(k) algebraically we need to
take the inverse DFТ of Н(т) in the form of
N/2
h(k) = LН(т)еi21tтk/N,
т= N /2)+1
(5-9)
where our time-domain index is k. Тhe solution to Eq, (5 9), derived in
Se<:tion 3.13 аБ Eq. (3 59), is repeated here as
Low-Pass F/R Filter Desigп 177
КfNШiii h( k)
.
.
. . . "
8 .j.. . .. .
.
I
. .
I. .... 8 ... 8 k
Figure 5-16 Time-domaln h{k) coefficlents obtalned Ьу evaluatlng Eq, (5-1 О),
h(k) = 2.. sin(rtkК / N) ,
N sin( 1tk / N)
(5-10)
If we evaluate Eq. (5 10) as а function of k, we get the sequence shown in
Figure 5 16 taking the form of the classic sin(x)/x function. Ву reviewing
the material in 5ection 3.13, it's easy to see the great deal of algebraic
manipulation required to arrive at Eq. (5 10) from Eq, (5-9).50 much alge
bra, in fact, with its тanу opportunities for making епоrs, that digital ш
ter designers like to avoid evaluating Eq, (5-9) algebraically. Тhey prefer
to use software routines to perform inverse DFTs (in the form of an
inverse FFТ) to determine h(k), and so will we.
We сап demonstrate the software inverse DFT method of FIR filter
design with an example. Let's say we need to design а low pass FIR ш
ter simulating the continuous frequency response shown in Figure
5 17(a). The discrete representation of the filter's frequency response
Н(т) is shown in Figure 5 17(b), where we've used N = 32 points to rep
resent the frequency domain variable Н(Л, Because it's equivalent to
Figure 5 17(b) but avoids the negative values of the frequency index m,
we represent the discrete frequency samples over the range О to 1. in
Figure 5 17(c), as opposed to the fs/2 to +1./2 range in Figure 5 17(b),
ОК, we're almost there, Using а 32 point inverse РРТ to implement а
32-point inverse DFT of the Н(т) sequence in Figure 5 17(c), we get the
32 h(k) values depicted Ьу the dots from k = 15 to k = 16 in Figure
5 18(a),t We have one more step to perform. Because we want our final
31-tap h(k) filter coefficients to Ье symmetrical with their peak value in
the center of the coefficient sample set, we drop the k = 16 sample and
shift the k index to the left from Figure 5-18(а) giving us the desired
t If уои want to use this FIR design method but on1y have а forward FFf software routine
available, Section 10,6 shows а slick way to perfonn an inverse FFf with the forward FFf
a1gorithm,
178 F/п/te /три/5е Re5poпse F//ter5
Filter Irequency
response
Н(')
(а)
/ Cutoff frequency
's/2
's/4 'J8
о
'J8
's/4
, s/2 Freq
1 . . . . . Н(т)
(Ь)
..............,...-...., ..........f...........
15 12 8 --4
HJ4) HJ8)
о
4
('s/8)
, ...........,.............................
16 т
('S/2)
8
('s/4)
Н(т) . . . . 1
. I!'
(с)
I
О
.............................................,....-......................................
4 8 1 16 ;0 ;4 ;8 31 I т
('s/8) ('s/4) ('s/2) (3's/4) ('s)
Flgure 5-17 An Ideallow-pass fllter: (а) eontinuous Irequeney response нef); (Ь)
diserete response нет) over the range 01 fs/2 to fs/2 Hz: (е) diserete
response нет) over the range О to " Hz,
0.219
. .
.' :.
Inverse OFТ 01 Н(т)
.
(а)
. . --в I!'
........... .... 8--4
..
i. . . . .. .
4 . Ii 12. · 11 k
. h(k) = Shifted inverse
.. !. OFТ 01 Н(т)
.; i i :.
(Ь) ...l.l!.. . 8. .. 1.; ,. 22 . .. за
.. 4 . . ' . !. 12 15 18. i ' . .
... k
Flgure 5-18 Inverse OFТ 01 the dlserete response In Figure S-17(e): (а) normal
inverse OFТ indexing lor 1<; (Ь) symmetrieal eoeffieients used for а
Зl-tар low-pass FIR fllter,
sin(x) I х form of h(k) as shown in Figure 5 18(Ь). This shift of the index
k wi11 not change the frequency magnitude response of our FIR fi1ter,
(Remember from our discussion of the DFT Shifting Тheorem in Section
З.6 that а shift in the time-domain manifests itself only as а linear phase
shift in the frequency domain with по change in the frequency domain
magnitude.) The sequence in Figure 5-18(Ь), then, is now the соеШ
Low-Poss F/R FJJter Desigп 179
cients we use in the convolution process of Figure 5 5 to implement а
10w-pass FIR fi1ter,
It's important to demonstrate that the more h(k} terms we use as filter
coefficients, the closer we'11 approximate our ideal 10w-pass fi1ter
response, Let's Ье conservative, just use the center ninе h(k} coefficients,
and see what our fi1ter response 100ks like. Again, our filter's magnitude
response in this case wil1 Ье the DFf of those nine coefficients as shown
оп the right side of Figure 5 19(a}, Тhe ideal filter's frequency response is
also shown for reference as the dashed curve. (То show the details of its
shape, we've used а continuous curve for I Н(т) I in Figure 5 19(a}, but
we have to remember that I Н(т} I is rea11y а sequence of discrete values,}
Notice that using nine coefficients gives us а 10w pass fi1ter, but it's cer
tainly far from ideal. Using more coefficients to improve our situation,
Figure 5 19(b} shows 19 coefficients and their corresponding frequency
magnitude response that is beginning to 100k more like our desired rec-
tangular response, Notice that magnitude fluctuations, or ripples, are evi
dent in the passband of our Н(т} fi1ter response, Continuing, using а1131
. h(k)
lН(т)1 lor а 9-tap lilter
.:
,.
ф
Сщо"
frequency
(а)
I!I :
;.
1\1 .
.
'./8
'./8
Freq
.8J!11 h(k)
I!I ; . : .
lН(т)1 lor а 19-tap Illter
(Ь)
.
! .
. i
i
l '
ф
'/8
Freq
l
. Ii . i.
. Ii
.-
Ii
..
';8
lН(т)1 lor а 31-1ар Illter
(с)
'./8 Freq
Flgure 5-19 Coeffieients ond frequeney responses of three low-pass filters: (о) 9-top
FIR filter; (Ь) о 19 top FIR filter; (е) frequency response of the full 31-top
FIR filter,
180 Fiпite /mpи/se Respoпse F//ters
of tbe h(k) values for our fi1ter coefficients resu1ts in the frequency
response in Figure 5 19(c). Our fi1ter's response is getting better
(approaching the ideal), but those conspicuous passband magnitude rip
ples are still present.
It's important that we understand why those passband ripples are in
the low-pass FIR fi1ter response in Figure 5 19, Recall the above discus-
sion of convolving the 5 tap averaging fi1ter coefficients, or impulse
response, with an input data sequence to obtain the averager' s output, We
estaыshedd that convolution in the tirne domain is equivalent to mu1tipli
cation in the frequency domain, that we symbolized with Eq, (5 8), and
repeat it here as
DFТ
h(k)*x(n) Н(т).Х(т).
IDFТ
(5 11)
Тhis association between convolution in the time domain and mu1tiplica
tion in the frequency domain, sketched in Figure 5 7, indicates that, if two
time domain sequences h(k) and х(п) have DFfs of Н(т) and Х(т), respec
tively, then the DFf of h(k) *х(п) is Н(т).Х(т), No restrictions whatsoev
er need Ье placed оп what the time domain sequences h(k) and х(п) in Eq.
(5-11) actually represent. As detai1ed later in Section 5.9, convolution in
one domain is equivalent to mu1tiplication in the other domain allowing
us to state that mu1tiplication in the tirne domain is equivalent to convo
lution in the frequency domain, or
DFТ
h(k).x(n) Н(т)* Х(т)
. IDFТ
( 12)
Now we're ready to understand why the magnitude ripples are present in
Figure 5 19,
Rewriting Eq, (5 12) and replacing the h(k) and х(п) expressions with
h (k) and w(k), respectively,
DFТ
h (k).w(k) H (m)*W(m)
IDFf
(5-13)
Let's say that h (k) represents an infinitely long sin(x)/x sequence of ideal
low-pass FIR fi1ter coefficients and that w(k) represents а window sequence
that we use to truncate the sin(x)/x terms as shown in Figure 5 20, Тhus,
the w(k) sequence is а finitt71ength set of unity values and its DFf is W(m).
Тhe length of w(k) is merely the number оЕ coefficients, or taps, we intend
Low Poss F/R Filter Desigп 181
. .
h (1<)
.
.
. .
. I!I .
. . . . .
.. . .
· i
1 . . . . . . . I!I I!I Иl(k)
. .. . .
. . . .
. . ........................
· · k
j
п
.
. · h(1<)
: :
.
, .
I .
.
. .
k
Flgure 5-20 Infinlte h (k) sequence wlndowed Ьу w(k) to define the final
filter coefficients h(k),
to use to implement our low pass FIR filter. With h (k) defined as such, the
product h (k) . w(k) represents the truncated set of filter coefficients h(k) in
Figures 5 19(a) and 5-19(Ь), 50, from Eq. (5-13), the FIR filter's true fre
quency response Н(т) is the convolution
Н(т) = H (т) * W(m) ,
(5 14)
We depict this convolution in Figure 5 21 where, to keep the figure from
being so busy, we show H (т) (the DFI of the h (k) coefficients) as the
dashed rectangle. Кеер in mind that it's reaHy а sequence of constant
amplitude sample values.
Let's look at Figure 5 21(a) very carefully to see why аН three I Н(т) I s
exhibit passband ripple in Figure 5 19, We сап view а particular sample
value of the Н(т) = H (т) * W(m) convolution as being the sum of the
products of H (т) and W(m) for а particular frequency shift of W(m).
H (т) and the unshifted W(m) are shown in Figure 5 21(a,) With an
assumed value of unity for аН of H (т), а particular Н(т) value is now
merely the sum of the W(m) samples that overlap the H (т) rectangle, 50,
with а W(m) frequency shift of О Нz, the sum of the W(m) samples that
overlap the H (т) rectangle in Figure 5 21(a) is the value of Н(т) at О Hz.
182 Fiпite /mpu/se Respoпse Filters
(а)
н'" (т) = DFТ 01 h (1<) , W(m) = DFТ 01
... rectangular w(n)
. [ - . i" . ' .. } ... . . '_._ ] ' 1
. : : : ;8
... i: : :; , II!I".
..o....i...i... . .. . .. ..
8.8 . О ..8 ...
Freq
.
LJm . '.. ==с>
. . .. . ." . . . . . .".. . .
." .
.! i : I!.
.... . . i О. i . i . : .1 .",;. . ....
.. ..
Freq
(Ь)
.
[ i
. ' . .
.1i!I. О . : i ...
,......L............. · .. ....................
_.. .' , . 8.-
.. 8.
Freq
(с)
.
· . · с::::{>.
L .I'
. : :8
О ... '! j \ .1i!I.
I .--'-0 .. ....,.. .
... ... . ... Freq
(d)
Figure 5-21
Convolutlon W(m)*H (m): (а) unshlfted W(m) and H (т);
(Ь) shlft 01 W(m) leadlng to rlpples wlthin Н(m)'s posltive
frequeney passband: (е) shift of W(m) eausing response roll-off
near Н(m)'s positive eutoff frequeney; (d) shift of W(m) eauslng
ripples beyond Н(m)'s pos/tlve eutoff frequeney.
As W(m) is shifted to the right to give us additional positive frequency
Н(т) va1ues, we can see that the sum оЕ the positive and negative values
оЕ W(m) under the rectangle oscil1ate during the shifting оЕ W(m), As the
convolution shift proceeds, Figure 5-21(Ь) shows why there ме ripples in
the passband оЕ H(m) again, the sum of the positive and negative W(m)
samples under the H (m) rectangle continue to vary as the W(m) function
is shifted, The W(m) frequency shift, indicated in Figure 5-21(с), where the
peak of W(m)'s таin lobe is now outside the H (m) rectangle, corresponds
to the frequency where H(m)'s passband begins to roH off, Figure 5 21(d)
shows that, as the W(m) shift continues, there will Ье ripples in Н(m)
Low-Pass F/R Filter Desigп 183
IH(m)1 Transition region
r
(а)
'5/8 О '5/8 Freq
IH(m)1 Transition region
r
(Ь)
'5/8
о
'5/8
Freq
Flgure 5-22 Passband ripple and transition regions: (а) for а 3 '-tap low pass
filter; (Ь) for а 63-tap low-pass filter,
beyond the positive cutoff frequency. t The point of all of this is that the rip-
ples in Н(т) are caused Ьу the side obes of W(m),
Figure 5 22 helps us answer the question: How many sin(x)/x соеШ-
cients do we have to use (or how wide must w(k) Ье) to get nice sharp
falling edges and по ripples in our Н(т) passband? The answer is that
we can't get there from here, It doesn't matter how many sin(x)/x coef-
ficients (filter taps) we use, there will always Ье fi1ter passband ripple,
As long as w(k) is а finite number of unity values (i.e" а rectangular win
dow of finite width) there will Ье sidelobe ripples in W(m), and this wil1
induce passband ripples in the final Н(т) frequency response, То il1us
trate that increasing the number of sin(x)/x coefficients doesn't reduce
passband ripple, we repeat the 31-tap, low-pass fi1ter response in Figure
5 22(a). The frequency response, using 63 coefficients, is shown in
t In Figure 5-21(Ь), had we started to shift W(т) to the left in order to determine the nega-
tive frequency portion of Н(т), we would have obtained the mirror image of the positive fre-
quency portion of Н(т),
184 F/п/te /mpu/se Respoпse F//ters
Figure 5 22(b), алd the раssЬалd ripple rеmаiлs, We сал make the ш-
ter's trалsitiол rеgiол more лаrrоw usiлg аdditiолаI h(k) fiIter соеЮ-
сiелts, but we cannot еIimiлаtе the раssЬалd ripple. Тhat ripple, kлоwл
ав Gibbs' рhелоmелол, mалifеsts itseIf алуtimе а fuлсtiол (w(k) iл this
саве) with а iлstалtалеоus disсолtiлuitу is rерrеsелtеd Ьу а Fourier
series[6 8]. No fiлitе set of sinusoids wiП Ье аЫе to сhалgе fast елоugh
to Ье exactly equal to ал iлstалtалеоus disсолtiлиity. Алоthеr way to
state this Gibbs' diIemma is that, ло matter how wide our w(k) window
is, its DFT of W(m) wiП always have sidelobe ripples, Ав shоwл iл
Figure 5-22(Ь), we сал use more соеffiсiелts Ьу ехtелdiлg the width оЕ
the rесtалgulаr w(k) to лаrrоw the fiIter trалsitiол rеgiол, bиt а wider
w(k) does лоt eIiminate the fiIter раssЬалd ripple лоr does it еvел reduce
their peak to-peak ripple mаgлitиdеs, ав lолg аБ w(k) has sиddел dis
солtinuitiеs.
5.3.2 Windows Used in FIR Filter Deslgn
ок. Тhe good леws is that we сал miлimizе FlR раssЬалd ripple with
wiлdоw functions the вате way we miлimizеd DFТ Ieakage in Sесtiол
3.9. Here's how, Looking back at Figure 5-20, Ьу truncating the iлfinitеlу
Iолg h (k) sеquелсе through multiрIiсаtiол Ьу the rесtалgulаr w(k), oиr
final h(k) exhibited ripples in the frequелсу-dоmаin раssЬалd. Figure 5-21
shows us that the раssЬалd ripples were caused Ьу W(m)'s sidelobes that,
in turл, were caused Ьу the suddел disсолtiлuitiеs from zero to оле алd
оле to zero in w(k). If we thiлk of w(k) in Figure 5 20 ав а rесtалgи1аr win
dow, thел, it is w(k)'s abrupt ampIitиde chалgеs that are the Boиrce of our
fiIter раssЬалd ripple, Тhe wiлdоw FlR dеsigл method is the tесhлiquе оЕ
reducing w(k)'s disсолtinuitiеs Ьу using window functions other thал the
rectangular window.
Consider Figure 5 23 to вее how а лолrесtалgulаr window funсtiол
сал Ье used to dеsigл Iow ripple FlR digital fiIters. Imagine if we replaced
Figure 5-20'в rесtалgulаr w(k) with the Вlасkmал window funсtiол whose
discrete values are defined as t
(()(k)=0.42 0.5COS( 2: )+O,08COS( } fork=0,1,2, .."N l. (5 15)
t As we mentioned in Section 3,9, specific expressions for window functions depend оп the
range of the sample index k, Had we defined k to cover the range N /2, < k < N /2, for ехат-
ple, the expression for the Blackman window would have а sign change and Ье
w(k) = 0.42 + O,5cos(21tk/N) + O,08cos(41tk/N),
Low Pass F/R Fliter Des/gп 185
.
I
h (k)
.
.
0'0 . .
.....78. ................. .
.
. .... ::.
.... .. k
Blackman window
function 1 .. . . . .
'" ./. ,. . i '.'. w(k)
./,... /' . , ! ' ' '.
./ : . '.
................ .... ...i... : j .... ... ........... ._........
k
{7
(а)
· .. h(k)
. .
.
.......................-.... .
.
."""":"""':'...................................
· · k
Transltlon '891оп
(Ь) x т)' 1 1[
....
'/8 О '/8 Freq
1 ' ' 1 Transition r89ion
' r
(с) ]
I .-
I
'з/8 О '/8 Freq
Flgure 5-23 Coeffleients and frequeney response of а ЗНар Blaekman-wlndowed
FIR Fllter: (а) deflnlng the windowed fllter eoeffieients h(k); (Ь) low ripple
ЗНар frequeney response; (е) low-ripple 6З-tар frequeney response,
Тhis situation is depicted for N = 31 in Figure 5-23(а) where Eq, (5-15)'s
w(k) looks very much like the Hanning window funсНоп in Figure 3-17(а),
Тhis Blackman window function resu1ts in the 31 smooth1y tapered h(k)
coefficients at the bottom of Figure 5-23(а), Notice two things about the
186 F/п/te lmpulse Respoпse Fllters
resulting Н(т) in Figure 5 23(b). First, the good news, Тhe passband rip
ples are greatly reduced from those evident in Figure 5 22(a) so our
Вlackman window function did its job. Second, the price we paid for
reduced passband ripple is а wider Н(т) transition region. We сап get а
steeper fi1ter response roll-off Ьу increasing the number of taps in our FIR
fi1ter. Figure 5-23(с) shows the improved frequency response had we used
а 63 coefficient Вlackman window fиnction for а 63 tap FIR filter, So иsing
а nonrectangular window function reduces passband ripple at the
expense of slower passband to stopband roll off.
А graphical comparison of the frequency responses for the rectangular
and Бlасkman windows is provided in Figure 5-24. (Тhe curves in Figure
5-24 were obtained for the window fиnctions defined Ьу 32 discrete sam
ples, to which 480 zeros were appended, applied to а 512 point DFТ.) Тhe
Linear window 'requency magnilude responses, IW(m)1
1 ',
\ . , Reclangular (dotled)
0,8
(а)
0,6 \
- - f> '... . Iackman (dashed)
0,4 ;, ....
. ,
\ ,..."...
I ',..."':<....... .....
. -",..., ..... е. ......... ........
о
......
.......
0,2
'.18
'./4
'./2 Freq
Logarilhmic window 'requency magnilude responses, W dB (m) in dB
-20
о ,,
.... , Rвctangular(dotled)
'. ,
-10 ,
/".', Blackman (dashed)
:( \/?:;!.\/.
;. " \:. . .
:: .. \ .е :.
.' \.:
... ""::..... .:: .';:" .. .. .::
,.
. ...
.1
:\
. .... . ."
.. ' " .
:...-1 .\.
.. ..
:.
.'.
: . \
" .
"
(Ь) -30
-40
,
"
,
....Т......-.
I ...
I \
I .\
..,
I,.....
I :'
. \
.'
.'
-50
-60
,
!
I :.
I . ,
,
/ ,
'.18
"'14
'./2 Freq
Flgure 5-24 Rectangular vs, Blackman wlndow frequency magnltude responses:
(а) I W(m) I оп а IInear scale; (Ь) normalized logarlthmic scale of WdB(m),
Low-Pass F/R F/lter Des/gп 187
sidelobe magnitudes of the Вlackman window's I W(m) I are too small to
see оп а linear scale. We сап see those sidelobe details Ьу plotting the two
windows' frequency responses оп а logarithmic scale and normalizing
each plot so that their main lobe peak values are both zero dB. For а given
window function, we сап get the log magnitude response of WdB(m) Ьу
using the expression
( 1 W(m) 1 )
W dB (m)=20.10g 10 I .
W(O) I
(5 16)
(The 1 W(O) 1 term in Eq. (5-16) is the magnitude of W(m) at the peak of the
main lobe when т = О.) Figure 5-24(Ь) shows us the great1y reduced side-
lobe levels of the Вlackman window and how that window's main lobe is
almost three times as wide as the rectangu1ar window' s main lobe.
Of course, we could have used any of the other window functions, dis
cussed in Section 3.9, for our low pass FIR fi1ter. That's why this FIR fil-
ter design technique is called the window design method. We pick а win
dow function and mu1tiply it Ьу the sin(x) / х values from H (т) in Figure
5 23(a) to get our final h(k) fi1ter coefficients. It's that simple. Before we
leave the window method of FIR filter design, let's introduce two other
interesting window functions.
A1though the Blackman window and those windows discussed in
Section 3,9 are useful in FIR fi1ter design, we have little control over their
frequency responses; that is, our only option is to select some window
function and accept its corresponding frequency response, Wouldn't it Ье
nice to have more flexibi1ity in trading off, or striking а compromise
between, а window's main lobe width and sidelobe levels? Fortunately,
there are two popular window functions that give us this opportunity,
Called the Chebyshev and the Kaiser window functions, they're defined
Ьу the following formidable expressions:
w(k) = the N point inverse DFТ of
СЬеЬузЬеу window:
(also called the Dolph
Chebyshev and the
Tchebyschev window)
со{ N. COS l[ а. cos( х )]]
1
coshfN .cosh (а)1
where а = coshl cosh 1 (lО У ») and т = О, 1,2,...,N 1, (5 17)
188 F/п/te /тpи/se Respoпse F//ters
Kaiser window:
(a15O calJed the
Kaise:r Вessel window)
+ ]
ro(k) Iоф)
for k = О, 1, 2, . . ., N l, and р = (N 1)/2.
(5 18)
Two typical Chebyshev and Kaiser window functions and their fre
quency magnitude responses are shown in Figure 5 25. For comparison,
the rectangu1ar and Blackman window functions are a1so shown in that
figure. (Again, the curves in Figure 5 25(b) were obtained for window
Window lunctions
о
о
5
Blackman (da8hed)
0,6
---------l---------..-- -
/1
Rectangular (dotled) I 1
/ /
I /
I
I /
I 1
1
, 1
I 1
I
I
I
//
0,8
(а)
0.4
0,2
10
15
20
25
зо
k
(Ь)
18/8
18/4
1812 Freq
Flgure 5-25 Typical wlndow functlons used wlth dlgltal fllters: (а) wlndow coefflclents
In the time domaln; (Ь) frequency-domaln magnltude responses In dB,
Low-Pass F/R F//ter Desigп 189
functions defined Ьу 32 discrete samples, with 480 zeros appended,
applied to а 512-point DFf.)
Equation (5 17) was originally based оп the analysis of antenna arrays
using the mathematics of Chebyshev polynomials[9 11]. Equation (5-18)
evolved from Кaiser's approximation of prolate spheroid functions using
zeroth order Bessel functions[12 13], Don't Ье intimidated Ьу the coт
plexity of Eqs. (5 17) and (5-18) at this point, we need not Ье concemed
with the mathematical details of their development, We just need to real-
ize that the coпtro/ parameters у and , in Eqs. (5 17) and (5-18), give us con-
trol over the windows' таin lobe widths and the sidelobe levels,
Let' s see how this works for Chebyshev window functions, having four
separate values of у, and their frequency responses shown in Figure 5-26,
FIR filter designers applying the window method typically use prede
fined software routines to obtain their Chebyshev window coefficients,
Commercial digital signal processing software packages allow the user to
specify three things: the window function (Chebyshev in this case), the
desired number of coefficients (the number of taps in the FIR filter), and
the value of у. 5electing different values for у enables us to adjust the side--
lobe levels and see what effect those values have оп таin lobe width, а
capability that we didn't have with the Blackman window or the window
functions discussed in 5ection 3.9. Тhe Chebyshev window function's
stopband attenuation, in decibels, is equal to
Atte11cheb = 2(Jy ,
(5 19)
50, for example, if we needed our sidelobe levels to Ье по greater than
--60 dB below the таin lobe, we use Eq. (5-19) to establish а у value of 3,0
and let the software generate the Chebyshev window coefficients,t
Тhe same process applies to the Kaiser window, as shown in Figure
5 27. Commercial software packages allow us to specify in Eq, (5 18)
and provide us with the associated window coefficients. The curves in
Figure 5 27(b), obtained for Kaiser window functions defined Ьу 32 dis-
crete samples, show that we сап select the desired sidelobe levels and
see what effect this has оп the main lobe width,
Chebyshev or Kaiser, which is the best window to use? It depends оп
the application, Retuming to Figure 5-25(Ь), notice that, unlike the con-
stant sidelobe peak levels of the Chebyshev window, the Kaiser window's
t Ву the way, some digita! signa1 processing software packages require that we specify
Atten Cheb in decibe!s instead of у, That way, we don't have to bother using Eq, (5-19) at аН,
190 Fiпite /три/5е Re5poп5e Filters
0,8
0,6
(а)
0.4
0,2
00
о
-10
-20
(Ь) -30
-40
-50
-60
-70
Chebyshev window lunctions
у = 1,5 (dotted) ,. ://.
/
у = 2,0 (da5hed) . /'
"'Х / I 1
'/1" 1.
.,/ I 11
\ ,/' /1 I
'.,/ ,./ I
, // /
.;' .;'
...
5
10
15
k
20
25
30
Chebyshev window Irequency magnitude responses, W dS (т)
у = 1,5 (dotted)
\ У = 2,0 (da5hsd)
' .. ' .. у = 2,5 (da5h dot)
\\ \ \ У= 3.0 (50Iid)
. ,
; \ \
\ \
,f'J ,"""'. " " ,"'\ I "
I 1\ \ ",' ,1 \ I I
I I ' , \ I I I ,( \ I
': "''': I \ ': 1.... ,\: I \ \ I! I
1, 11 :'\ ( 1 / 1: I \:,' 1 l' "
} '/:! 1; i.:!-..' '1. 1 ):/j,t '
11 :: I \; f l,' I I I I
ir I \ ........J 11 Т .\i I
\
15/8
15/4
15/2 Freq
Figure 5-26 Chebyshev wlndow functlons for vorious 'у volues: (о) window coefficients
in the time domoin; (Ь) frequency-domain mognltude responses in dB.
sidelobes decrease with increased frequency. However, the Kaiser side-
lobes are higher than the Chebyshev window's sidelobes near the main
lobe, Our primary trade-off here is trying to reduce the sidelobe levels
without broadening the main lobe too much, Digital filter designers typi-
саНу experiment with various values of у and for the Chebyshev and
Kaiser windows to get the optimum WdB(т) for а particular application.
(For that matter, the Вlackman window's very low sidelobe levels out-
weigh its wide таin lobe in many applications.) Different window func-
tions have their own individual advantages and disadvantages for FIR fil-
ter design, Regardless of the nonrectangular window function used, they
Baпdpass F/R Fi/fer Des/gп 191
Kai5er window coefficient5
О,В
0,6
(а)
0.4
0,2
00 5 10 15 20 25 3О k
Kal5er window Irequency magnitude re5pon5e5, W dB (m)
о
'5/8
15/4
15/2 Freq
-20
-40
(Ь)
-60
-ВО
-100
Flgure 5-27 Kaiser window functions for various values: (а) window coefficients in
the time domain; (Ь) frequency-domain magnitude responses In dB,
always decrease the FIR filter passband ripple over that of the rectangu
lar window, For the enthusiastic reader, а very thorough discussion of
window functions сап Ье found in reference [14].
5.4 Bandpass FIR Filter Design
The window method of low pass FIR filter design can Ье used as the first
step in designing а bandpass FIR filter. Let's say we want а ЗНар FIR ш-
ter with the frequency response shown in Figure 5-22(а), but instead of
being centered about zero Hz, we want the filter's passband to Ье centered
192 F/пite lтpuJse Respoпse F//ters
. . .
. /hIP(k)
; .
. I
. . ....!....!..... .
. . .
.
.
. . ....
.... -k
.
.
.
/ S-hlft(k)
i
(а)
.. .
..... ............ ......L.T. . ....L.......... .""""'!"""'......L..""""!"""". ............k
.
,;
I
.
Ii
.
.
..
,;
{7
h bp (k)
.
. .. :.- .
. . . ."""... ."""""".;. .Т-"""""'."""""'..."". "."'.k
.
(Ь)
Originallowpass
response IH 1p (т)1
"" ('
I
IHbp(m)1
r\
\
, .,
'/2
'5/4
о
'5/4
's /2 Freq
Flgure 5-28 Bondposs filter with frequency response centered ot '/4:
(о) generating Зl-tор filter coefflcients hbP(k); (Ь) frequency
mognltude response I ньр(т) I '
aboиt f/ 4 Hz. If we define а low pass FIR filter's coefficients as hip(k), our
problem is to find the 1;,p(k) coefficients of а bandpass FlR filter. As shown
in Figure 5 28, we сап shift Hip(т)'s frequency response Ьу multiplying the
filter's hip(k) low pass coefficients Ьу а sinusoid of fs/ 4 Нz, Тhat sinusoid
is represented Ьу the sshift(k) sequence in Figure 5 28(a) whose values are
а sinewave sampled at а rate of four samples per cycle. Our final Зl tар
1;,p(k) FIR bandpass filter coefficients are
1;,p(k) = hip(k)'Sshift(k) ,
(5-20)
Highpass F/R Fi/ter Desigп 193
whose frequency magnitude response I Ньр(т) I is shown as the solid
curves in Figure 5 28(b), Тhe actual magnitude оЕ I Ньр(т) I is half that
оЕ the original I Hip(т) I because half the values in hbp(k) are zero when
SShift(k) corresponds exactly to f/ 4. Тhis effect has an important practical
implication. It means that, when we design an N tap bandpass FIR fi1ter
centered at а frequency оЕ 1./4 Hz, we only need to perform approxi
mately N /2 mu1tiplications for each fi1ter output sample. (There's по
reason to multiply an input sample value, x(п k), Ьу zero before we sum
аН the products from Eq. (5 6) and Figure 5 13, right? We just don't both-
er to perform the unnecessary multiplications at аН,) ОЕ course, when
the bandpass FIRfi1ter's center frequency is other thanf./4, we're forced
to perform the ЕиН number оЕ N multiplications for each FIR fi1ter out
put sample.
Notice, here, that the hip(k) low pass coefficients in Figure 5 28(a) have
not been multip1ied Ьу anу window function. In practice, we'd use an
hip(k) that has been windowed prior to implementing Eq. (5 20) to reduce
the passband ripple. If we wanted to center the bandpass filter's response
at some frequency other thanf./4, we merely need to modify sshift(k) to
represent sampled values оЕ а sinusoid whose frequency is equal to the
desired bandpass center frequency. Тhat new sshift(k) sequence would then
Ье used in Eq, (5-20) to get the new hbp(k).
5.5 Highpass FIR Filter Design
Going one step further, we сап use the bandpass FIR fi1ter design tech-
nique to design а highpass FIR fi1ter, То obtain the coefficients for а high-
pass fi1ter, we need only modify the shifting sequence sShift(k) to make it
represent а sampled sinusoid whose frequency is 1./2. This process is
shown in Figure 5-29. Our final31 tap highpass FIR fi1ter's hhp(k) coeEH
cients are
hhp(k) = hip(k) -sshift(k)
= h ip (k)-(l, 1, 1, 1, 1, 1, etc.) , (5 21)
whose I Hhp(т) I frequency response is the so1id curve in Figure 5 29(b),
Because sshift(k) in Figure 5-29(а) has a1temating plus and minus ones, we
сan see that hh (k) is merely hip(k) with the sign changed for every other
coefficient. Ucl1ke I Ньр(т) I in Figure 5-28(Ь), the I Hhp(т) I response in
Figure 5 29(b) has the same amp1itude as the original I Hip(т) 1.
'94 Fiпite /тpиise Respoпse F//ters
.
/h1p(k)
.
I!I : 8
.... .... . . ...
. .
.
.... . .
. .. 8k
.
.
1
w /"""
i I!I .. I!I ·
.
i
.
.
I
.
.
(а)
.
oi .
oi
Ii
.
.
k
. . . . . . Ii . ..
{7
.
hhp (k)
.
. .... . .
. Ii · Ii
.
I
i
. . .... ....-....
.. .. k
. .
Ii .
(Ь)
Orlginallow-pass
response IH1p(т)1 IHhP(т)1
("'./
!
{'>}
's/2
's/4
о
,/4
, J2 Freq
Flgure 5-29 Hlghpass filter wlth frequency response centered at '5/2:
(а) generating 31-tap filter coefficlents hhp(k); (Ь) frequency
magnltude response I HhP(т) I '
Again, notice that the h'p(k) low pass coefficients in Figure 5-29(а) have
not been modified Ьу anу window function. In practice, we' d use а win
dowed h,pCk) to reduce the passband ripple before implementing Eq. (5-21),
5.6 Remez Exchange FIR Filter Design Method
Let's introduce one last FIR filter design technique that has found wide
acceptance in practice, Тhe Remez Exchange FIR filter design method
(also called the Parks McClellan, or Optimal method) is а popular tech
Reтez Exchaпge F/R Filter Desigп Method 195
1 + 8 р
ш.. JF
....шш. .....'"
1 8p
85
)/
о
о
'pass
k Passband !
'stop
!<
'" '5/2 Freq
.....05
Stopband
;.. '
Figure 5-30 Desired frequency response definition of а low-pass FIR filter
using the Reтez Exchange design тethod,
nique used to design high performance FIR filters,t То use this design
method, we have to visualize а desired frequency response Hd(m) like that
shown in Figure 5 ЗО,
We have to establish а desired passband cutoff frequency f pass and the
frequency where the attenuated stopband begins, fstop' In addition, we
must establish the variables 8 р and 85 that define our desired passband
and stopband ripple. Passband and stopband ripple, in decibels, are relat-
ed to 8 р and 85 Ьу[15]
Passband ripple = 20 .10glO(1 + 8 р ) ,
(5-22)
and
Stopband ripple = 20 .10glO(8 s ) ,
(5-22')
(Some of the early joumal papers describing the Remez design method
used the equally valid expression 20 . 10glO(8 p ) to define the passband
ripple in decibels, However, Eq, (5 22) is the most cornrnon form used
today,) Next, we apply these parameters to а computer software routine
that generates the filter's N time domain h(k) coefficients where N is the
minimum number of filter taps to achieve the desired filter response,
t Reтez is pronounced re-'ma, like "away,"
196 Fiп/te /три/sе Respoпse F//ters
4 Hd(т)
1......
I
; lIIi
11,
11
'.
о
о
f pass
, .-. . . .-. . .-.-.
'stop /2 Freq
!+ Passband i-<- TranSition... Stopband :
i I band i i
Figure 5-31 Alternate method for deftniпg the desired frequency response of
а low-pass FIR ftlter using the Remez Exchange technique,
Оп the other hand, some software Remez routines assume that we
want Ор and 05 to Ье as small as possible and require us only to define
the desired values of the Hd(m) response as shown Ьу the solid black
dots in Figure 5-31, The software then adjusts the values of the unde-
fined (shaded dots) values of Hd(m) to minimize the error between our
desired and actual frequency response while minimizing Ор and 05' The
filter designer has the option to define some of the Him) values in the
transition band, and the software ca1culates the remaining undefined
Hd(m) transition band values, With this version of the Remez algorithrn,
the issue of most importance becomes how we define the transition
region. We want to minimize its width while, at the same time, mini-
mizing passband and stopband ripple. So exactly how we design an FIR
filter using the Remez Exchange technique is specific to the available ш-
ter design software. Although the mathematics involved in the develop-
men.t of the Remez Exchange method is rather complicated, we don't
have to worry about that hеrе[lб-20], Just remember that the Remez
Exchange design method gives us а Chebyshev-type fi1ter whose actual
frequency response is as close as possible to the desired Hd(m) response
for а given number of fi1ter taps,
То i11ustrate the advantage of the Remez method, the solid curve in
Figure 5-32 shows the frequency response of а 31-tap FIR designed using
this technique. For comparison, Figure 5-32 also shows the frequency
responses of two 31-tap FIR fi1ters for the same passband width using the
Ha/f-Baпd F/R F//ters 197
Filter frequency magnitude responses (dB)
о
Remez (solid)
10
20
--40 .
..
...бо
70
О
0,05'.
0,1'
0,15'.
0,2'.
0,25'.
0,3'.
0,35'. Freq
Flgure 5-32 Frequency response comparison ot three 3 l-tap FIR fllters: Remez,
Chebyshev wlndowed, and Kaiser windowed,
Chebyshev and Кaiser windowing techniques. Notice how the three fH-
ters have rougbly the same stopband sidelobe levels, near the main lobe,
but that the Remez filter has the more desirable (steeper) transition band
roll off,
5.7 Half Band FIR Filters
Тhere's а specialized FIR filter that's proved useful in decimation appli
cations[21 25], Called а half-baпd FIRfilter, its frequency response is sym-
metrical about the 1./ 4 point as shown in Figure 5 33(a), As such, the sum
of Ipass and Istop is 1/2. Тhis symmetry has the beautiful property that the
time domain FIR impulse response has every other fHter coefficient being
zero, except at the peak. тhis enables us to avoid approximately haIf the
number of multiplications when implementing this kind of fi1ter. Ву way
of example, Figure 5 33(b) shows the coefficients for а 31-tap haIf band fH
ter where дf was defined to Ье approximately 1/32 using the Remez
Exchange method. (То preserve further symmetry, the parameters Ор and
05 were specified to Ье equal to each other.)
198 F/п/te /mpu/se Respoпse F//ters
Idealized frequency
response
1,0
'ра.. './4
'./2 Freq
1 + Ор
у
(а) 0,5
о
h(k)
1,0
.
0,6
(Ь)
. .
I ! !
. I , ' .
. . I I ; : I '" .
о ....., .,Т.,---.,Т.,........'i' '--т-'.........'Т' 'Т'.........'-............
2 i 1 3 5 7 9 11,13 15 17 119 21 23 25 27 29 k
, .
.4
0.2
0,6
0.4
+ IН(т)1
1,0 · i i . 1 i ' I ! i 1 i! ! ! I i
0,5 1 -1-1 Т. -,. 1- 1- r t ; 1 -1 -1.
I ,
(с)
3
5 7 9 11 13 15
! .
. . . . .-...........-............................
17 19 21 23 25 27 29 31 Freq
t
'./2
о
'./4
Flgure 5-33 Holf-Bond FIR ftlter: (о) ideolized continuous frequency response; (Ь)
coefficients for о Зl-tор holf-bond FIR fllter; (с) discrete frequency
mognltude response of о Зl-tор holf-bond FIR fllter,
Notice how the altemating h(k) coefficients are zero, so we have to per-
form only 17 multiplications for each filter output sample instead оЕ the
expected 31 multiplications, For the general case оЕ an 5 tap half band FIR
filter, we'll need perform only (5 + 1)/2 + 1 multiplications for each out-
put sample,t Taking the DFТ оЕ those h(k) coefficients gives us the half-
band FIR frequency response shown in Figure 5 33(c).
t Section 10.8 shows а technique to further reduce the number of necessary multiplications
for certain types of FIR filters including ha1f-band filters,
Phase Respoпse о' F/R Filters 199
5.8 Phase Response о' FIR Filters
Although we illustrated а couple of output phase shift examples for our
original averaging FIR fi1ter in Figure 5 10, the subject of FIR phase
response deserves additional attention, One of the dominant features of
FIR fi1ters is their 1inear phase response that we сап demonstrate Ьу
way of example, Given the 25 h(k) FIR fi1ter coefficients in Figure 5
34(а), we сап perform а DFT to determine the fi1ter's Н(т) frequency
response. The norma1ized real part, imaginary part, and magnitude of
о: t . , i
0 r: . "Т+-+-. I I I i I
. .
"'{),5 2 4 6 8 10 12
1 ... .... . .
. . I!I
.
(а)
j ! ! . . .
i i i I .-+--+-Т" , I I ..
· · k
14 16 18 20 22 24
0,75 . Аеаl part 01 Н(т)
0,5 [] lmaginary part 01 Н(т)
0,25
(Ь) О .
т
"'{),25 22 24 26 28 ЗО
"'{),5
"'{),75
1,O
(с)
! \Н(т)\
1,0.. ...
'''.... I I:
О 75 . i I . I ': 1: .
,5 : I I ! I i 1 I
0,25 ..!' i . I ! '
о
о 2 4 6 8 10 12 14 16 18 20 22 24 26 28 ЗО
16'8 '8
i . . 2 епсу = 128= 8
. ,..щ... #,........,..,...-.
т
Passband
;
>---;
;
Flgure 5-34 FIR fllter frequency response Н(т): (о) h(k) fllter coefficlenfs; (Ь) reol
ond Imoglnory ports of Н(т); (с) mognltude of Н(т),
200 F/п/te /тpи/se Respoпse FiJters
Н(т) are shown in Figures 5-34(Ь) and 5-34(с), respectively.t Being сот-
plex values, each Н(т) sample value сап Ье described Ьу its real and
imaginary parts, or equivalent1y, Ьу its magnitude I Н(т) I and its
phase Н,,(т) shown in Figure 5-35(а).
(а)
j Н.(т) /п dвgrees
200
.,
100 .......
2 4 ., 12 14 16 18 20 22 24 26 28 ЗО
° f '. · "'. ...
100 ".'8, 6 8 10 .'.... / '...., / '.'... 8'8,
.,. ,., .'. .'. ....
OO ·
D/sconffnUity induced Ьу
arctangent caJculation твthod
m
27rf'
(900)
(Ь)
168,750
Н.(О) = 00
Не(6) 202,50
(157,5")
101,250
OO
t Н.(т) ,п dвgrees
100
2 4 6 8 10 12 14 16 18 20 22 24 26 28 ЗО
О . , 1 I 1 I 1 I I I I I I I I I I
.,
.,
.,
.,
.,
.,
.,
.,
.'.
'.
'.
'.,
.,
.,
.
..
m
100
(с)
200
--400
500
LвgШmаtе
disconffnuities
'пН.(т)
--600
Passband= i
F/gure 5-35 F/R filter phase respoпse Н,,(т) /п degrees: (а) calculated Н,,(т);
(ь) polar plot of H,,(т)'s flrst ten phase angles in degrees;
(с) actual Н,,(т),
t Anу ОРТ size greater than the h(k) width of 25 is sufficient to obtain Н(т), The h(k)
sequence was padded with 103 zeros to take а 128-point ОРТ resulting in the Н(т) sample
values in Figure 5 34,
...
.. ,\
Phase Respoпse о( F/R$h"ters - 201 .
[- . ,
t-'i'-:'
The phase of а complex quantity is, of course, the arctangent (the
imaginary part divided Ьу the real part, or IZJ = tan l(imag/real), Тhц ,_, \,)'
the phase of Н",(т) is determined from the samples in Figure 5 34(b). "
The phase response in Figure 5 35(a) certainly looks linear over select-
ed frequency ranges, but what do we make of those sudden jumps, or dis-
continuities, in this phase response? If we were to plot the angles of Н",(т)
starting with the т = О sample оп а polar graph, using the nonzero real
part of Н(О), and the zero-valued imaginary part of Н(О), we'd get the
zero angled Н",(О) phasor shown оп the right side of Figure 5-35(Ь),
Continuing to use the real and imaginary parts of Н(т) to plot additional
phase angles resu1ts in the phasors going clockwise around the circle in
increments of 33,750, It's at the Н",(6) that we discover the cause of the
first discontinuity in Figure 5 35(a), Taking the real and imaginary parts
of Н(6), we'd plot our phasor oriented at an angle of 202.50, But Figure
5-35(а) shows that Н",(6) is equal to 157.50, The problem 1ies in the soft-
ware routine used to generate the arctangent values plotted in Figure
5-35(а). The software adds 3600 to any negative angles in the range of
1800 > IZJ 3600, i,e., angles in the upper half of the circle, This makes IZJ
а positive angle in the range of 00 < IZJ $; 1800 and that's what gets plotted,
(This apparent discontinuity between Н",(5) and Hi6) is called phase wrap-
piпg.) 50 the true Н",(6) of 202.50 is converted to а +157.50 as shown in
parentheses in Figure 5-35(Ь), If we continue our polar plot for additional
Н",(т) values, we'll see that their phase angles continue to decrease with
an angle increment of 33,750, If we compensate for the software's behav-
ior and plot phase angles more negative than 1800, Ьу uпwrappiпg the
phase, we get the true Н",(т) shown in Figure 5-35(c),t Notice that Н",(т)
is, indeed, linear over the passband of Н(т), It's at Н",(17) that our partic-
ular Н(т) experiences а polarity change of its real part while its imaginary
part remains negative-----this induces а true phase angle discontinuity that
really is а constituent of Н(т) at т = 17, (Additional phase discontinuities
occur each time the real part of Н(т) reverses polarity, as shown in Figure
5-35(с),) The reader mау wonder why we care about the linear phase
response of Н(т), The answer, an important one, requires us to introduce
the notion of group delay.
Group delay is defined as the derivative of the phase with respect to fre-
quency, or G = &:J/ N For FlR filters, then, group delay is the slope of the
Н",(т) response curve, When the group delay is constant, as it is over the
t When plotting fi1ter phase responses, if we encounter а phase angle sample '" that looks
like an unusua1 discontinuity, it's а good idea to add 3600 to '" when '" is negative, or 3600
when '" is positive, to see if that compensates for any software anomalies.
202 Fiпite /тpu/se Respoпse Fiiters
раssЪand of all FIR filters having symmetrical coefficients, all frequency
components of the filter input signal are delayed Ьу an equal amount of
time G before they reach the filter's output, This means that по phase dis
tortion is induced in the filter's desired output signal, and this is crucial in
communications signals, For amplitude modulation (АМ) signals, con
stant group delay preserves the time waveform shape of the signal's mod
ulation envelope. That's important because the modulation portion of an
АМ signal contains the signal' s information. Conversely, а nonlinear phase
will distort the audio of АМ broadcast signa1s, blur the edges оЕ television
video images, blunt the sharp edges оЕ received radar pulses, and increase
data errors in digital communication signals. (Group delay is sometimes
called eпve/ope delay because group delay was originally the subject оЕ
analysis due to its affect оп the envelope, or modulation signal, of атрН-
tude modulation АМ systems.) Of course we're not realIy concerned with
the group delay outside the passband because signal energy outside the
раssЪand is what we're trying to elirninate through fi1tering.
Over the passband frequency range for an S tap FIR digital filter, group
delay has Ьееn shown to Ье given Ьу
G = (5 l)t s
2 '
(5-23)
where t s is the sample period (l/fs).t This group delay is measured in sec-
onds, Eliminating the t s factor in Eq. (5 23) would change its dimensions
to samples, The value G, measured in samples, is always an integer for
odd tap FIR filters, and а non-integer for еуеn tap-filters,
Although we used а 128-point DFТ to obtain the frequency responses in
Figures 5 34 and 5-35, we could just as well have used N = 32-point or
N = 64-point DFТs. These smaller DFТs give us the phase response curves
shown in Figure 5-36(а) and 5-36(Ь). Notice how different the phase
response curves are when N = 32 in Figure 5 36(a) compared to when
N = 128 in Figure 5-36( с). The phase angle resolution is much finer in Figure
5 36(c). The passband phase angle resolution, or increment.10, is given Ьу
.10
c . 3600
N
(5-24)
t As derived in Section 3,4 of reference [15], and Section 5,7,3 of reference [19],
Phase Respoпse of F/R F//ters 203
where N is the number оЕ points in the DFТ. 50, for our 5 = 25 tap filter in
Figure 5 34(a), G = 12, and Ае is equal to 12 .3600/32 = 1350 in Figure
5 36(a), and de is 33.750 in Figure 5 36(c). If we look carefully at the sam
ple values in Figure 5 36(a), we'11 see that they're а11 included within the
samples in Figures 5 36(b) and 5 36(c),
Let's conclude this FIR phase discussion Ьу reiterating the meaning of
phase response. Тhe phase, or phase delay, at the output оЕ an FIR filter is
Н 0 (т) in degrees
(а)
100 . . . .
50 / //\ // ']/ ..
о · . . . . .
50' . . '//\ ;/ '\'I. .' / . ..... .... т
1OO .... \ ../..\/ .!.. о о,, . . '.' .,.
150 .. I
.
200
(Ь)
2001 Н 0 (т) in degrees .
:(\ "' c"': f;XiA!frjAf A ·
.= Vl'i# ' · 1. m
200 f
(с)
200 н.. (т) in degrees
.. .
150 . . .. . . .................
'1 12&-point OFТ . , . "1:' 1'1
50 · · t 1. ; f ' M ' J ' :. · ,
· · . н . . N 8 · \ .. .. \ ·
о, :' .". w . ... · :.+:. · :. :. · .... · ·
50 f 8 J . t J ': ' ... J J.\ ,. , .. . . "... т
. . . ' . . , ". .. \
100; 1. , \ :.. ... ,
150 · ' . . ... о. . . . .... . " " ... .. .....
. ..
200
Flgure 5-36 FIR fllter phase response Н 0 (т) In degrees: (а) ealeulated uslng а
32-polnt DFТ; (Ь) uslng а 64-polnt DFТ; (е) using а 128-point DFТ,
204 Fiп/te /mpu/se Respoпse Fi/ters
ТаЫе 5-2 Values Used lп Eq, (5-25) for the
Frequeпcy './32
DFT size, N Iпdex m H./mf/N)
32 1 1350
64 2 1350
128 4 1350
the phase of the first output sample relative to the phase of the filter's first
inpиt sample, Over the passband, that phase shift, of course, is а linear
function of frequency. Тhis will Ье Пuе only as long as the filter has sym
metrical coefficients, Figure 5-10 is а good illustration of an FIR filter's
output phase delay.
For FIR filters, the output phase shift measured in degrees, for the pass
band frequency f = mf/N, is expressed as
m.C.3600
phase delay= H,,(m..€ jN)= m. = .
N
(5--25)
We сап illustrate Eq. (5-25) and show the relationship between the phase
responses in Figure 5 36, Ьу considering the phase delay associated with
the &equency off/32 in ТаЫе 5 2. Тhe subject of group delay is described
further in Appendix F where ап example of envelope delay distortion,
due to а filter's nonlinear phase, is illustrated.
5.9 А Generic Description о' Discrete Convolution
Although convolution was originally an analysis tool used to prove соп-
tinuous signal processing theorems, we now know that convolution
affects every aspect of digital signal processing. Convolution influences
our results whenever we analyze or filter anу finite set of data samples
from а linear time invariant system. Convolution not оnlу constrains
DFТs to Ье just approximations of the continuous Fourier transform; it is
the reason that discrete spectra are periodic in the frequency domain. It's
interesting to note that, a1though we use the process of convolution to
implement FIR digital filters, convolution effects induce frequency
response ripple preventing us from ever building а perfect digital fi1ter. Its
influence is so pervasive that, to repeal the law of convolution, quoting а
А Geпeric Descriptioп о( Discrete Coпvoiutioп 205
N-1
Y j = I Pk"Oj-k , or
k=O
УО 00 °N-1 °N_2 01 РО
У1 01 00 °N_1 02 Р 1
У 2 02 01 00 ОЗ Р 2
=
Y N - 1
о N-1 О N-2 ОN_З
00
P N - 1
Theorem: if
Pj f-- DFТ --7 Ап ,
О j f-- DFТ --7 Вп , and
Yj f-- DFТ --7 сп
then,
Сп = N'An"8n
Flgure 5-37 One very efficlent. but perplexlng. way of definlng convolutlon,
phrase from Dr. Who, would "unravel the entire eausal nexus" of digital
signal proeessing.
Convolution has always Ьееп а somewhat difficu1t eoneept for the
beginner to grasp. That's not too surprising for several reasons.
Convolution's effeet оп diserete signal proeessing is not intuitively obvi
ous for those without experienee working with diserete signals, and the
mathematics of eonvolution does seem а litt1e puzzling at first. Moreover,
in their sometimes justified haste, тanу authors present the eonvolution
equation and abrupt1y start using it as an analysis tool without explaining
its origin and meaning. For example, this author опее eneountered what
was eal1ed а tutorial article оп the FFr in а professional journal that pro
eeeded to define coпvolutioп merely Ьу presenting something like that
shown in Figure 5 37 with по further explanation!
Unfortunately, few beginners еan gain an understanding of the eonvo
lution proeess from Figure 5 37 alone. Here, we avoid this dilemma Ьу
defining the proeess of eonvolution and gent1y proeeed through а eouple
of simple eonvolution examples. We eonelude this ehapter with а diseus
sion of the powerful eonvolution theorem and show why it's so useful as
а qualitative tool in diserete system analysis.
206 Fiп/te /тpu/se Respoпse F//ters
5.9.1 Discrete Convolution in the Time-Domaln
Discrete convolution is а process, whose input is two sequences, that pro
vides а single output sequence, Convolution inputs сап Ье two time
domain sequences giving а time--domain output, or two frequency domain
input sequences providing а frequency-domain result. (Although the two
input sequences must both Ье in the same domain for the process оЕ con
volution to have any practical meaning, their sequence lengths need not Ье
the same,) Let's say we have two input sequences h(k) оЕ length Р and x(k)
оЕ length Q in the time domain, Тhe output sequence у(п) оЕ the convolu
tion оЕ the two inputs is defined mathematically as
P+Q 2
у(п)= )(k)x(n k),
k O
(5 26)
Let's examine Eq. (5-26) Ьу way оЕ example using the h(k) and x(k)
sequences shown in Figure 5-38. In this example, we сап write the terms
for each у(п) in Eq. (5-26) as
у(О) = h(O)x(O O) + h(1)x(O l) + h(2)x(0 2) + h(3)x(0 3) + h(4)x(0 4) + h(5)x(0 5),
у(l) = h(O)x(l O) + h(1)x(1 I) + h(2)x(I 2) + h(з)х(I З) + h(4)x(I 4) + h(5)x(1 5),
у(2) = h(0)x(2 0) + h(l)x(2 1) + h(2)x(2 2) + h(3)х(2 З) + h(4)x(2 4) + h(5)x(2 5),
у(З) = h(O)x( O) + h(l)x( I) + h(2)x( 2) + h(3)x( 3) + h(4)x( 4) + h(5)x( 5),
у(4) = h(0)x(4---0) + h(l)x(4---1) + h(2)x(4---2) + h(з)х(4---3) + h(4)x(4---4) + h(5)x(4---5),
and
у(5) = h(O)x(S--О) + h(l)x(S--I) + h(2)x(S--2) + h(3)x(S--З) + h(4)x(S--4) + h(5)x(,5--.5).
(5 27)
With Р = 4 and Q = 3, we need evaluate only 4 + 3 1 = 6 individual у(п)
terms, Because h(4) and h(5) are zero, we сап eliminate some оЕ the terms
in Eq, (5-27) and evaluate the remaining x(n k) indices giving the follow-
ing expressions for у(п) as
у(О) = h(O)x(O) + h(l)xH) + h(2)x( 2) + h(3)x( 3) ,
y(l) = h(O)x(l) + h(l)x(O) + h(2)xH) + h(3)x( 2) ,
у(2) = h(0)x(2) + h(l)x(l) + h(2)x(0) + h(3)xH) ,
у(3) = h(0)x(3) + h(1)x(2) + h(2)x(1) + h(3)x(0) ,
у(4) = h(0)x(4) + h(1)x(3) + h(2)x(2) + h(3)x(1) ,
and
у(5) = h(0)x(5) + h(1)x(4) + h(2)x(3) + h(3)x(2) . (5 28)
А Geпeric Descriptioп of Discrete coпvo/ut/oп 207
2
1
О
. . .
J х(К)
З
2
1
О
.
j h(К)
...
2 1 О 1 2 З 4 5 6 k
.
, :
, i
...
З 2 1 О 1 2 З 4 5 6 k
(а)
(Ь)
Flgure 5-38 Convolution example input sequences: (а) flrst sequence h(k) of
length Р = 4; (Ь) second sequence x(k) of length Q = З,
. х(К)
з'
2
1
О
x( К)
.
зА
2
1
О
.
! .
i
...
.
.
.
, ; ...
2 1 О 1 2 З 4 5 6 k
З 2 1 О 1 2 З 4 5 6 k
(а)
(Ь)
Flgure 5-39 Convolution example input sequence: (а) second sequence
x(k) of length З; (Ь) reflection of the second sequence about
the k = О Index,
Looking at the indices of the h(k) and x(k) terms in Eq. (5-28), we see two
very important things occurring, First, convolution is merely the summa-
tion of а series of products so the process itself is not very complicated,
Second, notice that, for а given у(п), h(k)'s index is increasing as x(k)'s
index is decreasing. Тhis fact has led many authors to introduce а new
sequence хНе), and use that new sequence to graphically illustrate the
convolution process. Тhe x( k) sequence is simply our original x(k) ref1ect-
ed about the О index of the k axis as shown in Figure 5-39. Defining x( k)
as such enables us to depict the products and surnmations of Eq. (5-28)'s
convolution as in Figure 5-40; that is, we can now align the x( k) samples
with the samples of h(k) for а given п index to calcиlate у(п), As shown
in Figure 5-40(а), the alignment of h(k) and x(п k), for п = О, yields
у(о) = 1. Тhis is the result of the first line in Eq. (5-28) repeated оп the
right side of Figure 5-40(а). Тhe calculation of y(l), for п = 1, is depicted
208 Fiп/te /mpu/se Respoпse Filters
(а)
1 r h(1c) ....
о I I I I I I : I I I 11"-
...... --з --2 1 О 1 2 3 4 5 6 k
! >«(}.OIC) .
1 . < .
о I I I J I I I I I
...... --з --2 1 О 2 4 5 6 k
(Ь)
1 r h(1c) ....
О I I I I I i i I I I
4 --з --2 1 О 1 2 3 4 6 k
3 1 >(H" ·
2 ' .
1 j .
о I I I I I I I I I I 1"
...... --з --2 1 О 1 2 3 4 5 6 k
(с)
' r h(1c) ....
о I I I I I I I j I
...... --з --2 1 О 1 2 3 4
3 + >(2-1<) .
!! I I I f I : I
...... --з 2 1 О 1 2 3
(d)
' I h(J(J ....
о I I I I i ; I i I
...... --з 2 1 О 1 2 3 4
3 1 >(3-1<)
l I I I I I
...... --3--2 1 О
.
.
1 .
t i I I I 1"-
123456.
(е)
1 r h(k) · I!I
. ,
о I I I I I I I I I I 1'"
--4 --з --2 1 О 1 2 3 4 5 6 k
3 + >(....1<) .
lt I I I I I I ; r I ,.
...... --з --2 1 О 1 2 3 4 5 6 k
(1)
1 r h(k) .
О I I I I I I i I I I I
........з --2 1 О 1 2 3 4 5 6 k
3 + >(5-1<) .
1:t I I I I I i 1 1"
...... --з 2 1 О 3 4 5 6 k
convotutJon
wIIh п_О
=с>
соnvoluUo"
wlthп.1
=с>
I ..-
5 6 k
convolutlon
wlthп-2
с::с>
t ио) 8 1>(0)>(0) + h(11)( 1) + 1>(2)>(--2) + 1>(3)>(--3)
;t/0+0+081
.
О 1 I I I I I I I I ..
012345678 п
i х1) 8 1>(0)>(1) + 1>(1)>(0) + 1>(2)>( 1) + 1>(3)>( 2)
6 _2+1.0+0_3
4 /'
.
2 .
.
О I i I I I I I I I ..
012345678 п
f и2) 8 1>(0)>(2) + 1>(1)>(1) + 1>(2)>(0) + 1>(3)>( 1)
6 . .З+2+1+0.6
4
.
2
.
0 1
О
i I I I I I I
2345678
..
п
'
6 k
'1"
6 k
convoIu1ion
wIIhп_З
с::с>
f из) 8 1>(0)>(3) + 1>(1)>(2) + 1>(2)>(1) + 1>(3)>(0)
6 . . .О+З+2+1.6
4
2 · :
. i
О I i j I I I I I ..
О 234 5 678 п
f и4) = 1>(0)>(4) + 1>(1)>(3) + 1>(2)>(2) + 1>(3)>(')
6 .. #" ..0+0+3+2_5
.
4
2. ,
.
0 1
О
i I I I I I I
2345678
..
п
convotutlon
wlthп_4
=с>
convolution
wIIh п_5
=с>
! и5) = 1>(0)>(5) + 1>(1)>(4) + 1>(2)>(3) + 1>(3)>(2)
6 .. ..0+0+0+3_3
4 · /
. , .
2
О 1 I I I I I I I ..
02345 678 п
Flgure 5-40 Graphlcal deplctlon of the convolutlon of h(k) and x(k) in Figure &38,
in Figurе-З4(Ь) where x(п k) is shifted оnе element to the right, resulting
in у(1) = 3, We continue this x(п k) shifting and incrementing п until we
arrive at the last nonzero convolution resu1t of у(5) shown in Figure
5-40(f), 50, performing the convolution of h(k) and x(k) comprises
А Geпer/c Descript/oп о' Discrete coпvo/ut/on 209
Step 1: plotting both h(k) and x(k),
Step 2: flipping the x(k) sequence about the k = О value to get x( k),
Step 3: summing the products of h(k) and x(O k) for аН k to get у(О),
Step 4: shifting x( k) one sample to the right,
Step 5: summing the products of h(k) and x(l k) for all k to get у(l), and
Step 6: shifting and summing products until there's по overlap of h(k)
and the shifted x(n k), in which case all further y(n)s are zero and
we're done.
Тhe full convolution of our h(k) and x(k) is the у(п) sequence оп the right
side of Figure 5-40(f). We've scanned the x( k) sequence across the h(k)
sequence and summed the products where the sequences overlap. Ву the
way, notice that the у(п) sequence in Figure 5-40(f) has six elements where
h(k) had а length of four and x(k) was of length three. In the general case,
if h(k) is of length Р and x(k) is of length Q, the length of у(п) will have а
sequence length of L, where
L = (Р + Q 1) .
(5-29)
At this point, it's {аи for the beginner to ask, "ок, so what? What does
this strange convolution process have to do with digital signal process-
ing?" Тhe answer to that question lies in understanding the effects of the
convolution theorem.
5.9.2 The Convolution Тheorem
Тhe convolution theorem is а fundamental constituent of digital signal pro-
cessing, It impacts our results anytime we filter or Fourier transform dis-
crete data. То see why this is true, let's simplify the notation of Eq. (5-26)
and use the abbreviated form
у(п) = h(k) * x(k) ,
(5-30)
where, again, the "*" symbol mеаns convolution. Тhe convolution theo-
rem mау Ье stated as follows: If two time-domain sequences h(k) and x(k)
have DFТs of Н(т) and Х(т), respectively, then the OFТ of h(k) *x(k) is the
product Н(т) .Х(т). Likewise, the inverse DFТ of Н(т) .Х(т) is h(k) *x(k).
We сап represent this relationship with the expression
DFr
h(k)*x(k) Н(т).Х(т).
IDFr
(5-31)
21 О Fiп/te /три/sе Respoпse Filters
Sequence x(k)
Fourier
transform pair
\
Sequence h(k)
Time
domain
t
Х(т)
Frequency
domain
Figure 5-41 Relationships of the convolution theorem,
Equation (5-31) tells us that two sequences resulting from h(k) * x(k) and
Н(т) . Х(т) are Fourier trапsfопn pairs, 50, taking the DFТ of h(k) * x(k)
always gives us Н(т) .Х(т). Likewise, we can detennine h(k) *x(k) Ьу taking
the inverse DFТ of Н(т) .Х(т). The important point to learn from Eq. ( 31)
is that convolution in the tirne domain is equivalent to multipJication in the
frequency domain, (We won't derive the convolution theorem here ЬесаUБе
it's derivation is readily available to the interested rеаdеr[2б-...29],) То help us
appreciate this principle, Figure 5 41 sketches the relationship between con-
volution in the tirne domain and multipJication in the frequency domain.
We can easily illustrate the convolution theorem Ьу taking 8-point DFТs
of h(k) and x(k) to get Н(т) and Х(т), respectively, and listing these values as
in ТаЫе 5 3, (Of course, we have to pad h(k) and x(k) with zeros, so they both
have lengths of 8 to take 8-point DFТs,) Tabulating the inverse DFТ of the
product Н(т) . Х(т) allows us to verify Eq. (5 31), as listed in the last two
columns of ТаЫе 5-3, where the acronym ШFТ again теans inverse DFТ,
The values from ТаЫе 5 3 are shown in Figure 5 42. (For simplicity, only the
magnitиdes of Н(т), Х(т), and Н(т) . Х(т) are shown in the figure,) We
need to Ьесоте comfortable with convolution in the Нте domain ЬесаUБе,
А Geпer/c Descr/ptloп о' D/screfe Coпvo/uf/oп 211
ТаЫе 5-3 Convo/ution Values of h(k) and x(k) from Figure 5-38,
Iпdex DFT о! OFТ о! IDFT о!
korm h(k) x(k) h(k) = Н(m) x(k) = Х(m) Н(m) , Х(m) Н(m)' Х(m) h(k) 'x(k
О 1 1 4,0 + jO,O 6,0 + jO,O 24,0 + jO,O 1,0 + jO,O 1
1 1 2 1,00 j2.41 2,41 j4,41 ,24 j10,24 3,0 + jO,O 3
2 1 3 О 2,O j2,O О 6,0 + jO,O 6
3 1 О 1,00 jO.41 O,41 + j1.58 0,24 + j1,75 6,0 + jO,O 6
4 О О О 2,0 + jO.o О 5,0 + jO,O 5
5 О О 1,00 + jO,41 O,41 j1.58 0,24 j1,75 3,0 + jO,O 3
6 О О О 2.()() + j2,oo О 0,0 + jO,O О
7 О О 1,00 + j2,41 2,41 + j4,41 ,24 + j10,24 0,0 + jO,O О
S"QU""ce >(11)
:1. .
.
о ,--+--+--- ' I!I II!I ,,,,,,,
О 1 2 3 4 5 8 7 k
S"qU""C" h(1I)
o l .
о ' ' I!I I!I.
О 1 2 З .. 5 в 7 k
хп) = h(1I).>(1I)
: ,! . I!' .
Co"volullon 1" th"
time domain
. .
1) 1 I I f . ......
о 1 .. 5 6 :, п
=Ш
,,1 Frequency
, domaln
: L 'IН(m)l
. .
2 . i
1 ; . . :
О .+, ..----+..
о 1 2 3 .. 5 е 7 т
"-
[; ;: : :.
о .. .
о 1 2 3 .. 5 в 7 т
Multiplication in the
frequency domain
: h I lН(т)'Х{т)1
10 .
о \.. . ..,
............""""t'"'-.
О 1 2 3 .. 5 в 7
Flgure 5-42 Convolutlon relatlonshlps of h(k), x(k), Н(т), and Х(т) from Flgure 5-38,
212 Fiпite /mpulse Respoпse Filters
Sequence x(k)
Fourier
transform pair
\
Sequence h(k)
Time
domain
t
t
Х(т)
Frequency
domain
Convolution in the frequency
domain = Н(т).Х(т)
Flgure 5-43 Relatioпshlps of the convolution theorem related to multiplication in
the time domain,
as we've 1eamed, it's the process used in FIR ffiters. As detai1ed in Section
5,2, we perform discrete time domain FIR ffitering Ьу conv01ving an input
sequence, х(п) say, with the impи1se response x(k) of а ffiter, and, for FIR т-
ters that impu1se response happens to a1so Ье the ffiter's coefficients,t The
resu1t of that conv01ution is а ffitered time--domain sequence whose spec
trum is modified (mu1tiplied) Ьу the filter's frequency response Х(т),
Section 10.10 describes а c1ever scheme to perform FIR тtering efficient1y
using the FFТ a1gorithm to imp1ement conv01ution.
Because of the duality of the conv01ution theorem, we cou1d have
swapped the time and frequency domains in our discussion of conv01u-
tion and mu1tiplication being а Fourier transform pair, This means that,
similar to Eq, (5 31), we can a1so write
t As we'lI see in Chapter 6, the coefficients used for an infinite impulse response (IIR) fi!ter
are not equa! to that filter's impulse response.
А Geпeric Descriptioп of Discrete Coпvoiutioп 213
OFТ
h(k). x(k) Н(т)" Х(т)
ЮFТ
(5 32)
50 the convolution theorem сап Ье stated more generally as Convolution
in опе domain is equivalent to multiplication in tfre other domain. Figure 5 43
shows the relationship between multiplication in the time domain and
convolution in the frequency domain. Equation (5 32) is the iundamental
relationship used in the process of windowing time domain data to
reduce DFТ leakage, as discussed in 5ection 3.9.
5.9.3 Applying the Convolution Theorem
Тhe convolution theorem is useful as а qualitative tool in predicting the
affects of different operations in discrete linear time-invariant systerns. For
example, many authors use the convolution theorem to show why period
ic sampling of continuous signals results in discrete samples whose spec
tra are periodic in the frequency domain, Consider the real continuous
time domain waveform in Figure 5-44(а), with the one sided spectrum of
bandwidth В, Being а real signa1, of course, its spectrum is symmetrica1
about О Нz. (In Figure 5 44, the large right-pointing shaded arrows repre
sent Fourier transform operations,) 5ampling this waveform is equiva
lent to multiplying it Ьу а sequence of periodically spaced impulses,
Figure 5 44(b), whose values are unity. If we say that the sampling rate
is/. samples/second, then the sample period t. = 1/1. seconds, The
result of this multiplication is the sequence of discrete time domain
impulses shown in Figure 5 44(c). We can use the convolution theorem
to help us predict what the frequency domain effect is of this multipli-
cation in the time domain. From our theorem, we now realize that the
spectrum of the time domain product must Ье the convolution of the
original spectra. Well, we know what the spectrum of the original con
tinuous waveform is, What about the spectrum of the time domain
impulses? It has Ьееп shown that the spectrum of periodic impulses,
whose period is t. seconds, is also periodic impulses in the frequency
domain with а spacing of/. Hz as shown in Figure 5 44(b) [30].
Now, аН we have to do is convolve the two spectra. In this case, con
volution is straightforward because both of the frequency domain func
tions are symmetrica1 about the zero Нz point, and flipping опе of them
about zero Hz is superfluous. 50 we merely slide опе of the functions
across the other and plot the product of the two. Тhe convolution of the
original waveform spectrum and the spectral impulses results in replica
tions of the waveform spectrum every fs Hz, as shown in Figure 5-44(с).
214 F/п/te /mpu/se Respoпse Filters
(а) .., .. <? Lb ..
о Tlme --8:' 0'8 Frвquency
{}
(с)
о /. Тlт.
'Jlll111111L <? "UlLL
0'\ Тlт. FT 21/. 1II. О 111. 211s Fr.qu.ncy
/.
V V
..
2f. f. О 2 Frequency
(Ь)
Figure 5-44 Using convolution to predict the spectral replication effects of periodic
sampling,
This discussion reiterates the fact that the DFТ is always periodic with а
period of fs Hz.
Here's another example of how the convolution theorem сап соте in
handy when we try to understand digital signal processing operations.
This author опсе used the theorem to resolve the puzzling result, at the
time, of а triangular window fиnction having its first freqиency
response null at twice the frequency of the first null of а rectangular
window function. Тhe question was "Н а rectangular time domain func
tion of width Т has its first spectral null at l/Т Hz, why does а triangu
lar time-domain function of width Т have its first spectral null at 2/Т
Hz?" We сап answer this question Ьу considering convolution in the
time domain,
Look at the мо rectangular time-domain functions shown in Figures
5 45(a) and 5 45(b). Н their widths are each Т seconds, their spectra are
shown to have nu11s at l/Т нz as depicted in the frequency domain func
tions in Figures 5-45(а) and 5 45(b). We know that the frequency magnitude
А Geпeric Descriptioп of Discrete Coпvoiutioп 215
responses will Ье the absolute value of the classic sin(x)/x function,t If we
convolve those two rectangu1ar time-domain functions of width Т, we'll get
the triangu1ar function shown in Figure 5 45(c), Again, in this case, flipping
оnе rectangular function about the zero time axis is unnecessary, То соn-
volve them, we need only scan оnе function across the other and dеtепninе
(а)
ф
v
ф
v
, ф
(Ь)
(с)
(d)
Т12 О Т12
Time
1fT О 1fT
Freq
1fT О 1fT
Freq
v
1fT О НТ
Freq
( :tI\
'lJТ О 21Т Fr.q
Flgure 5-45 Uslng convolutlon to show that the Fourler transform of а triangular
function has its first null at twlce the frequency of the Fourler transform
of а rectangular function,
t The sin(x)/x function was introduced in our discussion of window functions in Section 3.9
and is covered in grea ter detail in Section 3,13,
216 F/п/te /тpи/se Respoпse F//ters
the area of their overlap, The time shift where they overlap the most мр-
реns to Ье а zero time shift. Thus, our resultant convolution has а peak at а
time shift of zero seconds because there' s 100 percent overlap, If we slide one
of the rectangular functions in either direction, theconvolution decreases
linearly toward zero, When the time shift is Т /2 seconds, the rectangular
functions have а 50 percent overlap, тhe convolution is zero when the time
shift is Т seconds-----that' s when the two rectangular functions сеаБе to over
lap,
Notice that the triangular convolution result has а width of 2Т, and
that's really the key to answering our question, Because convolution in
the time domain is equivalent to multiplication in the frequency
domain, the Fourier transform magnitude of our 2Т width triangular
function is the I sin(x)/x I in Figure 5-45(а) times the I sin(x)/x I in
Figure 5-45(Ь), or the (sin(x)/x)2 function in Figure 5-45(с), If а triangu-
lar function of width 2Т has its first frequency domain null at l/Т Hz,
thел the same function of width Т must have its first frequency null at
2/Т Hz as shown in Figure 5-45(d), and that's what we set out to show,
Cornparison of Figure 5-45(с) and Figure 5-45(d) illustrates а funda
mелtаl Fourier transform property that compressing а function in the
time domain results in an expansion of its corresponding frequency
domain representation,
References
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ОрНmиm FIR Linear Phase Digital Fi1ters," IЕЕЕ Trans, оп Circuit Theory, Vol,
СТ-20, November 1973,
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Signal Proc" Уоl, ASSP 25, No, 4, August 1977,
218 Flпlte /mpu/se Respoпse FlJters
[24] Crochiere, R. Е" and Rabiner, L, R. "Decimation and Interpolation of Digital
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[26] Oppenheim, А. V" Wil1sky, А. 5" and Young, 1, Т, Signals and Systeтs,
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Prentice-Hall, Englewood Cliffs, New Jersey, 1983, рр, 201,
CHAPTER SIX
Infinite Impulse
Response Filters
Infinite impulse response (IIR) digital filters are fundamentally different
from FIR filters because practical ПR fi1ters always require feedback.
Where FIR filter output samples depend only оп past input samples,
each ПR filter output sample depends оп previous input samples and
previous filter output samples, IIR filters' тeтory of past outputs is both
а blessing and а curse. Like аН feedback systems, perturbations at the
IIR fi1ter input could, depending оп the design, cause the fi1ter output to
Ьесоmе unstable and oscil1ate indefinitely, This characteristic of possi
Ыу having an infinite duration of nonzero output samples, even if the
input becomes аН zeros, is the origin of the phrase infinite iтpu/se
response. It's interesting at this point to know that, relative to FIR filters,
IIR filters have more complicated structures (block diagrams), are
harder to design and analyze, and do not have linear phase responses,
Why in the world, then, would anyone use an IIR filter? Because they
are very efficient, IIR filters require far fewer multiplications per filter
output sample to achieve а given frequency magnitude response, From
а hardware standpoint, this means that IIR fi1ters сап Ье very fast, allow-
ing us to build real-time IIR fi1ters that operate over much higher sam
ple rates than FIR fi1ters,+
То illustrate the utility of IIR filters, Figure 6 1 contrasts the frequency
magnitude responses of what's called а fourth order low pass ПR filter
and the 19-tap FIR fi1ter of Figure 5 19(b) from Chapter 5. Where the 19
tap FIR fi1ter in Figure 6 1 requires 19 multiplications per filter output
sample, the fourth order IIR fi1ter requires only 9 mu1tiplications for
each filter output sample. Not only does the IIR filter give us reduced
t At the end of this chapter, we briefly compare the advantages and disadvantages of IIR fil
ters relative to FIR filters,
219
220 /пfiпite /три/5е Re5poпse Fllter5
IH(т)1 for а 19-tap FIR filter
о
's/B
Freq(т)
"..........,
, s/B
Flgure 6-1 Comparison of the frequency magnltude responses of а 19-tap
low-pass FIR filter and а 4th-order low-pass IIR filter,
passband ripple and а sharper filter ro11-off; it does so with less than
half the multiplication workload of the FIR filter,
Reca11 from Section 5.3 that, to force an FIR fi1ter's frequency response
to have very steep transition regions, we had to design an FIR fi1ter with а
very long impulse response. Тhe longer the impulse response, the more
ideal our fi1ter frequency response wi11 Ьесоmе, From а hardware stand-
point, the maximum number of FIR fi1ter taps we can have (the length of
the impulse response) depends оп how fast Our hardware can perforrn the
required number of multiplications and additions to get а filter output
value before the next fi1ter input sample arrives, IIR fi1ters, however, can
Ье designed to have impulse responses that are longer than their number
of taps! Тhus, IIR filters сап give us much better fi1tering for а given nшn
ber of multiplications per output sample than FIR filters. With this in
mind, let's take а deep breath, flex our mathematical muscles, and leam
about IIR filters.
6.1 An Introduction to Infinite Impulse
Response Filters
IIR filters get their namе from the fact that some of the fi1ter's previous
output samples are used to calculate the current output sample. Given а
finite duration of nonzero input values, the effect is that an IIR fi1ter could
have а infinite duration of nonzero output samples. 50, if the IIR fi1ter's
input suddenly becomes а sequence of а11 zeros, the fi1ter's output could
conceivably remain nonzero forever. Тhis peculiar attribute of IIR filters
cornes about because of the way they're щlizеd, i,e., the feedback struc-
ture of their delay units, multipliers, and adders. Understanding IIR filter
Ап /пtrodисtiоп to /пflпite /трИ/5е Respoпse Filters 221
structures is straightforward if we start Ьу recalling the bиilding blocks о!
an FIR filter. Figиre 6 2(a) shows the now familiar structure о! а Нар FIR
digital filter that implements the time-domain FIR eqиation
у(п) = h(O)x(n) + h(l)x(n l) + h(2)x(n 2) + h(3)x(n 3) + h(4)x(n--4). (6-1)
Althoиgh not specifically called oиt as sиch in Chapter 5, Eq. (6-1) is
known as а difference equation. То appreciate how past filter oиtpиt sam-
ples are иsed in the structure о! ПR filters, let's begin Ьу reorienting our
FIR strиcture in Figиre 6-2(а) to that о! Figиre 6-2(Ь). Notice how the
structures in Figиre 6-2 are compиtationally identical, and both are imple
mentations, or realizations, о! Eq, (6-1).
(а) h(O)
х(n)
(Ь)
h(З)
у(n)
у(n)
Flgure 6-2 FIR digital filter structures: (а) traditiona\ FIR filter structure;
(Ь) rearranged, but equiva\ent, FIR filter structure,
222 /пfiпite /три/sе Respoпse Filters
We can now show how past filter output samples are combined with
past input samples Ьу using the IIR fi1ter structure in Figиre 6 3. Because
IIR fi1ters havetwo sets of coefficients, we'll use the standard notation of
the variables b(k) to denote the feed forward coefficients and the variables
a(k) to indicate the feedback coefficients in Figure 6 3, ОК, the difference
equation describing the lIR fi1ter in Figиre 6 3 is
у(п) = Ь(О)х(п) + Ь(l)х(п 1) + Ь(2)х(п 2) + Ь(3)х(п 3)
+ а(l)у(п 1) + а(2)у(п 2) + а(3)у(п 3) ,
(6 2)
Look at Figure 6-3 and Eq. (6-2) carefиl1y, It's irnportant to convince our
selves that Figure 6 3 really is а valid irnplementation ofEq, (6--2) and that,
conversely, difference equation Eq, (6 2) fиlly describes the IIR fi1ter struc-
ture in Figure 6-3. Кеер in mind now that the sequence у(п) in Figure 6-3
is not the same у(п) sequence that's shown in Figure 6 2, The d(n)
sequence in Figure 6 3 is equal to the у(п) sequence in Figиre 6 2.
х(п)
У(п)
Ь(3)
8(3)
Feed forward calculations
Feedback calculations
Ffgure 6-3 IIR dlgital filter structure showing feed forward and feedback
calculations,
The Lap/ace Traпsforт 223
Ву now you're probably wondering, "Just how do we determine those
a(k) and b(k) ПR fi1ter coefficients if we actually want to design ап ПR fil
ter?" Well, fasten your seat ЬеН because this is where we get serious about
understanding IIR fi1ters, Recal1 from the last chapter conceming the win-
dow method of low-pass FIR fi1ter design that we defined the frequency
response of our desired FIR filter, took the inverse Fourier transform of
that frequency response, and then shifted that transform resu1t to get the
fi1ter's time domain impulse response, Happily, due to the nature of
transversal FIR filters, the desired h(k) filter coefficients tumed out to Ье
exact1y equal to the impulse response sequence. Following that same pro-
cedure with IIR fi1ters, we could define the desired frequency response of
our ПR fi1ter and then take the inverse Fourier transform of that response
to yield the filter's time-domain impulse response. The bad news is that
there's по direct method for computing the ПR fi1ter's a(k) and b(k) соеЮ-
cients from the impulse response! Unfortunately, the FIR fi1ter design
techniques that we've learned so far simply cannot Ье used to design ПR
fi1ters, Fortunately for us, this wrink1e can Ье ironed out Ьу using опе of
several available methods of designing ПR fi1ters.
Standard IIR fi1ter design techniques fall into three basic classes: the
impulse invariance, bilinear transform, and optimization methods, These
design methods use а discrete sequence, mathematical transformation
process known as the z transform whose origin is the Laplace transform
traditionally used in the analyzing of continuous systems, With that in
mind, let's start this ПR fi1ter analysis and design discussion Ьу briefly
reacquainting ourselves with the fundamentals of the Laplace transform,
6.2 The Laplace Transform
The Laplace transform is а mathematical method of solving linear differ-
ential equations that has proved very useful in the fields of engineering
and physics, This transform technique, as it's used today, originated from
the work of the brilliant English physicist Oliver Heaviside,+ The funda
mental process of using the Laplace transform goes something like the
following:
t Heaviside (1850 1925), who was interested in electrical phenomena, developed аn efficient
algebraic process of solving differential equations, Не initially took а lot of heat from his
contemporaries because they thought his work was not sufficiently justified from а mathe-
matical standpoint, However, the discovered correlation of Heaviside's methods with the
rigorous mathematical treatment of the French mathematician Marquis Pierre Simon de
Laplace's (1749 1827) operational calculus verified the validity of Heaviside's techniques,
224 /пflпffe /mpu/se Respoпse F/fters
Step 1: А time-domain differentia1 equation is written that describes the
input/output re1ationship of а physica1 system (and we want to
find the output function that satisfies that equation with а given
input).
Step 2: Тhe differentia1 equation is Lap1ace transformed, converting it to
an a1gebraic equation,
Step 3: Standard a1gebraic techniques are used to determine the desired
output function's equation in the Lap1ace domain.
Step 4: Тhe desired Lap1ace output equation is, then, inverse Lap1ace
transformed to yie1d the desired time domain output function's
equation.
тhis procedure, at first, seems cumbersome because it forces us to go the
loпg way arouпd, instead of just solving а differentia1 equation directly. Тhe
justification for using the Lap1ace transform is that a1though solving dif
ferentia1 equations Ьу c1assica1 methods is а very powerfu1 ana1ysis tech
nique for а11 but the most simp1e systems, it can Ье tedious and (for some
of us) error prone. Тhe reduced comp1exity of using a1gebra outweighs the
extra effort needed to perform the required forward and inverse Lap1ace
transforrnations. Тhis is especia11y true now that tables of forward and
inverse Lap1ace transforms exist for most of the commonly encountered
time functions, Well known properties of the Lap1ace transform a1so allow
practitioners to decompose comp1icated time functions into combinations
of simp1er functions and, then, use the tables, (Tables of Lap1ace trans
forrn.s allow us to trans1ate quickly back and forth between а time function
and its Lap1ace transform ana1ogous to, say, а German English dictio
nary if we were studying the German 1anguage. t) Let' s briefly 100k at а few
of the more important characteristics of the Lap1ace transform that will
рroуе usefu1 as we make our way toward the discrete z-transform used to
design and ana1yze IIR digita1 filters.
The Lap1ace transform of а continuous time domain function f(t),
where f(t) is defined оnlу for positive time (t > О), is expressed mathe
та tica11y as
F(s) = f f(t)e st dt
о
( б-З)
t Although tables of common1y encountered Laplace transforms are included in almost
every system analysis textbook, very comprehensive tables are also аvаilаЫе[I З],
The Lap/ace Traпsforт 225
F(s) is called "the Laplace transform off(t)," and the variable s is the сот-
plex number
s = а + jro .
(6-4)
А more general expression for the Laplace transform, called the bilateral
or two-sided transform, uses negative infinity ( ) as the lower liтН of
integration, However, for the systems that we'll Ье interested in, where
system conditions for negative tirne (t < О) are not needed in our analysis,
the oпe-sided Eq. (6 З) applies. Тhose systems, often referred to as causal
systems, тау have initial conditions at t = О that must Ье taken into
account (velocity of а mass, charge оп а capacitor, temperature of а body,
etc,) but we don't need to know what the system was doing prior to t = О,
ln equation (6 4), а is а real number and ro is frequency in radians/
second. Because е....! is dirnensionless, the exponent term s must have the
dirnension of l/tirne, or frequency. Тhat's why the Laplace variable s is
often called а complex frequency.
То put Eq, (6 З) into words, we can say that it requires us to multiply,
point for point, the functionf(t) Ьу the complex fиnction e S ! for а given
value of s. (We'll soon see that using the function е....! here is not acciden-
tal; e--s! is used because it's the general form for the solution of linear dif
ferential equations.) After the point for point multiplications, we find the
area under the curve of the functionf(t)e....! Ьу summing аВ the products.
Тhat area, а complex number, represents the value of the Laplace trans-
form for the particular value of s = а + jro chosen for the original multi-
plications. If we were to go through this process for аВ values of s, we'd
have а fиll description of F(s) for every value of s,
1 like to think of the Laplace transform as а continuous function, where
the complex value of that function for а particular value of s is а correla
tion of f(t) and а damped complex e--s! sinusoid whose frequency is ro and
whose damping factor is а, What do these complex sinusoids look like?
WeB, they are rotating phasors described Ьу
j(J)!
e S! = e (a+j(J)! = е Й!е j(J)! =
еЙ!
(6-5)
From our knowledge of complex numbers, we know that e j(J)! is а unity
magnitude phasor rotating clockwise around the origin of а complex
plane at а frequency of ro radians per second. Тhe denominator of Eq. (6-5)
is а real number whose value is one at tirne t = о. As t increases, the denom-
inator еЙ! gets larger (when а is positive), and the complex e--s! phasor's
226 /пfiп/te /mpu/se Respoпse Filters
magnitude gets smaller as the phasor rotates оп the complex plane. Тhe
tip of that phasor traces out а curve spiraling in toward the origin of the
complex plane. One way to visualize а complex sinusoid is to consider its
real and imaginary parts individually. We do this Ьу expressing the coт
plex: e--s t sinusoid from Eq. (6-5) in rectangular form as
st e j(J)t cos((Ot) ,sin((Ot)
е ="""""Cit= J ,
е е е
(6--5')
Figure 6 4 shows the real parts (cosine) of several complex sinusoids with
different frequencies and different damping factors, In Figure 6-4(а), the
complex sinusoid's frequency is the arbltrary (о', and the damping factor
is the arbltrary 0',50 the real part of F(s), at s = о' + j(O', is equal to the cor-
relation of f(t) and the wave in Figure 6 4(a). For different values of s, we'll
correlate f(t) with different complex sinusoids as shown in Figure 6 4, (As
we'll see, this correlation is very much like the correlation off(t) with var-
ious sine and cosine waves when we were calculating the discrete Fourier
transform.) Again, the real part of F(s), for а particular value of s, is the
correlation of f(t) with а cosine wave of frequency (О and а damping fac
tor of о, and the imaginary part of F(s) is the correlation of f(t) with а
sinewave of frequency (О and а damping factor of о.
Now, if we associate each of the different values of the complex s vari-
аЫе with а point оп а complex plane, rightfully called the s plane, we
could plot the real part of the F(s) correlation as а surface above (or below)
that s-plane and generate а second plot of the imaginary part of the F(s)
correlation as а surface above (or below) the s plane, We can't plot the full
complex F(s) surface оп paper because that would require four dimen
sions. Тhat's because s is complex, requiring two dimensions, and F(s) is
itself complex and also requires two dimensions, What we сап do, how
ever, is graph the magnitude I F(s) I as а function of s because this graph
requires only three dimensions, Let's do that as we demonstrate this
notion of an I F(s) I surface Ьу illustrating the Laplace transform in а tan
gible way,
5ау, for example, that we have the linear system shown in Figure 6--5,
Also, let's assume that we can relate the x(t) input and the y(t) output of
the linear time invariant physical system in Figure 6-5 with the following
messy homogeneous constant coefficient differential equation
d 2 y(t) dy(t) dx(t)
а2 + аl + Qoy(t) = Ь 1 + box(t)
dt dt dt
(6 6)
The Lap/ace Traпsforт 227
(а)
r 00 = 00'
"", """ . ....,'"...,. (1 = (1'
-'. ..._....,-.. щ
\rv
.=00'
..--...'--.........,..... . "'" (1 = 2(1'
/""", Щ",
\../ Щ ..
Time
Time
(Ь)
(с)
Time
(d)
Time
Flgure 6-4 Real part (cosine) of various e'5t functions, where 5 = (! + jro, to Ье
correlated with f(t),
We'l1 use the Laplace transform toward our goal of figuring out how the
system wi11 behave when various types of input functions are applied,
i.e" what will the y(t) output Ье for any given x(t) input.
Let's slow down here and see exactly what Figure 6 5 and Eq, (6-6) are
telling us, First, if the system is time invariant, then the а п and Ь п соеШ-
cients in Eq, (6-6) are constant, They mау Ье positive or negative, zero,
real or complex, but they do not change with time, If the system is elec-
trical, the coefficients might Ье related to capacitance, inductance, and
resistance, If the system is mechanical with masses and springs, the coef-
ficients could Ье related to rnass, coefficient of damping, and coefficient of
228 /пfiпi1 /тpu/se Respoпse Fi/ters
x(t) I
System
уи)
Figure 6-5 Systeт descrlbed Ьу Eq. (6-6). The systeт's input and output are
the continuous tiтe functlons x(t) and уи) respectively.
resilience, Then, again, if the system is thermal with masses and insula-
tors, the coefficients would Ье related to thermal сарасНу and thermal
conductance. То keep this discussion general, though, we don't reaHy care
what the coefficients actuaHy represent,
ОК, Eq, (6-6) also indicates that, ignoring the coefficients for the
moment, the sum of the y(t) output plus derivatives of that output is equal
to tЬe sum of the x(t) input plus the derivative of that input. Our problem
is to determine exact1y what input and output functions satisfy the elabo
rate relationship in Eq. (6 6). (For the stout hearted, classical methods of
solving differential equations could Ье used here, but the Laplace trans
form makes the problem much simpler for our purposes,) Thanks to
Laplace, the complex exponential time function of 1ft is the one we'H use.
It has the beautifиl property that it сan Ье differentiated any number of
times without destroying its original form, Тhat is
d(e st ) = sest d 2 (e st ) = s2est d 3 (e st ) s3est dn(e st ) = sne st
dt 'dt 2 'dТ ,.." dt n
(6-7)
If we let x(t) and уи) Ье functions of e st , x(e st ) and y(lf t ), and use the prop
erties shown in Eq. (6 7), Eq. (6 6) becomes
a2 s2 y(e st ) + a1sy(e st ) + aoy(e st ) = b 1 sx(e st ) + box(e st ),
or
(a 2 s 2 + al s + ao)y(e st ) = (b1s + bo)x(e st ) .
(6-8)
Although it's simpler than Eq, (6 6), we сап further simplify the rela
tionship in the last line in Eq, (6-8) Ьу considering the ratio of y(lft) over
x(e"t) as the Laplace transfer function of our system in Figиre 6 5, If we саН
that ratio оЕ polynomials the transfer fиnction Н(s),
The Laplace Traп5forт 229
н(s)= y(e st ) = b 1 s+b o
x(e st ) a2 s2 +a 1 s+ao '
(6-9)
То indicate that the origina1 x(t) and y(t) have the identical functional form
of eS t , we can follow the standard Laplace notation of capitalletters and
show the transfer function as
Н( ) Y(s) Ь 1 нЬ о
S 2
X(S) a2s +als+aO '
(6-10)
where the OUtput Y(S) is given Ьу
ь s + Ь о
Y(S) = X(S) 21 = Х(s)Н(s)
a2 s +a 1 s+a O
(6 1l)
Equation (6--11) leads us to redraw the original system diagram in а form
that high1ights the definition of the transfer function Н(s) as shown in
Figure 6 6.
Тhe cautious reader тау Ье wondering, "ls it really va1id to use this
Laplace analysis technique when it's strict1y based оп the system's x(t)
input being some function of eS t , or x(eS t )?" Тhe answer is that the Laplace
analysis technique, based оп the complex exponential x(eS t ), is valid
because аll practical x(t) input functions can Ье represented with complex
exponentials. For example,
· а constant, с = ceO t ,
· sinusoids, sin(rot) = (ei oot e joot)/2j or cos(rot) = (ei oot + e joot)/2,
· а monotonic exponential, e at , and
· an exponentially varying sinusoid, e.... t cos(rot).
)«s) .1
System
H(s)
Y(S)
Flgure 6-6 Unear 5Y5tem descrlbed Ьу Eq5, (6-10) and (6-11), The 5Y5tem's
Input 15 the Laplace functlon Х(5), it5 output Is the Laplace
functloп У(5), and the system transfer functlon 15 Н(5),
230 /пf1пite /mрИ/sе Respoпse Fffters
With that said, if we know а system's transfer function Н(s), we сап
take the Laplace transform of any x(t) input to determine X(s), multiply
that X(s) Ьу Н(s) to get Y(s), and then inverse Laplace transform Y(s) to
yield the time domain expression for the output y(t). In practical situa
tions, however, we usual1y don't go through аН those analytical steps
because it's the system's transfer function H(s) in which we're most inter-
ested, Being аЫе to express H(s) mathematicaHy or graph the surface
I H(s) I as а function of s will teH us the two most important properties we
need to know about the system under analysis: Is the system stable, and
if so, what is its frequency response?
"But wait а minute," уои say. "Equations (6 10) and (6 1l) indicate that
we have to know the Y(s) output before we сап determine Н(s)!" Not
really, АН we reaHy need to know is the time domain differential equation
like that in Eq. (6 6). Next we take the Laplace transform of that differen
tial equation and rearrange the terms to get the H(s) ratio in the form of
Eq. (6-10), With practice, systems designers сап look at а diagram (block,
circuit, mechanical, whatever) of their system and promptly write the
Laplace expression for H(s), Let's use the concept of the Laplace transfer
fиnction Н(s) to determine the stabi1ity and frequency response of simple
continuous systems,
6.2.1 Poles and Zeros оп the s-plane and Stability
Onе of the most important characteristics of any system involves the con-
cept of stabi1ity, We can think of а system as stable Н, given any boooded
input, the output will always Ье boooded, Тhis sounds like an easy con
dition to achieve because most systems we encoooter in our dai1y lives
are indeed stable. Nevertheless, we have аН experienced instability in а
system containing feedback, Recal1 the annoying how/ when а public
address system's microphone is placed too close to the loudspeaker, А
sensational example of an unstable system occurred in western
Washington when the first Тасоmа Narrows Bridge began oscillating оп
the afternoon of November 7th, 1940. Тhose oscillations, caused Ьу 42
mph winds, grew in amplitude ootil the bridge destroyed itself. For IIR
digital fi1ters with their bui1t-in feedback, instability would resu1t in а fil
ter output that's not at аН representative of the fi1ter input; that is, our fil
ter output samples would not Ье а filtered version of the input; they' d Ье
sorne strange oscillating or pseudorandom values, А situation we' d like to
avoid ifwe can, right? Let's see how.
We сап determine а continuous system's stability Ьу examining sev
era1 different examples of H(s) transfer fиnctions associated with linear
The Lop/oce Troп5forт 231
time-invariant systems, Assume that we have а system whose Laplace
transfer fиnction is of the form of Eq. (6 10), the coefficients are аН real,
and the coefficients Ь 1 and а 2 are equal to zero. We'll саН that Laplace
transfer function H 1 (s), where
Н ( ) Ь о Ь о / а 1
1 S
a 1 s+a O s+a o / а 1
(6-12)
Notice that if s = ao/a1' the denominator in Eq. (6-12) equals zero and
H 1 (s) would have аn infinite magnitude. This s = ao/a1 point оп the
s-plane is called а pole, and that pole's location is shown Ьу the "х" in
Figиre 6 7(a), Notice that the pole is located ехасНу оп the negative por-
tion of the real (j axis, If the system described Ьу Н 1 were at rest and we
disturbed it with an impulse like x(t) input at time t = О, its continuous
time domain y(t) output would Ье the damped exponential curve
shown in Figure 6 7(b), We сап see that H 1 (s) is stable because its y(t)
output approaches zero as Нте passes, Ву the way, the distance of the
pole from the (j = о axis, а о / а 1 for our Н 1 (s), gives the decay rate of the
y(t) impulse response. То illustrate why the term pole is appropriate,
Figure 6 8(b) depicts the three-dimensional surface of I H 1 (s) I above the
s-plane. Look at Figure 6-8(Ь) carefиlly and see how we've reoriented
the s plane axis, Тhis new axis orientation allows us to see how the H 1 (s)
system's frequency magnitude response сап Ье determined from its
three-dimensional s-plane surface. If we examine the I H 1 (s) I surface at
(j = О, we get the bold curve in Figure 6 8(b). Тhat bold curve, the inter
section of the vertical (j = о plane (the jm axis plane) and the I Н 1 (s) I sur-
face, gives us the frequency magnitude response I Н 1 (т) I of the
system and that's one of the things we're after here, Тhe bold I Н 1 (т) I
<1 = aO/a1
\
э-рlапе
уи) = (bo/a,)e 80/a1 t
I
jro
cr
о
Time
(а)
(Ь)
Flgure 6-7 Descrlptions of Н, (5): (о) pole locoted ot s = а + jro = OoIOl + jO оп the
5-plone; (Ь) tlme-domoln y(t) Impulse response of the system,
232 /пl1n1te /mpu/se Respoпse F//ters
jOJ
(а)
(J = aolal
cr
..
+
(Ь)
IH I (5)1 surface
jOJ
(с)
о
jOJ
Flgure 6-8 Further depletlons of H 1 (s): (а) pole /oeated at (J = ao/al оп the
s-plane; (ь) I H)(s) I surfaee; (е) eurve showlng the Interseetion of the
I H 1 (s) I surfaee and the vertleal (J = о plane, Тhls Is the eonventlonal
depietion ofthe I Н 1 (ю) I frequency magnltude responsв,
curve in Figure 6 8(b) is shown in а more conventional way in Figure
6 8(c), Figures 6 8(b) and 6-8(с) higblight the very important property
that the Laplace transform is а more general case of the Fourier
transform because if (1 = О, then 5 = j(j), In this case, the I Н) (5) I curve for
The Lap/ace Traпsforт 233
cr = о above the s plane becomes the I Н 1 (оо) I curve above the jm axis in
Figure 6-8(с).
Another common system transfer function leads to an impulse
response that oscillates. Let's think about an alternate system whose
Laplace transfer function is of the form of Eq, (6-10), the coefficient Ь о
equals zero, and the coefficients lead to complex terms when the denom-
inator polynomial is factored. We'll саll this particular second-order trans-
fer fиnction H 2 (s), where
Н ( ) b 1 s
2 S 2
a 2 s + al s + ао
(Ь 1 / a 2 )s
S2 +(al/ a 2)s+a O /a 2
(ИЗ)
(Ву the way, when а transfer fиnction has the Laplace variable s in both
the numerator and denominator, the order of the overall function is
defined Ьу the largest exponential order of s in the denominator polyno-
mial. 50 our H 2 (s) is а second-order transfer function.) То keep the fol
lowing equations from becoming too messy, let's factor its denominator
and rewrite Eq, (6 IЗ) as
As
H 2 (s) = .
(s+p)(s+p)
( 6-14)
where А = Ь/а 2 , Р = Preal + jPirnag' and р' = P rea ! jPimag (complex conju
gate of р). Notice that, if s is equal to P or p', one of the polynomial
roots in the denominator of Eq. (6-14) will equal zero, and H 2 (s) will
have an infinite magnitude. Тhose two complex poles, shown in Figure
6-9(а), are located off the negative portion of the real cr axis, If the Н 2 sys
tem were at rest and we disturbed it with an impulselike хи) input at
time t = О, its continuous time-domain y(t) output would Ье the damped
sinusoidal curve shown in Figure 6-9(Ь), We see that H 2 (s) is stable
because its oscil1ating уи) output, like а plucked guitar string,
approaches zero as Нmе increases. Again, the distance of the poles from
the cr = О axis ( Prea!) gives the decay rate of the sinusoidal y(t) impulse
response, Likewise, the distance of the poles from the jm = О axis (:tPimag)
gives the frequency of the sinusoidal y(t) impulse response, Notice
something new in Figure 6 9(a). When s = О, the numerator of Eq, (6-14)
is zero, making the transfer fиnction H 2 (s) equal to zero. Any value of s
where H 2 (s) = О is sometimes of interest and is usually plotted оп the s-
plane as the litt1e circle, cal1ed а "zero," shown in Figure 6 9(a), At this
point we're not very interested in knowing exactly what Р and р' are in
234 /пflпlte /тpu/se Respoпse Filters
s = Pr.., +jPimag jro
x
y(l) t
y(l) = 2IAle Pr.., COS(Pimag 1+ 6)
"х 6=tan-'(A,m.g / ......')
/Х
s Preal jpimag
'\ cr
'zero"atl>"'O
Time
(а)
(Ь)
Flgure 6-9 Descriptlons of H 2 (s): (а) poles located at 5 = Р,ео! :!: JP;mag оп the
s-plane; (Ь) tlme domaln y(t) Impulse response of the sysrem,
terms of the coefficients in the denominator of Eq. (6 13). However, аn
energetic reader could determine the values of р and р" in terms of ао, а 1 ,
and а 2 Ьу using the following well-known quadratic factorization for
mula: Given the second-order polynomialf(s) = as 2 + bs + с, thenf(s) сап
Ье factored as
2 [ ь 2 4ac ) [ Ь 2 4ac )
j(s) = as +bs+c = s+ + 2. s+ 2
2а 4а 2а 4а
. (6-15)
Figure 6-10(Ь) illustrates the I H 2 (s) I surface аЬоуе the s-plane.
Again, the bold I Н 2 (0)) I curve in Figure 6-10(Ь) is shown in the соn-
ventional way in Figure 6 10(c) to indicate the frequency magnitude
response of the system described Ьу Eq, (6-13). A1though the three
dimensional surfaces in Figures 6 8(b) and 6-10(Ь) are informative,
they're also unwieldy and unnecessary, We сап determine а system's
stabllity merely Ьу looking at the locations of the poles оп the two
dimensional s-plane.
Т() further i11ustrate the concept of system stability, Figure 6-11 shows
the s plane pole locations of several example Laplace transfer functions
and their corresponding time-domain impulse responses, We recognize
Figures 6-11(а) and 6-11(Ь), from our previous discussion, as indicative
of stable systems. When disturbed from their at-rest condition they
respond and, at some later time, retum to that initial condition. Тhe sin
gle pole location at s = О in Figure 6-11(с) is indicative of the l/s trans-
fer function of а single element of а linear system, In аn electrical
syst€m, this l/s transfer function could Ье а capacitor that was charged
with аn impulse of current, and there's по discharge path in the circuit.
The Lop/oce Troпsform 235
jw
х
(а)
о'
х
+
(s)=O,ats=O
(Ь)
(с)
jw
о jw
Flgure 6-10 Further deplctions of H 2 (s): (о) poles ond zero locotions оп the
s--plone; (Ь) I H 2 (s) I surfoce; (с) I Н 2 (ro) I frequency mognitude
response curve,
For а mechanical system, Figure 6-11(c) would describe а kind of spring
that's compressed with ап impulse of force and, for some reason,
remains under compression, Notice, in Figure 6-11(d), that, if ап н(s)
transfer function has conjugate poles located exact1y оп the jro axis
236 /пffп/te /тpu/se Respoпse FI/ters
+
I W '
,
i W ' ,
+
(а)
+
'
-.-.----..."....,.
А. .ш...
V "-../ ............. . ......;;;
..
t
..
t
(Ь)
(с)
jro
(d)
..
t
а
1 ,", /
..............
/'Ccl\j\
..
t
(е)
(1)
Figure 6-11 Various н(s) pole loeatloпs aпd their tlтe-doтaln lтpulse responses:
(а) single pole at а < О; (Ь) eonjugate poles at а < О; (е) slngle pole
loeated at а = О; (d) conjugate poles loeated at а = О; (е) slngle pole
at а > О; (f) eonjugate poles at а > О,
(а = О), the system will go into oscillation when disturbed from its ini
tial condition, This situation, called conditional stability, happens to
describe the intentional transfer function of electronic oscillators,
Instability is indicated in Figures 6 ll(е) and 6-11(f), Here, the poles lie to
the right of the jro axis, When disturbed from their initial at rest condition
The Lap/ace Traпsforт 237
Conditionally stabIe
Flgure 6-12 The Laplace s plane showlng the regions of stability and
instabillty for pole locations for Ilпеш continuous systems,
Ьу an impulse input, their outputs grow without bound,t 5ее how the
value of 0', the real part of s, for the pole locations is the key here? When
о' < О, the system is we11 behaved and stable; when о' = О, the system is
conditiona11y stable; and when о' > О the system is unstable, 50 we сап
say that, when о' is located оп the right half of the s plane, the system is
unstable, We show this characteristic of linear continuous systems in
Figure 6 12. Кеер in mind that real world systems often have more than
two poles, and а system is on1y as stable as its least stable pole, For а sys
tem to Ье stable, а11 of its transfer function poles must Не оп the left half
of the s plane,
То consolidate what we've learned so far: Н(s) is determined Ьу writing
а linear system's time domain differential equation and taking the Laplace
transform of that equation to оЫаin а Laplace expression in terms of X(s),
Y(s), s, and the system's coefficients. Next we rearrange the Laplace
expression terms to get the Н(s) ratio in the form of Eq, (6-10). (The rea11y
slick part is that we do not have to know what the time domain x(t) input
t Impulse response testing in а !aboratory сап Ье an important part of the system design
process, The difficu!t part is generating а true impu!selike input, If the system is e!ectrica!,
for ехатр!е, a!though somewhat difficu1t to imp!ement, the input x(t) impu!se wou!d Ье а
very short duration vo!tage or current ри!Бе, If, however, the system were mechanica!, а
whack with а hammer wou!d suffice as an x(t) impu!se inрщ. For digita! systems, оп the
other hand, an impu!se input is easy to generate; it's а sing!e unity-va!ued samp!e preceded
and followed Ьу all zero-va!ued samp!es,
238 /пfiпite /тpu/se Respoпse Filters
is to analyze а linear system!) We сап get the expression for the continuous
frequency response of а system just Ьу substitutingjro for s in the H(s) equa
tion. То determine system stability, the denominator polynomial of H(s) is
factored to find each of its roots, Еат root is set equal to zero and solved
for s to find the location of the system poles оп the s plane. Any pole located
to the right оЕ the jro axis оп the s-plane will indicate an unstable system,
ок, returning to our original goal of understanding the z-transform,
the process of analyzing IIR fi1ter systems requires us to replace the
Laplace transform with the z-transform and to replace the s-plane with а
z-plane, Let's introduce the z-transform, determinewhat this new z-plane
is, discuss the stabi1ity of I1R filters, and design and analyze а few simple
IIR fi1ters,
6.3 The z- Transform
The z-transform is the discrete time cousin of the continuous Laplace
transform,t WlU1e the Laplace transform is used to simplify the analysis
of continuous differential equations, the z-transform facilitates the analy
sis of discrete difference equations, Let's define the z-transform and
explore its important characteristics to see how it's used in analyzing I1R
d igi tal fil ters.
The z-transform of а discrete sequence h(n), expressed as H(z), is
defined as
H(z) = Lh(n)z п ,
(6 16)
п=---оо
where the variable z is complex. Where Eq, (6 З) al10wed us to take the
Laplace transform of а continuous signal, the z-transform is performed
оп а discrete h(n) sequence, converting that sequence into а continuous
function H(z) of the continuous complex variable z, Similarly, as the func-
tion e st is the general form for the solution of linear differential equa
tions, z п is the general form for the solution of linear difference
equations, Moreover, as а Laplace function F(s) is а continuous surface
t In the ear!y 1960s, James Kaiser, after whom the Kaiser window function is named, con-
solidated the theory of digita! filters using а mathematica! description known as the
z-trаnsfоrm[4,S], Unti! that time, the use of the z-transform had generally been restricted to
the fie!d of discrete contro! systems[ 9],
I
The z-Traпsforт 239
above the s plane, the z transform function H(z) is а continuous surface
above а z-plane. То whet your appetite, we'll now state that, if H(z) rep-
resents an IIR fi1ter's z-domain transfer function, evaluating the H(z) sur-
[асе will give us the fi1ter's frequency magnitude response, and H(z)'s
pole and zero locations wi1l determine the stabi1ity of the filter,
We сап determine the frequency response of an IIR digital fi1ter Ьу
expressing z in polar form as z = re jOJ , where r is а magnitude and ro is the
angle. In this [опn, the z-transform equation becomes
Н(z)=Н(rе jOJ )= Lh(n)(rejOJ) п = Lh(n)r п(e jOJп)
(6-17)
п \XI
п= oo
Equation (6-17) can Ье interpreted as the Fourier transform of the product
of the original sequence h(n) and the exponential sequence r п. When r
equals one, Eq, (6-17) simplifies to the Fourier transform. Тhus оп the
z-plane, the contour of the н(z) surface for those values where I z I = 1 is
the Fourier transform of h(n), If h(n) represents а ffiter impulse response
sequence, evaluating the H(z) transfer function for I z I = 1 yields the Пе-
quency response of the fi1ter, 50 where оп the z plane is I z I = 1? It's а cir-
cle with а radius of one, centered about the z = О point, This circle, so
important that it's been given the name unit circle, is shown in Figure 6 lЗ,
Recall that the jro frequency axis оп the continuous Laplace s-plane was
linear and ranged from --00 to +00 radians/s, Тhe ro frequency axis оп the
complex z-plane, however, spans only the range пот п to +п radians,
With this relationship between the jro axis оп the Laplace s-plane and the
unit circle оп the z-plane, we сап see that the z plane frequency axis is
Zimag
At this point,
/ z= еiro
Zreal
Unit circle
(where Izl = 1)
Figure 6-13 Unit circle оп the сотр/ех z..plane,
.
240 /пfiпite /mpu/se Respoпse F//ters
Figure 6-14 Mapping of the Laplace s--plane to the z--plane, АН frequency values
are In radlans/s,
equivalent tосоШng the s-plane's jO) axis about the unit circle оп the
z-plane as shown in Figure 6-14,
Then, frequency о) оп the z plane, is not а distance along а straight line,
but rather an angle around а circle. With о) in Figure 6--13 being а general
normalized angle in radians ranging from п to +п, we сап relate о) to an
equivalent f. sampling rate Ьу defining а new frequency variable 0). = 2п!.
in radians/ s. The periodicity of discrete frequency representations, with а
period of 0). = 21tf. radians/ s or f. Hz, is indicated for the z plane in Figure
6 14, Where а walk along the jO) frequency axis оп the s plane cou1d take
us to infinity in either direction, а trip оп the о) frequency path оп the
z plane leads us in circles (оп the unit circle), Figure 6 14 shows us that
on1y the п!. to +п!. radians/s frequency range for о) can Ье accounted for
оп the z plane, and this is another example of the universal periodicity of
the discrete frequency domain, (Of course the 1tf. to +1tfs radians/s range
corresponds to а сусliс frequency range of f.l2 to +f.l2.) With the
perimeter of the unit circle being z = e jro , later, we'll show exactly how to
substitute ejro for z in а fi1ter's н(z) transfer fиnction, giving us the fi1ter's
frequency response.
6.3.1 Poles and Zeros оп the z plane and Stability
One of the most important characteristics of the z plane is that the region
of fi1ter stability is mapped to the inside of the unit circle оп the z plane.
Given the н(z) transfer fиnction of а digital filter, we сап examine that
fиnction's pole locations to determine filter stabi1ity. If аН poles are
located inside the unit circle, the filter will Ье stable. on the other hand, if
(а)
(Ь)
(с)
(d)
(е)
Thв z-Troпsform 241
4 У (n)
' 'i : . . ..........
.1 у(п)
1 1Ifо
. r > 2 . ".
, . ..l....l.J"'"""7". . .", '""" ' .:!...!..!:.ii" ;'-'..
';j . . ..
.'.
.
n
п
4 У (n)
ti :... ; 11 .
1. 'llшш ш . шш' E 1
....
п
Y(n) ..' . . , "1
I ....." "",'
i ............. .: :' , ': ,:: :' ", .п
l У(n)ш. ...... . ..;
..."J!..!..I.iiirn. . . . . ................
1 .. . . ' . ' п
.1
Ii
F\gure 6-15 Various 1-/(1) pole loeations and their diserete time-domain impulse
responses: (а) single pole inside the unit eirele; (Ь) eon]ugate poies
loeated inside the unlt elrele; (е) eon]ugate poles loeated оп the
unit eirele; (d) single pole outside the unit elrele; (е) eonJugate
poles loeated outside the unit eirele,
аnу pole is located outside the unit circle, Фе fi1ter will Ье unstable. Figure
6-15 shows the z-plane pole locations of several example z-domain tranS-
fer functions and their corresponding discrete time domain impulse
responses. It' s а good idea for the reader to compare the z-plane and
discrete-time responses of Figure 6-15 with the s-plane and the continuous
time responses of Figure 6 11. Тhe у(п) outputs in Figures 6-15(d) and
242 /пflпfte /mpUlse I<espoпse rflTers
(е) show examples of how unstable filter outputs increase in amplitude,
as time passes, whenever their х(п) inputs are nonzero. То avoid this sit
uation, any IIR digital filter that we design should have an н(z) transfer
fиnction with аll of its individual poles inside the unit circle. Like а chain
that's only as strong as its weakest link, an IIR filter is only as stable as its
least stable pole.
The (00 oscillation frequency of the impulse responses in Figures 6-15(с)
and (е) is, of course, proportional to the angle of the conjugate pole pairs
from the zreal axis, or (00 radians/s corresponding to 10 = (О/2тс Hz.
Because the intersection of the zreal axis and the unit circle, point z = 1,
corresponds to 1t radians (or п!. radians/s = 1/2 Hz), the (00 angle of 'ТС/4
in Figure 6-15 means that 10 = 1/8 and our у(п) мll have eight samples per
cycle ofl o '
6.3.2 Using the z-transform to Analyze IIR Filters
We have one last concept to consider before we can add the z-transform to
our collection of digital signal processing too1s, We need to determine what
the time delay operation in Figure 6-3 is relative to the z transform. То do
this, assume we have а sequence х(п) whose z-transform is X(z) and а
sequence у(п) = x(п l) whose z transform is Y(z) as shown in Figure 6 16.
Тhe Hransform of у(п) is, Ьу definition,
Y(z) = Ly(n)z п = Lx(n l)z п
(6 18)
п=---оо п=---оо
Now if we let k = п l, then, Y(z) becomes
Y(z) = Lx(k)z (k+l) = Lx(k)z kZ l ,
k==---oo k=
(6 19)
X( I У(п) =х(п--1)
Delay
X(z) Y(z)
Flgure 6-16 Output sequence у(п) equal to а uп/t de/ayed version of the
input х(п) sequence,
The z- Traпsforт 243
which we can write аа
Y(Z) = Z l Lx(k)ZH) = Z l[X(Z)]
k= oo
(6-20)
ТhиБ, the effect of а single unit of time delay is to mu1tiply the z trапsfопn
Ьу Z l.
Interpreting а unit time delay to Ье equivalent to the r 1 operator leads
иБ to the relationship shown in Figure 6-17, where we сап аау
X(z)ZO = X(z) is the z-trапsfопn of х(п), X(z)r 1 is the z trапsfопn of х(п)
delayed Ьу опе sample, X(z)r 2 is the z-transform of х(п) delayed Ьу two
samples, and X(z)Z k is the z transform of х(п) delayed Ьу k samples, 50 а
transfer function of Z-k is equivalent to а delay of kt s seconds from the
instant when t = О, where t s is the period between data samples, or опе
over the sample rate. 5pecifically, t s = 1/ fs. ВесаиБе а delay of опе sample
is equivalent to the factor r 1 , the unit time delay symbol used in Figures
6-2 and 6 3 is usually indicated Ьу the r 1 operator,
Let's раиБе for а moment and consider where we stand БО far, Oиr
acquaintance with the Laplace trапsfопn with its s plane, the concept оЕ
stability based оп H(s) pole locations, the introduction of the z-trаnsfопn
with its z-plane poles, and the concept of the r 1 operator signifying а аin-
gle unit оЕ time delay has led иБ to our goal: the ability to inspect an IIR
fi1ter difference equation or filter structure and immediately write the fil-
ter's z-domain transfer function Н(z), Accordingly, Ьу evaluating an IIR
filter's H(z) transfer function appropriately, we сап dеtепninе the filter's
frequency response and its stability. With those ambitious thoughts in
mind, let's develop the z domain equations we need to analyze IIR filters,
Using the relationships оЕ Figure 6 17, we redraw Figure 6-3 аа а gen-
eral Mth-order IIR fi1ter using the Z l operator аа shown in Figure 6-18. (In
hardware, those r 1 operations are shift registers holding successive filter
Time
domain
Flgure 6-17 Correspondence of the delay operatlon In the tlтe doтaln
and the Z-k operatlon in the z-doтain,
244 /пfiп/te /mpu/se Respoпse F//ters
input and output sample values, When implementing аn ПR fi1ter in а
software routine, the r 1 operation merely indicates sequential memory
locations where input and output sequences are stored.) The ПR fi1ter
structure in Figure 6 18 is often called the Direct Farm 1 structure,
The time-domain difference equation describing the general Mth-order
ПR filter, having N feed forward stages and М feedback stages, in Figure
6-18 is
у(п) = Ь(О)х(п) + Ь(l)х(п 1) + Ь(2)х(п 2) + ,., + b(N)x(n N)
r- +a(1)y(п 1)+a(2)y(п 2)+..,+a(M)y(п M) (6 21)
Time domain expression for an Mth-order IIR filter
In tЛе z domain, that ПR fi1ter's output can Ье expressed Ьу
Y(z) = b(O)X(z) + b(l)X(z)Z l + Ь(2)х(ф 2 + ... + b(N)X(z)z N
+ a(l)Y(z)z l + a(2)Y(z)z 2 + ..' + a(M)Y(z)z M , (6-22)
where Y(z) and X(z) represent the z transform of у(п) and х(п), Look Eqs.
(6 21) and (6 22) over carefully and see how the unit time delays translate
х(п)
у(п)
Ь(2}
а(2}
Ь(М
а(М)
Flgure 6.18 Generol (Dlrect Form 1) structure of оп Mth-order IIR fllter. hovlng
N feed forword stoges ond М feedbock stoges. wlth the z 1
operotor indlcotlng о unlt time deloy,
The z-Traпsforт 245
to negative powers of z in the z domain expression. А more compact form
for Y(z) is
z-domain expression for an N М
Mth-order IIR filter: Y(z) = X(z) Lb(k)z k + Y(z) La(k)z k . (6 23)
k=O k=l
ок, now we've arrived at the point where we сап describe the transfer
function of а general ПR fi1ter, Rеапаnging Eq. (6 23) to collect like terms,
Y(Z{l a(k)z k]=X(Z) b(k)z k. (6-24)
Finally, we define the fi1ter'sz-domain transfer functionasНCz) = Y(z)/X(z),
where НCz) is given Ьу
z.domain transfer function
of an Mth-order IIR filter:
N
Lb(k)z k
НCz) = Y(z) = k=O
X ( z ) М
1 La(k)z k
k=1
(6 25)
Gust like Laplace transfer functions, the order of our z domain transfer
function is defined Ьу the largest exponential order of z in the denominator,
in this case М.) Equation (&-25) tel1s us all we need to know about an ПR ы-
ter. We can evaluate the denominator of Eq. (&-25) to determine the posi-
tions of the filter's poles оп the z plane indicating the filter's stability,
Remember, now, just as the Laplace transfer function H(s) in Eq. (6-9)
was а complex-valued surface оп the s plane, H(z) is а complex-valued
surface above, or below, the z plane. The intersection of that H(z) surface
and the perimeter of а cylinder representing the z = &00 unit circle is the ш-
ter's complex frequency response. This means that substituting ejOO for z in
Eq. (6-25)'s transfer function gives us the expression for the filter's НпR(О))
frequency response as
Frequency response of an
Mth-order IIR filter filter
(exponential form):
N
Lb(k)e jkOO
Н ПR (О)) = H(Z)!z=e i " = k= . (6-26)
1 La(k)e jkOO
k=l
246 /пfiпitе /mpu/se Respoпse F//ters
Let's a1ter the form of Eq, (6 26) to obtain more useful expressions for
Н1ш(оо)'s frequency magnitude and phase responses. Because а typical ПR
fi1ter frequency response НпR(оо) is the ratio of two complex functions, we
сап express НпR(оо) in its equivalent rectangular form as
N
2,.b(k). [cos(koo) j sin(koo)]
Н ( ) k=O
ПR 00 = м
1 2,.a(k). [cos(koo) j sin(koo)]
k=l
(6-27)
or
Frequency response
of an Mth-order IIR
filter (rectangular
form):
N N
2,.b(k). cos(koo) j2,.b(k). sin(koo)
Н ( r,, ) k=O k=O
ПRUI М М
1 2,.a(k).cos(koo)+ j2,.a(k).sin(koo)
k=l k=l
(6-28)
H's usually easier and, certainly, more useful, to consider the complex
frequency response expression in terms of its magnitude and phase. Let's
do this Ьу representing the nиmerator and denominator in Eq. (6 28) as
two complex functions of radian frequency 00, Calling the nиmerator of
Eq, (6-28) Num(oo), then,
Num(oo) = Numreai(oo) + jNumimag(oo) ,
(6-29)
where
N
Numreai (00) = 2,.b(k). cos(koo) ,
k=O
and
N
Numimag(oo) = 2,.b(k). sin(koo)
k=O
(6-30)
Likewise, the denominator in Eq. (6-28) сап Ье expressed as
Deп(oo) = Deпreai(oo) + jDeпimag(oo) ,
(6-31)
The z-Troпsforт 247
where
м
Deпr.al(ro)=I La(k)oCOS(kro),
k;l
and
м
Denimag(ro) = La(k) о sin(kro)
k;l
(6 32)
Тhese Num(ro) and Den(ro) definitions allow us to represent H IIR (ro) in the
more sirnple forms of
н (ro) = Num(ro)
lIR Den(ro)
Numrea!(ro)+ jNumimag(ro)
Denrea!(ro)+ jDeпimag(ro)
(6-33)
= I Num(ro) I L0Num(ro)
I Deп(ro) I L0Den(ro)
(6-33')
Given the form in Eq, (6--33) and the rules for dividing оnе complex пит-
ber Ьу another, provided Ьу Eqs. (А-2) and (А-19') inAppendixA, the fre-
quency magnitude response of а general IIR filter is
I Num(ro) I [Numreal (ro)] 1 + [Numimag(ro)] 1
I НпR(ro) 1= = (6 34)
lDen(ro) I [Denrea!(ro)] 1 + [Denimag(ro)] 2
Furthermore, the filter's phase response 0 пR (ro) is the phase of the numer
ator minus the phase of the denominator, or
в'ПR(ro) = 0Num(ro) 0Den(ro)
l ( NUmimag(ro) ) l ( Denimag(ro) )
= tan tan
Num rea ! (ro) Den rea ! (ro)
( 6--35)
248 /пfiпi1e /тpи/se Respoпse Fllters
То reiterate our intent here, we've gone through the above algebraic gym
nastics to develop expressions for ап IIR filter's frequency magnitude
response I H1IR(0)) I and phase response "'IIR(O)) in terms of the filter coef
ficients in Eqs. (6 30) and (6 32), 5hortly, we'll use these expressions to
anal yze ап actual IIR filter.
Pausing а moment to gather our thoughts, we realize that we can use
Eqs. (6-34) and (6 35) to compute the magnitude and phase response of
IIR filters as а function of the frequency 00. And again, just what is О)? It's
the I10rmalized radian frequency represented Ьу the angle around the unit
circle in Figure 6-13, having а range о! 7t $; О) $; +7t, In terms of а discrete
sampling frequency 0). measured in radians/s, from Figure &-14, we see
that О) covers the range 0)./2 $; О) $; +0)/2, In terrns of ош old friend f. Нz,
Eqs. (6 34) and (6 35) apply over the equivalent frequency range of fs/2
to +1./2 Hz. 50, for example, if digital data is arriving at the filter's input
at а rate off s = 1000 samples/s, we could use Eq, (6 34) to plot the fi1ter's
frequency magnitude response over the frequency range of 500 Hz to
+500 Hz,
Although the equations describing the transfer function НпR(О)), its
magnitude response I H1IR(0)) 1, and phase response "'ПR(О)) look some
what complicated at first glance, let's illustrate their simplicity and utility
Ьу analyzing the simple second order low pass IIR filter in Figure 6 19
whose positive cutoff frequency is 0)./10, Ву inspection, we can write the
filter's time domain difference equation as
х(п)
0,0605
....(),436
Flgure 6-19 Second order low pass IIR filter example,
The z-Traпsforт 24
у(п) = 0.0605. х(п) + 0.121. х(п 1) + 0,0605. х(п 2)
+l,194'y(п 1) O,436'y(п 2) ,
(6--36)
or the z domain expression as
Y(z) = 0.0605. X(z) + 0.121. X(Z)Z 1 + 0,0605. X(z)z 2
+ 1.194. Y(Z)Z 1 0.436. Y(z)z 2 .
(6 37)
Using Eq. (6 25), we write the z-domain transfer function H(z) as
н(z) = Y(z) = 0,0605. zO + 0.121. Z 1 + 0.0605. z 2
X(z) 1 1,194. Z 1 + 0.436. Z 2
(6 38)
Replacing z with ei ro , we see that the frequency response of our example
IIR fil ter is
н O,0605'e joro +0.121'e jlro +O,0605.e j2ro
ПR (оо) 1 1.194. e jlro + 0.436. e j2ro
( 6 39)
We're a1most there, Remembering Euler's equations and that cos(O) = 1
and sin(O) = О, we сап write the rectangular form of НпR(ro) as
н (00)= O,0605+0,121'cos(loo)+O,0605'cos(200) j[O,121,sin(lOO)+O,0605'sin(200)] , (6-40)
ПR 1 1,194, cos(1ro) + 0,436' cos(2oo)+ j[l,194' sin(loo) 0,436 .sin(2oo)]
Equation (6 40) is what we're after here, and if we calculate its magnitude
over the frequency range of п::;; оо::;; п, we get the ! НпR(ro) I shown as the
solid curve in Figure 6 20(a). For comparison purposes we also show а 5
tap low pass FIR filter magnitude response in Figure 6 20(a). Although
both filters require the same computational workload, five mu1tiplications
per filter output sample, the 10w pass IIR filter has the superior frequency
magnitude response, Notice the steeper roll off and 10wer sidelobes of the
IIR filter relative to the FIR fi1ter,t
t То make this ПR and FIR filter comparison valid, the coeffidents used for both filters were
chosen so that еасЬ filter would approximate the ideallow-pass frequency response shown
in Figure 5-17(а),
250 /пflпne /mpu/se Respoпse Filters
(а)
IH 11R (w)1 for the IIR filter
, IH F1R (т)1 for а 5-tap FIR filter
\/
,«,««А
OJ=4t
HJ2)
OJ = --ТrJ2
( /4)
о w =1tI4 OJ =1f12
('s/8) ( /4)
OJ =1t OJ
(f 5/2)
(Ь)
"'IR (т) for the 11 R filter
у'
"" '
I ',,- ;
I ',- .
"
l ""
OJ = 'JJ OJ = --ТrJ2
Hi 2 ) ( /4)
Degrees
180
90
(5 14 )
OJ =1f12
(fs/2)
I"''P =1t
J I
! '
/
"FIR (т) for а 5-tap FIR filter
OJ
180
Flgure 6-20 Frequency responses of the example IIR filter (solid liпe) in Flgure 6-19
and а 5-tap FIR filter (dashed line): (а) magnitude responses; (Ь) phase
responses,
А word of waming here. It' s easy to reverse some of the signs for the
terms in the denorninator of Eq. (6-40), so Ье careful if уои attempt these
ca1culations at home. 50те authors avoid this problem Ьу showing the
a(k) coefficients in Figure 18 as negative values, so that the summation
in the denominator of Eq. (6-25) is always positive. Moreover, some сот-
mercial software IIR design routines provide a(k) coefficients whose signs
must Ье reversed before they сan Ье applied to the IIR structure in Figure
18. (Н, while using software routines to design or analyze IIR filters,
your results are very strange or unexpected, the first thing to do is reverse
the signs of the a(k) coefficients and see if that doesn't solve the problem.)
The solid curve in Figure 6-20(Ь) is our IIR filter's "'ПR(ro) phase
response, Notice its nonlinearity relative to the FIR filter's phase
response. (Remember now, we're only interested in the filter phase
responses over the low-pass filters' passband, 50 those phase discontinu-
ities for the FIR filter are of по consequence.) Phase nonlinearity is inher-
ent in IIR filters and, based оп the ill effects of non1inear phase introduced
in the group delay discussion о! 5ection 5,8, we must carefully consider
its implications whenever we decide to use an IIR fi1ter instead of an FIR
The z- Traпsforт 251
filter in any given application. Тhe question anу filter designer must ask
and answer, is "How тиф phase distortion сan 1 tolerate to realize the
benefits of the reduced computationalload and high data rates afforded
Ьу IIR filters?"
We've arrived at an appropriate point to illustrate an important rela-
tionship between IIR and FIR filters. We сап use our IIR analysis equa-
tions to characterize FIR filters. Notice that, if there were по feedback
coefficients in Figure 6 18, equivalent to setting а(l) through а(М) in Eq.
(6-21) а11 to zero, we'd have an N tap FIR filter. Тhis simplifies Eqs. (6-25)
through (6 28) because their denominators would Ье equal to one. In
addition, Den(O)) in Eq. (6-33) and I Den(O)) I in Eq. (6-34) would both Ье
unity, and "'Оеn(О)) would Ье zero in Eq. (6-35). Тhus, for an FIR filter, with
the a(k) coefficients in Figure 6 18 being а11 zeros, we сап rewrite Eqs.
(6-26) and (6-28) as
Frequency response of an
N-stage FIR filter
(exponential form):
N
H FIR (О)) = Lb(k)e jkro,
k=O
(6 41)
and
Frequency response
of an N-stage FlR
filter (rectangular
form):
N N
H F1R (O)) = Lb(k)'cos(kO)) jLb(k)'sin(kO)) .
k=O k=O
( 6 42)
Just as in Eqs. (6-34) and (6-35), the о) in Eqs. (6-41) and (6-42), is the
normalized radian frequency represented Ьу the angle around the unit
circle in Figure 6-13, having а range of л о) +Л, corresponding to the
equivalent frequency range of fJ2 to +fJ2 Hz.
6.3.3 А" Improved IIR Fllter Structure
Тhe structure of the I1R filter in Figure 6-18 сan Ье converted to а more
efficient and popular form. It's easy to explore this idea Ьу assuming that
there are an equal number of feed forward and feedback stages, letting
М = N in Figure 6-18 and thinking of the feed forward and feedback
stages as two separate filters, Specifica11y, the b(k) coefficients are used to
calculate the sequence d(n) from х(п), and the a(k) coefficients are used to
calculate the filter output у(п) sequence from d(n). Because the two halves
of the filter are both linear, we сап swap them, as shown in Figure 6 21(a),
with по change in the finаl у(п) output, Of course, after this reorientation,
252 /пfiп/te /три/sе Respoпse FlJters
а(""
Ь(""
(а)
а(""
Ь(""
(Ь)
Flgure 6.21 Rearranged Mh order IIR fllter structure: (а) feed forward and feedback
paths swapped from the structure in Figure 6-3: (Ь) flnal slmplifled Direct
Form 11 fllter structure,
а new seqиence that we'll call g(п) is generated, The two identical delay
paths in Figиre 6 21(a) provide the motivation for this reorientation.
Becaиse the seqиence g(п) is being shifted down along both delay paths
in Figиre 6--21(а), we сan eliminate оnе of the paths and arrive at the sim-
plified Direct Forт II filter structure shown in Figиre 6-21(Ь). Notice that
eliminating а delay path simplifies the strиctиre, so that а hardware
implementation would need only half the number of delay storage regis-
ters reqиired Ьу the originallIR structиre in Figиre 6-18.
Althoиgh the filter strиctиre depicted in Figure 6-21(Ь) is very popиlar,
in the literatиre we'll also encoиnter the eqиivalent representation shown
in Figure 6-22, It's interesting to compare Figиre 6--22's IIR structиre with
the FIR strиcture in Figиre 6-2(а). If the feedback coefficients a(k) in Figиre
6-22 were all zeros, for example, then this ПR filter would Ье eqиivalent
to the FIR filter in Figиre 6-2(а) as long as the b(k) I1R coefficients were
equal to the h(k) FIR coefficients. It's very important to keep in mind that
Eq. (6-21), associated with the IIR structure in Figиre 6-18, does not ехрНс-
Юу describe the filter structures in Figures 6-21(Ь) and 6-22. However, the
freqиency-domain expressions in Eqs. (6--25) through (6-35) сап Ье иsed to
analyze the performance of the popиlar filter structиres in Figures 6-21(Ь)
and 6-22,
The z-Traпsforт 25,
g( N)
Flgure 6-22 Alternate depiction of the IIR fllter Dlrect Form 11 structure in Figure tr21 (Ь),
Ву the way, because о! the feedback nature о! ПR filters, they're some-
times rеfепеd to as recursive filters. Similarly, FIR filters are often called
пoпrecursive filters. (Тhe term recursive originally signified filters with
both poles and zeros оп the complex z plane whereas пoпrecursive signi
fied filters with only zeros оп the z-plane.) We'll avoid the terms recursive
and пoпrecursive, however, because it's been shown that filters with only
zeros сап Ье designed with recursive structures and that filters with both
poles and zeros оп the z plane сап Ье designed with nonrecursive struc
tures[10]. So the terms recursive and пoпrecursive more correctly apply to
the structures used to implement а digital filter, and the terms IIR and FIR
better describe whether а filter has only zeros (FIR) or both poles and
zeros (ПR) оп the z-plane[l1].
Now that we have а feeling for the performance and implementation
structures о! I1R filters, let's briefly introduce three filter design tech-
niques. Тhese I1R design methods go Ьу the impressive names of impulse
iпvariaпce, biliпear traпsform, and optimized methods, Тhe first two methods
use aпalytical techniques to design digital filters that approximate con-
tinuous analog filters. t Тhe impulse invariance and bilinear transform
I1R design methods both start with а closed form Laplace equation
describing the desired analog filter and yield z transform expressions for
the approximating I1R digital filter, Тhe filter designer then solves the
z transform equations to obtain the a(k) and b(k) coefficients for use in the
I1R structures in Figures 6 21(b) or 6 22, Because analog filter design
t Due to its popularity, throughout the rest of this chapter we'lI use the phrase aпa/og fi/ter to
designate those filters made ир of resistors, capacitors, and inductors, designed to operate
оп continuous signals,
254 /пfiпJte /тpu/se Respoпse F//ters
methods are very well understood, designers сап take advantage of an
abundant variety of analog fi1ter design techniques to design, say, а
Butterworth fi1ter with its very f1at passband response or, perhaps, go
with а Chebyshev fi1ter with its f1uctuating passband response and
sharper passband to--stopband cutoff characteristics,
6.4 Impulse Invariance IIR Filter Design Method
Тhe impulse invariance method of IIR filter design is based upon the
notion that we сап design а discrete fi1ter whose time domain impulse
response is а sampled version of the impulse response of а continuous
analog fi1ter. If that analog fi1ter (often called the prototype filter) has some
desired frequency response, then our IIR fi1ter will yield а discrete
approximation of that desired response, The impulse response equiva-
lence of this design method is depicted in Figure 6-23 where we use the
conventional notation of о to represent an impulse function and hP) is
the analog fi1ter's impulse response. We use the subscript "с" in Figure
6 23(a) to emphasize the continuous nature of the analog fi1ter, Figure
6 23(Ъ) il1ustrates the definition of the discrete fi1ter's impulse response:
the filter's time domain output sequence when the input is а single
unity-valued sample (impulse) preceded and followed Ьу аН zero valued
samples. our goal is to design а digital blter whose impulse response is а
sampled version of the analog blter's continuous impulse response.lmplied
in the correspondence of the continuous and discrete impulse responses is
the property that we can тар each pole оп the s plane for the analog blter' s
f\ ..
Analog \.../ Continuous
(а) Filter time
*)=0(1)
Hc(s) Impulse response
output = hJ!)
. :II!I,
i . : .
................:............................ IIR Digital .-. . ...-.....................
· '8 · Discrete
(Ь) Filter time
ж(n) = б(n) H(z) Impulse response
output = h(n) = hc(nls)
Figure 6-23 Impulse invariance design equivalence of (а) analog filter continuous
impulse response; (Ь) digital filter dlscrete impulse response,
/mpu/se /пvarioпee IfR Filter Deslgп Method 255
Hc(s) transfer function to а pole оп the z plane for the discrete ПR filter's
H(z) transfer function. What designers have found is that the impulse
invariance method does yield useful ПR filters, as long as the sampling
rate is high relative to the bandwidth оЕ the signal to Ье filtered. In other
words, IIR fi1ters designed using the impulse invariance method are sиs
ceptible to aliasing problems becaиse practical analog fi1ters cannot Ье
perfect1y band limited. Aliasing wil1 occиr in an ПR filter's freqиency
response as shown in Figure 6 24.
From what we've learned in Chapter 2 aboиt the spectral replicating
effects оЕ sampling, if Figure 6 4(a) is the spectrum оЕ the continиoиs hc(t)
impиlse response, then the spectrиm оЕ the discrete hc(пt s ) sample
seqиence is the replicated spectra in Figure 6 24(b).
(а)
А
о
...
m
,
(Ь)
2"
( 2'5)
--1t
о
1t
211 m
(2'5)
Н 5 )
((5)
,
(с)
2"
( 2'5)
--1t
Н 5 )
о
1t
2" m
(2'5)
('5)
Flgure 6-24 Allosing In the Impulse invorlanee design method: (о) prototype
onolog filter mognitude response; (Ь) replieoted mognltude responses
where H11R(m) is the diserete Fourler tronsform of h(п) = hc(пt s ); (е)
potentiol resu/tont IIR filter mognltude response wlth ollosing effeets,
256 lпfiпife Impulse Respoпse Fllters
In Figure 6-24(с) we show the possible effect of a1iasing where the
dashed curve is а desired НпR(ro) frequency magnitude response.
However, the actual frequency magnitude response, indicated Ьу the
solid curve, сап occur when we use the impulse invariance design
method. For this reason, we prefer to make the sample frequency fs as
large as possible to minimize the overlap between the primary frequency
response curve and its rep1icated images spaced at multiples of :r.fs Hz, То
see how a1iasing can affect IIR filters designed with this method, let's 1ist
the necessary impulse invariance design steps and then go through а т-
ter design example,
There are two different methods for designing IIR filters using impulse
invariance, Тhe first method, which we'll саН Method 1, requires that an
inverse Laplace transform as weH as а z-transform Ье performed[12,13],
Тhe second impulse invariance design technique, Method 2, uses а direct
substitution process to avoid the inverse Laplace and z-transformations at
the ехреnsе of needing partial fraction expansion algebra necessary to
handle polynomials[14 17], Both of these methods seem comp1icated
when described in words, but they're really not as difficult as they sound,
Let's compare the two methods Ьу 1isting the steps required for each of
them, Тhe impulse invariance design Method 1 goes like this:
Method 1, Step 1: Design (or have someone design for уои) а prototype
analog fi1ter with the desired frequency response,t
Тhe result of this step is а continuous Laplace trans-
fer function H/s) expressed as the ratio of two poly-
nomials, such as,
N
L b(k)sk
Н (s)= b(N)SN +b(N l)SN l+..,+b(l)s+b(O) k:O
с a(M)sM + а(М l)SM-l +... + a(l)s + а(О) k k
k.J а( )s
k=O
(6-43)
which is the general form of Eq, (6-10) with N < М,
and a(k) and b(k) are constants.
Method 1, Step 2: Determine the analog filter's continuous time-
domain impulse response hc(t) from the Hc(s) Laplace
t In а low-pass filter design, for example, the filter type (Chebyshev, Butterworth, elliptic), fil-
ter order (number of poles), and the cutoff frequency are parameters to Ье defined in this step,
/тpиlse /пvariaпce J/R Fi/ter Desigп Method 257
transfer function. 1 hope, this сan Ье done using
Laplace tables as opposed to actually evaluating an
inverse Laplace transform equation,
Method 1, Step 3: Determine the digital filter's sampling frequency 1.,
and calculate the sample period as t. = 1/1., The 1.
sampling rate is chosen based оп the absolute fre
quency, in Hz, оЕ the prototype analog filter, Because
of the aliasing problems associated with this impulse
invariance design method, later, we'll see why 1.
should Ье made as large as practical,
Method 1, Step 4: Find the z transform оЕ the continuous hc(t) to obtain
the IIR filter's z domain transfer function H(z) in the
form of а ratio оЕ polynomials in z.
Method 1, Step 5: 5ubstitute the value (not the variable) t. for the con
tinuous variable t in the H(z) transfer function
obtained in 5tep 4, In performing this step, we are
ensuring, as in Figure 6-23, that the IIR fiIter's dis
crete h(n) impulse response is а sampled version of
the continuous filter's hc(t) impulse response so that
h(n) = hc(nt.), for О S; п S; 00.
Method 1, Step 6: Our н(z) from 5tep 5 will now Ье оЕ the general form
N
Lb(k)z k
н(z) = b(N)z N + b(N l)Z (N l) +... + b(l)z l + Ь(О) = k=O , (6-44)
a(M)z M+ а(М l)z (M l) +... + a(l)z l + а(О) k
1 ""La(k)z
k=l
Because the process of sampling the continuous
impulse response results in а digital filter frequency
response that's scaled Ьу а factor of l/t., тanу fiIter
designers find it appropriate to include the t. factor in
Eq. (6-44). 50 we сan rewrite Eq. (6-44) as
N
<Lb(k)Z k
Н(z) = Y(z) = k=O
X ( z ) М
1 La(k)Z k
k=l
(6 4S)
258 /пfiп/te /mpu/se Respoпse FlJters
1ncorporating the value of ' . in Eq. (6 45), then,
makes the 1IR fi1ter time response scaling indepen
dent of the sampling rate, and the discrete filter wi11
have the same gain as the prototype analog filter. t
Method 1, Step 7: Because Eq. (6 44) is in the form of Eq. (6-25), Ьу
.inspection, we сап express the filter' s time domain
difference equation in the general form of Eq. (6 21) as
у(п) = Ь(О)х(п) + b(l)x(n 1) + Ь(2)х(п 2) + '.' + b(N)x(n N)
+ a(l)y(n 1) +а(2)у(п 2) + ... + а(М)у(п М) .
(6--46)
Choosing to incorporate t s ' as in Eq. (6 45), to make
the digital filter's gain equal to the prototype analog
filter's gain, Ьу multiplying the b(k) coefficients Ьу
the sample period t s ' leads to an IIR filter time-
domain expression of the [олn
у(п) = t s . [Ь(о)х(п) + b(l)x(n 1) + Ь(2)х(п 2) + '" + b(N)x(n N)]
+ а(l)у(п 1) + а(2)у(п 2) + ...н(М)у(п М). (6--47)
Notice how the signs changed for the a(k) соеШ
cients from Eqs, (6-44) and (6 45) to Eqs. (6 46) and
(б 47). These sign changes always seem to cause
problems for beginners, so watch out. AIso, keep in
mind that the time-domain expressions in Eq,
(6--46) and Eq. (6 47) apply only to the filter struc
ture in Figure 6-18. The a(k) and b(k), or t s . b(k), coef-
ficients, however, сап Ье applied to the improved
ПR structure shown in Figure 6 22 to complete our
design.
Before we go through an actual example of this design process, let's dis
cuss the other impulse invariance design method,
t Some authors have chosen to include the t s factor in the discrete h(n) impuJse response in
the above Step 4; that is, make h(n) = tЛ(пt s ) [14, 18], Тhe final result of this, of course, is the
same as that obtaтed Ьу including t s as described in Step 5,
-
/тpu/se /пvariaпce I/R Filter Desigп Method 259
М-роlе a"alog prototype II"er
хи)
N r M
Lb(k)sk :[B{k)sk
О О
уЩ
n
PartiaJ lractio" expa"d
\1
ф
Х(п)
X(t)
Perform
substitutio"
Х(п)
n
Algebraically сотЫ"е
V
t. b(k)z k/ f.B{k)z k
о ;7 о
М-роlе discrete НА filter
у(п)
Flgure 6-25 Mathematlcal ftow ot the impulse Invarlance design Method 2,
The impulse invariance Design Method 2, also cal1ed the standard
z-transfoпn method, takes а different approach, It mathematical1y parti-
tions the prototype analog filter into mu1tiple single pole continuous filters
and then approximates each one of those Ьу а single-pole digital filter, The
set of М single-pole digital filters is then algebraically combined to form an
M-pole, Mth-ordered IIR filter, Тhis process of breaking the analog filter to
discrete filter approximation into manageable pieces is shown in Figure 6
25, The steps necessary to perform an impulse invariance Method 2
design are
Method 2, Step 1: Obtain the Laplace transfer function Hc(s) for the pro-
totype analog filter in the [oпn of Eq, (6 43), (Same as
Method 1, Step 1.)
Method 2, Step 2: Se1ect an appropriate sampling frequency f. and cal-
culate the sample period as t 5 = 1//5' (Same as
Method 1, Step 3,)
.
260 /пfiпite /тpu/se Respoпse Filters
Method 2, Step 3: Express the analog fi1ter's Laplace transfer function
His) as the sum of single-pole filters, Тhis requires us
to use partial fraction expansion methods to express
the ratio of polynomials in Eq, (6 43) in the form of
н (s) = b(N)sN + b(N l)sN l + ...+ b(l)s + Ь(О)
с a(M)sM +a(M l)sM l +..,+a(l)s+a(O)
А А А А
= = +.............L.+...+............м.., (6 48)
k;l s+Pk s+Pl s+P2 s+PM
where the individual Ak factors are constants and the
kth pole is located at Pk оп the s plane, We'll denote
the kth single-pole analog filter as Hk(s), or
А
Hk(S)= .
S+Pk
(6--49)
Method 2, Step 4: 5ubstitute 1 e Pkt, Z l for s + Pk in Eq. (6-48). This
mapping of each Hk(s) pole, located at s = Pk оп the
s-plane, to the z = e Pkt, location оп the z-plane is how
we approximate the impulse response of each single--
pole analog filter Ьу а single--pole digital filter. (Тhe
reader сап find the derivation of this 1 e-р 'Z l sub
stitution, illustrated in our Figure 6 25, in references
[14] through [16].) 50, the kth analog single--pole filter
Hk(s) is approximated Ьу а single pole digital fi1ter
whose z-domain transfer function is
н ( z ) Ak
k 1 Pk t 1
e 'Z
( 6 50)
The final combined discrete fi1ter transfer function
H(z) is the sum of the single--poled discrete filters, or
м м А
Н(z) = LHk(Z) = L p t, 1 ' (6 51)
k;l k;l 1 е z
/тpu/se /пvariaпce I/R Filter Desigп Method 261
Кеер in mind that the above н(z) is not а funetion of
time, Тhe t s faetor in Eq. (6 51) is а eonstant equal to
the discrete time sample period.
Method 2, Step 5: Calculate the z domain transfer ЕипеНоп of the sum
of the М single pole digital filters in the form of а
ratio of two polynomials in z, Beeause the H(z) in
Eq. (6 51) will Ье а series of fraetions, we'll have to
еотЬinе those fraetions over а еоттоп denomina-
tor to get а single ratio of polynomials in the [aтH
iar form of
N
L b(k)z k
Н(z) = Y(z) = k='h
X(z) 1 L a(k)z k
k=l
(6 52)
Method 2, Step б: Just as in Method 1 Step 6, Ьу inspeetion, we еап
express thE! filter's time--domain equation in the gen
eral form of
у(п) = Ь(О)х(п) + Ь(1)х(п 1) + Ь(2)х(п 2) + ..' + b(N)x(n N)
+ а(1)у(п 1) + а(2)у(п 2) +.., +а(М)у(п М) . (6 53)
Again, поНее the a(k) eoeffieient sign ehanges from
Eq. (6 52) to Eq, (6 53), As deseribed in Method 1
Steps 6 and 7, if we ehoose to make the digital filter's
gain equal to the prototype analog filter's gain Ьу
multiplying the b(k) eoeffieients Ьу the sample period
t., then the IIR filter's time domain expression will Ье
in the form
у(п) = t. . [Ь(О)х(п) + Ь(1)х(п 1) + Ь(2)х(п 2) +.., + b(N)x(n N)]
+a(1)y(п 1)+a(2)y(п 2)+,.,+a(M)y(п M), (6--54)
yielding а final Н(z) z domain transfer funetion of
262 /пfiпffe /три/5е Re5poп5e Filter5
N
t s . Lb(k)z k
н(z) = Y(z) = k O
X ( z ) М
1 La(k)z k
k l
(6 54')
Finally, we сап implement the improved IIR structure
shown in Figure 6 22 using the a(k) and b(k) соеЮ
cients from Eq. (6 53) or the a(k) and t s ' b(k) соеШ-
cients from Eq, (6 54).
То provide а more meaningful comparison between the two impulse
invariance design methods, let's dive in and go through an I1R filter
design example using both methods,
6.4. 1 Impulse Invarlance Design Method 1 Example
Assиme that we need to design an I1R fi1ter that approximates а second-
order Chebyshev prototype analog low-pass fi1ter whose passband ripple
is 1 dБ, Our f. sampling rate is 100 Hz (ts = 0,01), and the fi1ter's 1 dБ cut
off frequency is 20 Hz, Our prototype analog fi1ter will have а frequency
magnitude response like that shown in Figure 6 26.
Given the аЬоуе fi1ter requirements, assume that the analog prototype
fi1ter design effort results in the Hc(s) Laplace transfer function of
Hc(s) = 17410.145
s2 + 137.94536s + 17410.145
(6 55 )
It's the transfer function in Eq. (6-55) that we intend to approximate with
our discrete ПR fi1ter, То find the analog fi1ter's impulse response, we'd
IHdB(f)1
OdB
о
20Нz
50Hz
((5/2)
Freq
Flgure 6-26 Frequency magnitude respoпse of the example prototype
analog fllter,
/тpи/se /пvor/oпce I/R Fllter Desigп Method 263
like to get Н с (5) into а form that allows us to use Laplace transform tables
to find hc(t). Searching through systems analysis textbooks we find the fol-
lowing Laplace transform pair:
Х(5), Laplace traпsform о! x(t) :
А<о
(5 + ai + <02
x(t) :
Ae lX! . sin( <о t) .
(6-56)
our intent, then, is to modify Eq. (6-55) to get it into the form оп the left
side о! Eq. (6 56). We do this Ьу realizing that the Laplace transform
expression in Eq. (6 56) сап Ье rewritten as
А<о
(5+ai +<02
А<о
52 +2а5+а.2 +<02
(6-57)
If we set Eq. (6-55) equal to the right side о! Eq. (6-57), we сап solve for А,
а, and <о. Doing that,
н 5 17410,145
с( ) 52 + 137.945365 + 17410.145
А<о
52 +2а5+а 2 +<02
(6-58)
Solving Eq. (6-58) for А, а, and <о, we first find
а = 137.94536 = 68.972680 ;
2
(6 59)
а 2 + <02 = 17410,145 ,
(6-60)
so
<о = .,J 17410.145 a2 = 112.485173;
(6-61)
and
А = 17410.145 = 154.77724 .
<о
(6-62 )
ок, we сап now express Н с (5) in the desired form о! the left side о! Eq.
(6-57) as
264 /пflnite /mpu/se Respoпse F//ters
н 5 (154.77724)(112,485173)
. с( ) (5 + 68.972680)2 + (112.48517з)2
(6 3)
Using the Laplace transform pair in Eq. (&-56), the time domain impulse
response of the prototype analog filter becomes
hc(t) = Ae al . sin(cot) = 154,77724e ,9726801 . sin(112.485173t) . (6 64)
OK we're ready to perform Method 1, Step 4, to determine the discrete
IIR filter's z domain transfer function н(z) Ьу performing the z transform
of hc(t). Again, scanning through digital signal processing textbooks or а
good math reference book, we find the following z-transform pair where
the time domain expression is in the same form as Eq, (6-64)'s hc(t)
impulse response:
x(t) :
X(z), z transform of x(t) :
Се a! . sin( ф t)z 1
1 2[e al 'СОS(фt)]z l +e 2alz 2
(6-65)
Се a! . sin( фt)
Remember now, the а and ф in Eq. (6 65) are generic and are not related
to the а and ф values in Eq. (6 59) and Eq. (6 61). Substituting the con-
stants from Eq. (6 64) into the right side of Eq. (6 65), we get the
z transform of the IIR fi1ter as
н 154. 77724е 68,9726801 . sin(112.485173t)z 1
(z) = 1 2[e ,9726801 . cos(112.485173t)]Z 1 +e 2'68'972680IZ 2
(6-66)
Performing Method 1, Step 5, we substitute the t s value of 0,01 for the con-
tinuous variable t in Eq. (6 66), yielding the final н(z) transfer function of
н 154.77724e ,972680'O,Ol . sin(1l2.48S173. 0.01)Z 1
(z) = 1 2[e 68.972680,oт . cos(1l2,485173. O,Ol)]Z l +e 2'68.972680'0.D1Z 2
154.77724 е ..о,68972680. sin(l.l248517з)z 1
= 1 2[е..о,68972680 . соs(1.1248517з)]z 1 +e 2'O,68972680z 2
Y(z) 70.059517z 1
X(z) 1 O,43278805z 1 +O,25171605z 2
(&-67)
/mpu/se /пvariaпce J/R Filter Desigп Method 265
ОК, hang in there, we're almost finished. Here are the final steps of
Method 1, Because of the transfer function H(z) = Y(z)/X(z), we сап crOss-
multiply the denominators to rewrite the bottom line of Eq, (6-67) as
Y(z). (1 0.43278805z 1 + 0.25171605z 2) = X(z). (70,059517z 1) ,
or
Y(z) = 70,059517. X(Z)Z l + 0.43278805. Y(Z)Z l 0,25171605' y(z)z 2, (6 68)
Ву inspection of Eq, (6 68), we сап now get the time-domain expression
for our IIR filter. Performing Method 1, Steps 6 and 7, we mu1tiply the
x(п l) coefficient Ьу the sample period value of t s = 0,01 to allow for
proper scaling as
у(п) = 0,01.70,059517. х(п 1) + 0.43278805. у(п 1) 0,25171605' у(п 2)
= 0,70059517. х(п 1) +0,43278805. у(п 1) O,25171605. у(п 2), (6-69)
and there we (Нnаllу) are. The coefficients from Eq, (6-69) are what we use
in implementing the improved I1R structure shown in Figure 6-22 to
approximate the original second order Chebyshev analog low-pass filter,
Let's see if we get the same resu1t if we use the impulse invariance
design Method 2 to approximate the example prototype analog filter,
6.4.2 Impulse Invariance Design Method 2 Example
Given the original prototype filter's Laplace transfer function as
н 17410,145
с(5) = i + 137.945365 + 17410.145 '
(6 70)
and the value of t s = 0.01 for the sample period, we're ready to proceed
with Method 2's Step 3, То express Нс(5) as the sum of single-pole filters,
we'll have to factor the denominator of Eq, (6 70) and use partial fraction
expansion methods, For соnуеniеnсе, let's start Ьу replacing the constants
in Eq, (6-70) with variables in the form of
с
нс(s) = 2 '
5 + Ь5 + с
(6-71)
266 /пflпПе /mpu/se Respoпse F//ters
where Ь::: 137.94536, and С::: 17410,145. Next, using Eq, (6-15) with а::: 1,
we сап factor the quadratic denominator of Eq, (6-71) into
He(s)::: [ ) С [ ) .
ь ь 2 4с Ь ь 2 4с
B+2+ 4 . B+2 ) 4
(6-72)
If we substitute the vaIues for Ь and С in Eq, (6-72), we'lI find that the
quantity under the radicaI sign is negative, Тhis means that the factors in
the denominator of Eq. (6-72) are compIex, Because we have lots of aIge-
bra ahead of us, Iet's repIace the radicals in Eq. (6-72) with the imaginary
termjR, wherej::: Н and R::: I (1J2 4c)/41, such that
С
Не(В) =
(в+Ь/2+ jR)(s+b/2 jR)
(6-73)
ок, partial fraction expansion methods alIow us to partition Eq. (6-73)
into two separate fractions of the form .
С
Н (В)=
е (НЬ/2+ jR)(s+b/2 jR)
К} К 2
= + ,
(в+Ь/2+ jR) (B+b/2 jR)
(6-74)
where the К} constant сan Ье found to Ье equaI to jc/2R and constant К 2
is the compIex conjugate of К}I or К 2 = jc/2R. (То Ieam the details of par-
tiaI fraction expansion methods, the interested reader shouId investigate
standard coIIege algebra or engineering mathematics textbooks.) Тhus,
Не(в) сап Ье of the form in Eq. (6-48) or
н s jc/2R jc/2R
е( ) (НЬ/2+ jR) + (B+b/2 jR)
(6-75)
We сап see from Eq. (6-75) that our second-order prototype fiIter has two
poIes, опе Iocated at р} = b/2 jR and the other at Р2 ::: Ы2 + jR. We're
now ready to тар those two poIes from the s-plane to the z-рIапе as
caIled out in Method 2, Step 4. Making our 1 e P*',Z'"'} substitиtion for the
S +Pk terms in Eq. (6-75), we have the foIIowing expression for the
z-domain singIe-роIе digital filters,
I
/mpu/se /пvar/aпce //R F//ter Des/gп Method 267
Н J 'c/2R J 'c/2R
(z) +
1 e (b/2+jR)t'z 1 1 e (b/2 JR)t,z 1'
(6-76)
Our objective in Method 2, Step 5 is to massage Eq. (6 76) into the form of
Eq. (6 52), so that We can determine the IIR filter's feed forward and feed-
back coefficients. Putting both fractions in Eq. (6 76) over а common
denominator gives us
ис /2R}(1 e (b/2 jR)t, Z-I) ис /2R)(l e (b!2+jR)t, z-l)
H(z) (1 e (b/2+jR)t'z 1}(1 e (b/2 jR)t'z 1) ,
(6 77)
Collecting like terms in the numerator and multiplying out the denomi
nator gives us
ис /2R}(1 e (b/2 jR)t, Z 1 1 + e (b!2+jR)t, Z I)
H(z) = l е (Ь/2 jR)t'z-1 e (b/2+jR)t'z 1 +e bt,z-' (6 78)
Factoring the exponentials and collecting like terms of powers of z in
Eq. (6 78),
(jc /2R)(e (b!2+jR)t, e (b/2 jR)t, )Z l
H(z) 1 (e (b/2 jR)t, + e (b!2+jR)t, )Z l + e Ы, Z 2
(6-79)
Continuing to simplify our Н(z) expression Ьу factoring out the real part
of the exponentials,
(jc/2R)e bt, !2(e jRt, ejRt, )Z l
Н(z)= "
1 e bt,/2(eJRt, +e JRt, )Z l +e bt'z 2
(6 80)
We now have Н(z) in а form with all the like powers of z combined into sin
gle terms, and Eq, (6 80) looks something like the desired form of Eq.
(6 52), Кnowingthatthe final coefficients of our IIR filter must Ье real num-
bers, the question is "What do we do with those imaginary j terms in Eq.
(6 80)?" Once again, Euler to the rescue,+ Using Euler's equations for sinu
soids, we can eliminate the imaginary exponentials and Eq. ( 80) becomes
+ From Euler, we know that sin(0) = (J0 e j0)/2j, and COS(0) = (ei'" + e j0)/2,
I
268 /nfJnite /тpu/se Respoпse F/lters
( J 'C / 2R ) e bt. /2[ 2 J ' sin ( Rt ) ]Z l
н(z) == s
1 e bt. /2[2cos(Rts)Jz 1 +e Ы. Z 2
(С/ R)e bt./2[sin(Rts)Jz 1
1 e bt. /2[2cos(Rts)Jz 1 + e Ы. Z 2
(&-81)
If we plug the values С == 17410.145, Ь == 137.94536, R == 112.48517, aпd
t s == 0.01 into Eq. (6-81), we get the following I1R filter transfer function:
н (z) == (154.77724) (0.50171312) (0.902203655)z 1
1 (0.50171312) (0.86262058)z 1 + 0.25171605z 2
70.059517z 1
1 О,4З278805z 1 +0.25171605z 2
(6 82)
Весаше the transfer function н(z) == Y(z)/X(z), we сап again cross mu1tiply
the denominators to rewrite Eq. (&-82) as
Y(z). (1 О,4З278805z 1 +0.25171605z 2) = X(z). (70.o59517z 1) ,
or
Y(z) == 70,059517 . х(ф 1 + 0.43278805. Y(Z)Z l 0,25171605. у(ф 2 . (6 83)
Now we take the inverse z transform of Eq. (6 83), Ьу inspection, to get
the time domain expression for our I1R fi1ter as
у(п) == 70.059517. x(п 1) + 0.43278805. y(п 1) 0,25171605. у(п 2) , (&-84)
Onе final step remains, То force the IIR fi1ter gain equal to the prototype
analog fi1ter's gain, we mиlНрlу the x(п l) coefficient Ьу the sample
period t s as suggested in Method 2, Step 6. In this case, there's only оnе
х(п) coefficient, giving us
у(п) == 0.01.70.059517' х(п 1) + 0.43278805. у(п 1) 0,25171605' у(п 2)
== 0,70059517. х(п 1) + 0.43278805' у(п 1) 0.25171605. у(п 2), (&-85)
.
/три/5в /пvariaпce J/R FiJter De5/gп Method 269
plane z-plane
Imag(s) Imag(Z)
Stabl. Unstable
region region
* ., +jR
(а) -,
Ы2 Real( s) ReaI(Z)
)(. jR
'Н( I in dB
о
-2
-4
(Ь) -6
-8
О 20 Hz
50 Hz Freq
(',12)
Flgure 6-27 Impulse Invariance design example fllter characteristics: (а) s-plane pole
locatlons of prototype analog fllter and z-plane pole locatlons of discrete
IIR fllter: (ь) frequency magnltude response of the d;screte IIR fllter,
that compares wel1 with the Method 1 result in Eq. (6-69). (1sn't it сот-
forting to work а problem two different ways and get the same result?)
Figure 6-27 shows, in graphical form, the result о! our ПR design ехат-
ple, The s-plane pole locations о! the prototype filter and the z-plane poles
о! the ПR filter are shown in Figure 6-27(а), Because the s-plane poles are
to the left о! the origin and the z-plane poles are inside the unit circle, both
the prototype analog and the discrete llR filters are stable, We find the
prototype filter's s-plane pole locations Ьу evaluating Hc(s) in Eq, (6-75),
When s = b/2 jR, the denominator о! the first term in Eq, (6 75)
becomes zero and H/s) is infinitely large, That s = b/2 jR value is the
location о! the lower s-plane pole in Figure 6-27(а). When s = b /2 + jR the
denominator о! the second term in Eq, (6-75) becomes zero and
s = b/2 + jR is the location о! the second s-plane pole,
The 1IR filter's z-plane pole locations are found from Eq, (6-76). If we
multiply the numerators and denominators о! Eq, (6 76) Ьу z,
270 lпflnlte lтриlsе Respoпse Filters
н(z),!..= z(jc/2R) + z( jc/2R)
z z(1 e (b/2+JR)t'z 1) z(1 e (bI2 JR)t'z 1)
(jc/2R)z ( jc/2R)z
z e (bI2+jR)t, + z e (b/2 jR)t,
(6-86)
In Eq, (6-86), when z is set equal to e( b!2 + jR)t" the denominator оЕ the first
term in Eq, (6-86) becomes zero and н(z) becomes infinitely large. The
value of z оЕ
z=e (bI2+jR)t, =e bt,/2e jRt, =e bt,/2L Rt radians
s
(6-87)
defines the location оЕ the lower z-plane pole in Figure 6-27(а),
SpecificalIy, this lower pole is located at а distance оЕ с Ы ,/2 = 0.5017 from
the origin, at an angle оЕ е = Rts radians, or --64.450, Being conjugate
poles, the upper z-plane роlе is located the same distance from the origin
at an angle оЕ е = Rt s radians, or +64.450, Figure 6--27(Ь) illustrates the fre-
quency magnitude response of the ПR filter in Hz,
Two different implementations оЕ our I1R filter are shown in Figure
6-28, Figure 6-28(а) is an implementation оЕ our second-order I1R filter
based оп the generallIR structure given in Figure 6-22, and Figure
6-28(Ь) shows the second-order IIR filter implementation based оп the
alternate structure from Figure 6-21(Ь), Кnowing that the Ь(О) coefficient
оп the left side оЕ Figure 6-28(Ь) is zero, we arrive at the simp1ified struc-
ture оп the right side оЕ Figure 6-28(Ь), Looking carefulIy at Figure
6-28(а) and the right side оЕ Figure 6-28(Ь), we сап see that they are
equivalent.
Although both impulse invariance design methods are covered in the
literature, we might ask, "Which one is preferred?" There's по definite
answer to that question because it depends оп the HC<s) оЕ the prototype
analog filter, Although our Method 2 ехатрlе аЬоуе required more
algebra than Method 1, if the prototype filter's s-domain poles were
located on1y оп the real axis, Method 2 would have been much simpler
because there would Ье по сотрlех variables to manipulate, In general,
Method 2 is more popular for two reasons: (1) the inverse Laplace and
z-transformations, although straightforward in our Method 1 example,
сап Ье very difficult for higher-order filters, and (2) unlike Method 1,
Method 2 сап Ье coded in а software routine or а computer spreadsheet,
Upon examining the frequency magnitude response in Figure 6-27(Ь),
we сап see that this second-order IIR filter's roll-off is not particularly
/тpu/se /пvariaпce J/R Filter Desigп Method 271
(а)
Ь(1)
(Ь)
а(2)
Ь(1) = 0.70059517
а(1) = 0.43278805
а(2) = -0,25171605
у(п)
у(п)
F/gure 6-28 lтpleтentations of the lтpulse invariaпce deslgn example filter,
а(2)
steep, This is, admittedly, а simple low order filter, but it's attenuation
slope is БО gradual that it doesn't appear to Ье оЕ much use аБ а low pass
filter,t We сan also Бее that the filter's passband ripple is greater than the
desired value оЕ 1 dB in Figиre 6 26, What we'll find is that it's not the low
order оЕ the fiIter that contributes to it's poor performance, but the sam
pling rate used, Тhat second order IIR filter response is repeated аБ the
shaded curve in Figure 6 29, If we increased the sampling rate to 200 Hz,
we' d get the frequency response shown Ьу the dashed curve in Figure
6.29, Increasing the sampling rate to 400 Hz results in the тит improved
frequency response indicated Ьу the solid linе in the figure, Sampling rate
changes do not affect our filter order or implementation structure,
Remember, if we change the sampling rate, only the sample period t s
changes in our design equations, resulting in а different set оЕ filter
t А piece of advice. Whenever уои encounter аnу frequency representation (ье it а digita! fil-
ter magnitude response or а signa! spectrum) that has nonzero values at +1.12, Ье suspi-
cious----be very suspicious----that aliasing is taking р!асе,
. 272 /пffпlte /mpu/se Respoпse F/Jters
IH(f)1
1
0,9
0.8
0,7
0,6
0,5
0.4
0,3
0,2
0,1
О
О
.,.....
>(' '8=100Hz
,
, "-
" '8=200Hz
,/ .---- ,
"
"
"
'....
..........................
.........................................
20 Hz when
'8= 400 Hz
20 Hz when
'8= 100 Hz
'8/2 Freq
Figure 6-29 IIR fllter frequency magnitude response, оп а Ilnear scale, at three
separate sampllng rates, Notice how the fllter's absolute cutoff
frequency of 20 Hz shifts relative to the dlfferent 's sampling rates,
coefficients for each new sampling rate. 50 we сап see that the smal1er we
make t s (larger 1.) the better the resulting fi1ter when either impulse
inуапаnсе design method is used because the replicated spectral overlap
indicated in Figure 6 24(b) is reduced due to the larger fs sampling rate.
The bottom linе here is that impulse invariance I1R filter design tech
niques are most appropriate for narrowband filters; that is, low pass fil
ters whose cutoff frequencies are much smal1er than the sampling rate.
Тhe second analytical technique for analog fi1ter approximation, the
bilinear transform method, alleviates the impulse invariance method's
aliasing problems at the expense of what's cal1ed frequency warping.
5pecifical1y, there' s а nonlinear distortion between the prototype analog
filter' s frequency scale and the frequency scale of the approximating I1R
filter designed using the bilinear transform. Let's see why.
6.5 Bilinear Transform IIR Filfer Design Mefhod
There's а popular analytical IIR fi1ter design technique known as the ыип-
ear traпsform method. Like the impulse invariance method, this design tech
nique approximates а prototype analog filter defined Ьу the continuous
Laplace transfer function H/s) with а discrete filter whose transfer function
is H(z). However, the bilinear transform method has great utility ЬесаUБе
.
Blliпear Traпsforт I/R Fllter Des/gп Method 273
· it a1lows us to simply substitute а fиnction of z for s in Hc(s) to get H(z),
thankfully, eliminating the need for Laplace and z-transformations as
well as anу necessity for partial fraction expansion a1gebra;
· it maps the entire s-plane to the z plane, enabling us to completely
avoid the frequency domain aliasing problems we had with the
impulse invariance design method; and
· it induces а nonlinear distortion of H(z)'s frequencyaxis, relative to
the original prototype analog filter's frequencyaxis, that sharpens the
final roll off of digitallow-pass fi1ters.
Don't worry, We'l1 explain each оnе of these characteristics and see
exact1y what they теan to us as we go about designing аn I1R fi1ter,
If the transfer fиnction of а prototype analog fi1ter is Hc(s), then we сап
obtain the discrete I1R filter z domain transfer fиnction н(z) Ьу substitut-
ing the following for s in Hc(s)
2 ( 1 Z 1 )
s=t; 1+Z 1 '
( 6 88)
where, again, t s is the discrete filter's sampling period (1//.). Just as in the
impulse invariance design method, when using the bilinear transform
method, we're interested in where the analog filter's poles end ир оп the
z plane after the transformation, Тhis s plane to z plane mapping behav-
ior is exactly what makes the bilinear transform such аn attractive design
technique,+
Let's investigate the major characteristics of the bilinear transform's
s plane to z plane mapping, First we'l1 show that аnу pole оп the left
side of the s-plane will тар to the inside of the иnН circle in the z plane,
It's easy to show this Ьу solving Eq, (6-88) for z in terms of s,
Multiplying Eq. (6 88) Ьу (t/2)(1 + Z l) and collecting like terms of z
leads us to
1 + (ts /2)s
z=
1 (ts /2)s
( 6 89)
+ The bilinear transforrn is а technique in the theory of complex variables for mapping а
fиnction оп the complex plane of оnе variable to the complex plane of another variable, It
maps circles and straight lines to straight lines and circles, respectively,
274 /пflпitе /mpuise Respoпse F//ters
If we designate the real and imaginary parts of s as
s = cr + jФ а '
(6-90)
where the subscript in the radian frequency Фа signifies aпalog, Eq, (6 89)
becomes
z = 1 +CTt s /2+ jФ.ts /2 = (1 + cr t s /2) + jФаt s / 2 (6 91)
l cтts /2 jФаt s /2 (l crts /2) jФаt s /2
We Бее in Eq, (6 91) that z is complex, comprising the ratio of two complex
expressions. As such, if we denote z as а magnitude at an angle in the
form of z = 1 z 1 Le z ' we know that the magnitude of z is given Ьу
1 z 1= Magnumerator = (1 + ст t s /2)2 + (Ф.ts /2)2
MagdenonUnator (1 ст t s /2)2 + (Фаt s /2)2
(6-92)
ОК, if cr is negative (а < О) the numerator of the ratio оп the right side of
Eq. (6 92) will Ье less than the denominator, and I z 1 will Ье less than 1,
Оп the other hand, if cr is positive (а > О), the numerator will Ье larger
than the denominator, and I z I will Ье greater than 1. Тhis confirms that,
when using the bilinear transform defined Ьу Eq. (6 88), any pole located
оп the left side of the s plane (а < О) will mар to а z plane location inside
the unit circle. Тhis characteristic ensures that any stable s-plane pole of а
prototype analog fi1ter will mар to а stable z plane pole for our discrete
IIR filter. Likewise, any analog fi1ter pole located оп the right side of the
s plane (а > О) will mар to а z plane location outside the unit circle when
using the bilinear transform, Тhis reinforces our notion that, to avoid fil-
ter instability, during IIR filter design, we should avoid allowing any
z plane poles to lie outside the unit circle,
Next, let's show that the jФ а axis of the s plane maps to the perimeter of
the unit circle in the z plane. We сап do this Ьу setting cr = О in Eq, (6-91)
to get
1 + jФ.ts /2
z=
1 jФаt s /2
(6 93 )
Here, again, we see in Eq. (6 93) that z is а complex number compris-
ing the ratio of two complex numbers, and we know the magnitude of this
z is given Ьу
B/l/пear Traпsform I/R Filter Des/gп Method 275
I z '<1;0 = Magnumerator
Magdenominator
(1)2 + (йJat s /2)2
(1)2 + (йJat s /2)2
(6-94)
The magnitиde of z in Eq, (6 94) is always 1, 50, as we stated, when using
the bilinear transform, the jйJ a axis of the s plane maps to the perimeter of
the unit circle in the z-plane. However, this frequency mapping from the
s-plane to the unit circle in the z-plane is not linear, It' s important to know
why this frequency nonlinearity, or warping, occurs and to understand its
effects, 50 we shall, Ьу showing the relationship between the s plane fre
quency and the z-plane frequency that we'll designate as йJ d .
If we define z оп the unit circle in polar form as z = re--j(fJd as we did for
Figиre 6-13, where r is 1 and йJ d is the angle, we can substitиte z = ej(fJd in
Eq, (6 88) as
2 ( 1 e j(fJd )
s=t; l+e j(fJd '
( 6 95)
If we show s in its rectangular form and partition the ratio in brackets into
half angle expressions,
, 2 e j(fJd/2(ej(fJd/2 e j(fJd/2)
s = о' +]йJ = .
а t s e j(fJd!2(ej(fJd!2+e j(fJd!2)
(6-96)
Using Euler's relationships of sin(0) = (&О e j0)/2j and
COS(0) = (&О + e-- j 0)/2, we сап convert the right side of Eq, (6-96) to rectan-
gиlar form as
, 2 e j(fJd!2[2jsin(йJd/2)]
s = о' +]йJ = . ,
а t s e J(fJd!2[2cos(йJd /2)]
2 2e j(fJd/2 jsin(йJd /2)
= t;' 2e j(fJd!2 cos(йJd /2)
'2
= Ltan(йJd /2).
t s
( 6 97)
If we now equate the real and imaginary parts of Eq, (6-97), we see that
cr = О, and
276 /пflпite /mpu/se Respoпse F//ters
Q). = tan ( Q)d ) .
t s 2
(6-98)
Let's rearrange Eq. ( 98) to give us the useful expression for the z domain
frequency O)d' in terms of the s-domain frequency 0)., of
2t l ( Q).tS )
Q)d an ,
2
( 99)
Тhe important relationship in Eq, (6 99), which accounts for the so called
frequency warping due to the bilinear transform, is illustrated in Figure
6-ЗО. Notice that, because tan 1(0).t/2) approaches х/2 as 0). gets large, O)d
must, then, approach twice that value, or Х. Тhis means that по matter
how large 0). gets, the z-plane O)d will never Ье greater than х.
Remember how we considered Figure 14 and stated that only the 1t/s
to +1t/ s radians/s frequency range for 0). сап Ье accounted for оп the
z-plane? Well, our new mapping from the bilinear transform maps the
entire s-plane to the z-plane, not just the primary strip of the s plane
shown in Figure 14. Now, just as а wa1k along the jO). frequency axis оп
the s-plane takes us to infinity in either direction, а trip ha1fway around
the unit circle in а counterclockwise direction takes us from 0). = О to
0). == +00 radians/s. As such, the bilinear transform maps the s-plane's
entire jO). axis onto the unit circle in the z-plane. We il1ustrate these bilin
ear transform mapping properties in Figure 6-З1.
О)а
1t
Figure 6-30 Nonllnear relatlonship between the z-domaln frequency Ш d and
the s-domain frequency Ша'
Blliпear Traпsforт J/R FJ/ter Des/gп Method 277
Flgure 6.31 Billnear transform mapplng of the .rplane to the z-plane,
То show the practical implications оЕ this frequency warping, let's
relate the s plane and z-plane frequencies to the more practical measure
оЕ the 15 sampling frequency, We do this Ьу remembering the funda
mental relationship between radians/s and Hz оЕ ro = 2пl and solving
for 1 to get
ro
I=
2п
Applying Eq, (&-100) to Eq, (&-99),
(6-100)
21tld = 2tan 1( 2 t5 )
(6 101)
Substituting l/fs for t 5 , we solve Eq, (&-101) forl d to get
{ ( 2 ) 1 ( 2п!. / /S ) tan \1tJ. / /S)
Jd tan ,
2п 2 1t
(&-102)
Equation (&-102), the relationsblp between the s plane frequency f. in Hz
and the z plane frequency Id in Hz, is plotted in Figure &-32(а) as а func-
tion оЕ the IIR filter's sampling ratefs,
The distortion of the la frequency scale, as it translates to the Id
frequency scale, is illustrated in Figure 6 32(b) where an s-plane
bandpass frequency magnitude response I Наиа) I is subjected to
278 /пfiпite lтpulse Respoпse FHters
'd
ОЫ.
0,4'.
0.3'.
(а)
0,2'.
О,Н.
О
О 0,5'. '.
1,5'. 2'. 2,5'. 3,," 3,Ы. 4,," 4,Ы. ы.'.
"
'd= tan 1 (1tf"/,.)/1t
.;
(Ь)
....."
,H."." V
О
'.
'"
'.
Flgure 6-32 Nonlinear relatlonshlp between the f d and f с frequencles: (а) frequency
warplng curve scaled In terтs of the IIR fllter's fs saтpllng rate;
(Ь) s-doтaln frequency response Hc(fJ transforтation to а z-doтaln
frequency responsв Hd(f d ),
frequency compression as it is transformed to I Hd(fd) 1. Notice how the
low frequency portion of the IIR filter's I Hd(fd) I response is reasonably
linear, but the higher frequency portion has been squeezed in toward
zero Hz. Тhat's frequency warping, Тhis figure shows why по IIR filter
aliasing сап occur with the bilinear transform design method, No matter
what the shape or bandwidth of the I Ha(fa) I prototype analog fi1ter,
попе of its spectral content сап extend beyond ha1f the sampling rate of
ВШпеаr Traпsforт IIR FiJter Des/gп Method 27
1./2 in I Hd(fd) and that's what makes the bilinear transform design
method ав popular ав it is,
The steps necessary to perform an IIR filter design using the bilinear
transform method are ав follows:
Step 1: Obtain the Laplace transfer function Hc(s) for the prototype ana-
log filter in the form of Eq, (6 43),
Step 2: Determine the digital filter's equivalent sampling frequency 1.
and establish the sample period t s = 1/1..
Step з: In the Laplace Hc(s) transfer function, substitute the expression
2 ( l Z l J
t: 1 H 1
(6-103)
for the variable s to get the 1IR filter's H(z) transfer function,
Step 4: Multiply the numerator and denominator of н(z) Ьу the appro
priate power of (1 + r 1 ) and grind through the algebra to collect
terms of like powers of z in the form
N
Lb(k)z k
н(z) = k=
1 La(k)z k
k=l
(6-104)
Step 5: Just as in the impulse invariance design methods, Ьу inspection,
we Сan express the IIR filter's time domain equation in the gen
eral form of
у(п) = Ь(О)х(п) + Ь(l)х(п 1) + Ь(2)х(п 2) +... + b(N)x(n N)
+ а(l)у(п 1) + а(2)у(п 2) + '.. + а(М)у(п М) .
(6 105)
Although the expression in Eq. (6-105) only applies to the filter
structure in Figure 6 18, to complete our design, we сап apply the
a(k) and b(k) coefficients to the improved IIR structure shown in
Figure 6-22.
280 /пflпfte /три/5е Re5poпse F/lters
То show just how straightforward the bilinear transform design
metЬod really is, let's use it to solve the ПR filter design problem first pre-
sented for the impu1se invariance design method,
6.5. 1 Blllnear Transform Deslgn Example
Again, our goal is to design an ПR filter that approximates the second-
order Chebyshev prototype analog low pass filter, shown in Figure 6 26,
whose passband ripple is 1 dB, Тhe fs sampling rate is 100 нz и. = 0.01),
and the filter's.1 dB cutoff frequency is 20 Нz, As before, given the origi
nal prototype filter's Laplace transfer function as
н 8 17410,145
с( ) 82 + 137.945368 + 17410,145 '
(6 106)
and the value of t. = 0,01 for the sample period, we're ready to proceed
with Step 3. For convenience, let's replace the constants in Eq. ( 106) with
variables in the form of
с
Н с (8) = 2 '
8 +Ь8+С
(6 107)
where Ь = 137,94536 and с = 17410,145, Performing the substitution of Eq,
(6-103) in Eq. (6 107),
с
н(z)= 2
( 2 1 Z 1 ) 2 ( 1 Z 1 )
. +b +с
t. 1 + Z l t s 1 + Z l
(6 108)
То simplify our algebra а little, let's substitute the variable а for the frac-
tion 2ft. to give
с
Н(z)= 2
( 1 l ) ( 1 l )
а 2 z l + аЬ z l + с
1+z 1+z
(6 109)
Proceeding with Step 4, we mиltiply Eq, (109)'s numerator and denomi
nator Ьу (1 + Z-1)2 to yield
Biliпear Traпsforт I/R Filter Desigп Method 281
н(z)=
а 2 (1 z 1)2 + аЬ(l + Z l )(1 z l) + с(l + z 1)2
с(l + z 1)2
(б-11 О)
Multiplying through Ьу the factors in the denominator of Eq. (б 110), and
collecting like powers of z,
н(z) = с(1 + 2Z 1 + Z 2)
(а 2 + аЬ + с) + (2с 2a2)z 1 + (а 2 + с ab)z 2
(б-111)
We're almost there. То get Eq. (б 111) into the form of Eq. (б 104) with а
constant term of one in the denominator, we divide Eq. (б 111)'s numera
tor and denominator Ьу (а 2 + аЬ + с) giving us
2 С (1+2z 1 +Z 2)
н(z) = (а + аЬ 2 +с) 2
1 (2c 2a) 1 (а +c ab) 2
+ z + z
(а 2 +аЬ+с) (а 2 +аЬ+с)
(б-112)
We now have H(z) in а form with all the like powers of z combined into sin-
gle terms, and Eq. (б 112) looks something 1ike the desired form of
Eq, (б 104), If we plug the values а = 2fts = 200, Ь = 137,9453б, and
с = 17410,145 into Eq. ( 112) we get the following IIR filter transfer fиnction:
н 0.20482712(1 + 2Z 1 + Z 2)
(z) = 1 0.53153089 Z l + 0.35083938 Z 2
0,20482712 +0.409б5424z 1 + O,20482712z 2
1 0,53153089 Z l + 0.35083938z 2
(б 113)
and there we are. Now, Ьу inspection of Eq. (б-113), we get the time
domain expression for our IIR filter as
у(п) = 0.20482712 + 0.409б5424. х(п 1) + 0.20482712. х(п 2)
+ 0,53153089. у(п 1) 0,35083938. у(п 2) . (б 114)
The frequency magnitude response of our bi1inear transform IIR
design example is shown as the dark curve in Figure б 33(а), where, for
282 /пfiпite /тpu/se Respoпse F//ters
IHd('d)1
"
/ mpulse inverience
0.6 design
.... . / 0.4 J ".
....., '.......
0.2 Bilineer lransform
design
(а)
50 Hz 20 Hz О 20 Hz 50 Hz Freq
('./2) ((,,12)
Degrees
180
140
100
60
(Ь) 20
--50 Hz 50 Hz Freq
Hsl2) ('./2)
Flgure 6-33 Comparison of the bIllnear transform and impulse invariance design IIR
filters: (а) frequency magnltude responses; (Ь) phase of the bIllnear
transform IIR filter,
comparison, we've shown the resu1t оЕ that impulse invariance design
example as the shaded curve. Notice how the bilinear transform designed
filter's magnitude response approaches zero at the folding frequency оЕ
1/2 = 50 Hz, Тhis is as it should b that's the whole purpose оЕ the bilin-
ear transform design method. Figure 6-33(Ь) illustrates the nonlinear
phase response оЕ the bilinear transform designed IIR filter.
We might Ье tempted to think that not only is the bilinear transform
design method easier to perform than the impulse invariance design
method, but that it gives us а тисЬ sharper rol1-off for our low-pass filter,
Well, the frequency warping оЕ the bilinear transform method does сот-
press (sharpen) the rol1-off portion оЕ а low-pass filter, as we saw in Figure
6-32, but аn additional reason for the improved response is the price we
рау in terms оЕ the additional complexity оЕ the implementation оЕ our
IIR fi1ter, We see this Ьу examining the implementation оЕ our IIR filter as
shown in Figure 6-34. Notice that our new filter requires five multiplica-
tions per fi1ter output sample where the impulse invariance design fi1ter
in Figure 6-28(а) required on1y three multiplications per fi1ter output sam-
ple, Тhe additional mu1tiplications are, оЕ course, required Ьу the addi-
tional feed forward z terms in the numerator of Eq, (6-113). Тhese added
Biliпear Traпsforт I/R Filter Des/gп Method 283
1>(0) = 0.20482712
1>(1) = 0.40965424
1>(2) . 0.20482712
а(1).0.53153О89
а(2) . -О.З508З9З8
у(п)
Flgure 6-34 Implementatlon of the bIllnear transform design example filter,
b(k) coefficient terms in the H(z) transfer function сопеsроnd to zeros in
the z-plane created Ьу the bilinear trаnsfопn that did not occur in the
irnpulse invariance design method.
Because our example prototype analog low-pass filter had а cutoff fre-
quency that was 1.15, we don't see а great deal of frequency warping in
the bilinear transform curve in Figure 6-33. (In fact, Kaiser has shown that,
when 1. is large, the irnpulse invariance and bilinear transform design
methods result in essentially identical H(z) transfer functions[19],) Had
our cutoff frequency been а larger percentage of fs' bilinear trаnsfопn
warping would have been more serious, and our resultant I Hif d ) I cutoff
frequency would have been below the desired value. What the pros do to
avoid this is to prewarp the prototype analog filter's cutoff frequency
requirement before the analog Hc(s) transfer function is derived in Step 1,
In that way, they compensate for the bilinear transfопn' s frequency warp-
ing before it happens, We can use Eq. (6-98) to determine the prewarped pro-
totype analog filter low-pass cutoff frequency that we want mapped to the
desired ПR low-pass cutoff frequency. We plug the desired ПR cutoff fre
quency ffid in Eq, (6--98) to calculate the prototype analog Юа cutoff frequency
used to derive the prototype analog filter's HC<s) transfer function.
Although we explained how the bilinear transform design method
avoided the impulse invariance method's inherent frequency response
aliasing, it' s irnportant to remember that we stШ have to avoid filter input
data aliasing. No matter what kind of digital filter or filter design method
is used, the original input signal data must always Ье obtained using а
sampling scheme that avoids the aliasing described in Chapter 2, If the
original input data contains errors due to sample rate aliasing, по filter
сап remove those епоrs.
284 /пfiпite /mpu/se Respoпse F//ters
Oиr introdиctions to the impиlse invariance and bilinear transform
design techniqиes have, Ьу necessity, presented only the essentials of
those two design methods, Althoиgh rigoroиs mathematical treatment of
the impиlse invariance and bilinear transform design methods is inap-
propriate for an introdиctory text sиch as this, more detailed coverage is
available to the interested reader[1 16], References [13] and [15], Ьу the
way; have excellent material оп the varioиs prototype analog filter types
иsed as а basis for the analytical ПR filter design methods, Althoиgh oиr
examples of ПR filter design иsing the impиlse invariance and bilinear
transform techniqиes approximated analog low-pass filters, it's important
to remember that these techniqиes apply eqиally well to designing band-
pass and highpass IIR filters, То design а highpass ПR filter, for example,
we'd merely start oиr design with а Laplace transfer fиnction for the pro--
totype analog highpass filter, our ПR digital filter design woиld then pro
ceed to approximate that prototype highpass filter.
As we have seen, the impиlse invariance and bilinear transform design
techniqиes are both powerful and а bit difficult to perform. The mathe-
matics is intricate and the evalиation of the design eqиations is ardиoиs
for аН bиt the simplest filters, As sиch, we'll introdиce а third class of ПR
filter design methods based оп software roиtines that take advantage of
iterative optimizatioп compиting techniques. In this case, the designer
defines the desired filter freqиency response, and the algorithm begins
generating sиccessive approximations until the ПR filter coefficients con-
verge (hopefully) to an optimized design.
6.6 Optimized IIR Filter Design Method
The final class of ПR filter design methods we'll introdиce are broadly cat-
egorized as optimization methods. These ПR filter design techniqиes were
developed for the situation when the desired ПR filter freqиency response
was not of the standard low-pass, bandpass, or highpass form. When the
desired response has an arbitrary shape, closed-form expressions for the
filter's z transform do not exist, and we have по explicit eqиations to work
with to determine the ПR filter's coefficients. For this general ПR filter
design problem, algorithms were developed to solve sets of linear, or non
linear, eqиations оп а computer, These software roиtines mandate that the
designer describe, in some way, the desired ПR filter freqиency response.
The algorithms, then, assиme а filter transfer fиnction Н(z) as а ratio of
polynomials in z and start to calcиlate the filter's freqиency response.
Баsed оп some error criteria, the algorithm begins iteratively adjиsting the
filter's coefficients to minimize the error between the desired and the
асшаl filter freqиency response, The process ends when the error cannot
Opt/mJzed I/R Filter Desigп Method 285
Ье fиrther minimized, or а predefined number of iterations has occurred,
and the final filter coefficients are presented to the filter designer. Although
these optimization algorithrns are too rnathernatically cornplex to cover in
any detail here, descriptions of the rnost popular optirnization schernes are
readily available in the literature[14,16,20 25].
The reader rnау ask, "Н we're not going to cover optimization rnethods
in any detail, why introduce the subject here at аШ" The answer is that if
we spend rnuch tirne at аll designing IIR filters, we'll end ир using opti-
rnization techniques in the forrn of cornputer software routines rnost of
the tirne, The vast rnajority of cornmercially available digital signal pro--
cessing software packages include one or rnore IIR filter design routines
that are based оп optirnization rnethods, When а cornputer aided design
technique is available, fi1ter designers are inclined to use it to design the
sirnpler low pass, bandpass, or highpass forrns even though analytical
techniques exist. With аll due respect to Laplace, Heaviside, and Kaiser,
why plow through аll the z transforrn design equations when the desired
frequency response сап Ье applied to а software routine to yield accept
able filter coefficients in а few seconds?
As it turns out, using cornrnercially available optirnized IIR filter
design routines is very straightforward, Although they соrnе in several
flavors, rnost optirnization routines only require the designer to specify
а few key arnplitude and frequency values, the desired order of the IIR
filter (the nurnber of feedback taps), and the software cornputes the
final feed forward and feedback coefficients, In specifying а low pass,
IIR filter for exarnple, а software design routine rnight require us to
specify the values for Ор' 0''/1' andf2 shown in Figure 6 З5. Sorne opti
rnization design routines require the user to specify the order of the IIR
IHd(f)1
i
i
v;
Os
j/
шш,
1 Ор
о
о
l'
'1 '2
's/2 Freq
Flgure 6-35 Example low-pass IIR ftlter deslgn parameters requlred for the
optimized IIR ftlter deslgn method,
286 /пfiп/te /три/sе Respoпse F//ters
fi1ter, Those routines then compute the fi1ter coefficients that best
approach the required frequency response, Some software routines, оп
the other hand! don't require the user to specify the fi1ter order, They
compute the minimum order of the fi1ter that actua11y meets the desired
frequency response,
6.7 Piffalls in Building IIR Digital Filters
Тhere's аn old saying in engineering: "It's оnе thing to design а system оп
paper, and another thing to actua11y build оnе and make it work." (Reca11
the Тасота Narrows Bridge episode!) Fabricating а working system
based оп theoretical designs сап Ье difficu1t in practice, Let's see why this
is often true for IIR digital fi1ters,
Again, the IIR filter structures in Figures 6-18, 6 21(b), and 6 22 are
ca11ed Direct Form implementations of an IIR fi1ter because they're а11
equivalent to direct1y implementing the general time-domain expression
for an Mth-order IIR filter given in Eq, (6--21), As it turns out, there Сan Ье
stability probIems and frequency response distortion errors when direct
form implementations are used. Such probIems arise because we're forced
to represent the IIR filter coefficients and the resu1ts of intermediate fi1ter
ca1cи1ations with binary numbers having а finite number of bits, These
finite word length effects are particularly serious if а direct form IIR filter's
н(z) transfer function polynomial in z is of high order (i.e., а filter having
а large number of delay elements), There are three major categories of
finite word length errors that plagиe IIR filter implementations: coefficient
quantization, overflow errors, and roundoff errors, Although Chapter 9
discusses the nature and effects of using finite word lengths in more detail,
we'll briefly discuss those error types as they relate to ПR filters and see
what precautions сап Ье taken to minimize their effects.
Remember, а11 of our IIR filter ca1culations, thus far, have resu1ted in
very accurate fi1ter coefficient values, In fact, the fractional coefficient val
ues in our previous IIR design examples were accurate to eight decimal
digits as indicated in Eq. (6 114), Кnowing that it takes four binary bits to
store а single decimal digit, we сап say that the coefficient values in Eq.
(6-114) require the equivalent of 32 bits of accuracy. Consider the situation
where we try to actua11y build the IIR fi1ter in Figиre 6--34, but we're con
strained to store our a(k) and b(k) coefficients in four bit hardware regis
ters, We11, we сan, indeed, represent the fractional coefficients with only
оnе decimal digit, but they lose some of their precision. For example, we
сап represent Eq, (6 1l4)'s original Ь(О) = 0,20482712 coefficient as
Ь(О) = 0,2 and lose some accuracy due to coefficient quantization.
PJtfa//s ;п Bu//d;пg //R Dlg;tal Filters 287
Limited precision coefficients will result in slight filter pole and zero
shifting оп the z plane, and а frequency magnitude response that тау not
meet our requirements.
For example, if we used only four bits (one decirn.al digit) to represent
the coefficients in Figure 6 34, our reduced-accuracy filter magnitude
response is shown Ьу the solid сшvе in Figure 6-36. For contrast, we've
also shown the original high-accuracy coefficient response as the shaded
curve in Figure 6-36. Sure enough, using four bit coefficients distorted our
original second-order IIR frequency response, Had our filter cutoff fre-
quency (20 Hz) been higher relative to the sample rate (100 Hz), соет-
cient quantization would resu1t in а frequency response that was even
more skewed from the original, As it tums out, higher order filters are
very sensitive to coefficient accuracy and, for them, coefficient quantiza-
tion is more likely to cause unacceptable frequency response distortion.
An even bigger problem сап occur in high order polynomials of H(z)
when quantized coefficients are rounded up to the nearest quantization
level. After factoring а high-order error containing н(z) polynomial into
its single-order factors, this сап result in actually moving а pole outside
the unit circle, yielding an unstable filter,
Overflow, the second finite word length effect that troubles digital fil
ters, is what happens when the resu1t of an arithmetic operation is too
large to Ье represented in the fixed length hardware registers designed
to contain that result, Because we perform so many additions when we
implement IIR filters, overflow is always а potential problem. With по
precautions being made to handle overflow, large nonlinearity errors
IHd('d)1
' '" Full accuracy
/ < coefficients
Four-blt '
coefficients
Four-bil coefficienls:
Ь(О) = 0,2
Ь(1) = 0.4
Ь(2) = 0,2
а(1) =0,5
а(2) = О,З
0,8
0,6
0.4
0.2
о
20 Hz
50 Hz
('5/2)
Freq
Flgure 6-36 Comparlson of the billnear transform deslgned ftlter magnitude
responses using full-accuracy coefficients (shaded curve) and
four-bIt coefficients (solld си rve) ,
288 /пfiпite /тpuJse Respoпse Fllters
сап result in our filter output samples often in the form of overflow
oscil1ations,
Thе most coinmon way of dealing with binary overflow errors is caHed
roundoff, or rounding, where а data value is represented Ьу, or rounded
оН to, the b bit binary number that's nearest the unrounded data value,
It's usuaHy valid to treat roundoff errors as а random process, but condi-
tions occur in ПR filters where rounding сап cause the filter output to
oscillate forever, even when the filter input sequence is аН zeros. Тhis sit
uation, going Ьу the names /iтit cyc/es and deadband effects, has been weH
analyzed in the literature[26,27], We can demonstrate limit cycles Ьу con
sidering the second order IIR filter in Figure 6 З7(а) whose time-domain
expression is
у(п) = х(п) + l,Зу(п 1) О,76у(п 2) ,
(6 115)
Let's assume that this filter rounds the adder's output to the nearest inte
gral value, If the situation ever arises where y( 2) = О, y( l) = 8, and х(О)
and аН successive х(п) inputs are zero, the filter output goes into endless
oscil1ation, as shown in Figure 6 З7(Ь). If this filter were to Ье used in an
audio application, when the input signal went silent, the listener could
end ир hearing an audio tone instead of silence. The shaded line in Figure
6 З7(Ь) shows the filter's stable response to this particular situation if по
rounding is used. With rounding, however, this IIR filter certainly lives ир
to its пате.
There are several ways to reduce the ill effects of quantization errors
and limit cycles. We can increase the word widths of the hardware regis-
ters that contain the results of intermediate calculations, Because round-
ОН lirnit cycles affect the least significant bits of an arithmetic result, larger
word sizes diminish the impact of limit cycles should they occиr, То avoid
filter input sequences of аН zeros, some practitioners add а dither sequence
to the filter's input signal sequence. А dither sequence is а sequence of
low amplitude pseudorandom numbers that interferes with an ПR filter's
roundoff error generating tendency, aHowing the filter output to reach
zero should the input signal remain at zero. Dithering, however, decreases
the effective signal-to--noise ratio of the filter output[l1J, FinaHy, to avoid
limit cycle problems, we can just use an FlR filter. Because FIR filters, Ьу
definition, have finite length impulse responses and have по feedback
paths, they cannot support output oscillations of any kind,
As for overflow oscillations, we сап eliminate them if we increase the
word width of hardware registers, so that overflow never takes place in
the IIR filter. Altematively, it has been shown that the effects of overflow
oscillations can Ье avoided Ьу modifying hardware adders so that their
P/tfa//s /п Bu//diпg //R Dig/ta/ Fi/ters 289
(а)
у(п)
"(),76
i у(п)
108:
8
6 "
4
(Ь) 2 т-
О
8;
2 ,;
4 8
' /
Response with
по rounding
.' . ,/ . . . .
" , , . . .........
" . . ..... 18 . 24 Тiтe
8
8
Flgure 6-37 Limlt cycle osclllotlons due to rounding In оп IIR ftlter: (о) second-
order IIR ftlter wlth rounding toking ploce following the odder;
(Ь) one possibIe tlme-domoln response of the IIR ftlter,
sum saturates to а fixed limit when an overflow condition is detected[28,
29]. It mау Ье usefиl for the reader to keep in mind that, when the signal
data is represented in two's complement arithmetic, multiple summations
resu1ting in intermediate overflow errors cause по prob1ems if we сап
guarantee that the finаl magnitude of the sum of the numbers is not too
large for the final accumulator register, тhis potential relief from overflow
errors, as discussed under the data overflow topic in Section 9,3, is one
reason for the popularity of the two' s complement format for binary num
bers. Some digital filters incorporate additional control circuitry to detect
overflow conditions and allow the designer to control (Ьу scaling down)
the amplitudes оЕ signals at critical points in the system[14,17]. If your
budget can afford floating-point hardware, it has been shown that both
standard floating point and block floating point data formats сап greatly
reduce the errors associated with overflow oscillations and limit
cycles[30], (We discuss floating-point number formats in Section 9.4)
Perhaps the most роршar technique for minimizing the епоrs associated
with finite word widths is to design filters comprising а cascade string, or
para1lel combination, of low order filters. The next section tells us why
290 /пflп/te /тpu/se Respoпse F//ters
6.8 CascaQe and Parallel Combinations о'
DigitalFilters
In practice, we're likely to encounter multiple digital filters connected in
cascade or parallellike those in Figure 38, As indicated in Figиre 6 38(a),
the resultant transfer function of two cascaded filter transfer functions is
the product of those functions, or
Hcascade(z) = H 1 (z)H 2 (z) ,
( 116)
If the filters in Figиre 38(a) are linear, we сan swap their order, having
H 2 (z) precede H 1 (z), with по change in the Y(z) output. Ву the way, this
property of the combined transfer function equaling the product of the
two individualtransfer functions applies to any combination of linear
time invariant IIR and FIR filters, As shown in Figиre 38(b), the coт
bined transfer function of two parallel filters is the sum of their transfer
functions, or
Hparallel (z) = H 1 (z) + H 2 (z) ,
( 117)
Hardware filter designers routinely partition high order digital filters
into а string of lower order filters arranged in cascade because the cas
caded system usually requires fewer multiplications for а given filter fre
quency response, It's common practice to subdivide а high--order digital
(а)
Y(z) = Н 1 (z) (z)X(z)
(Ь)
X(z)
Y(z) = [Н 1 (z) + (z)]X(z)
Figure 6-38 CoтbInotions of two dlgltol fllters: (о) coscoded fllters; (Ь) porallel
fllters,
Cascade aпd Ратаllе! Coтbiпatioпs о, Digitai Fllters 291
filter into multiple second order building blocks Ьесаше second order ш
ters are easier to design, are less susceptible to coefficient quantization
and roundoff errors, and their implementations allow easier data word
scaling to reduce the potential overflow effects of data word size growth.
Optimizing the partitioning оЕ а high order filter into multiple second
order filter sections is а challenging task, however. For example, say we
have the sixth order filter in Figure 6--39(а) that we want to partition into
three second order sections. In factoring the sixth order filter's H(z) poly
nomial, we could get ир to three separate sets of feedback coefficients in
the factored H(z) numerator: а'(п), а"(п), and а'''(п), Likewise, we could
have ир to three separate sets of feed forward coefficients in the factored
Х(п)
(а)
(Ь)
0(6) Ь(6)
n
Partilio" i"IO seco"d-order filters
\;
0'(2)
Ь'(2)
Ь'''(2)
Flgure 6-39 IIR fllter partitlonlng: (а) original sixth-order IIR fllter; (Ь) three second-
order sections,
292 /пflпite /три/sе Respoпse Fllters
denominator: Ь'(n), Ь"(n), and Ь'''(n), Because there are three second
ordered sectioos, there are three factorial, or six, ways of pairing the sets of
coefficients. Notice in Figure 6-39(Ь) how the first section uses the а'(n)
and Ь'(n) coefficients, and the second section uses the а"(n) and Ь"(n) coef-
ficients. We could just as well have interchanged the sets of coefficients so
that the first second-order section uses the а'(n) and Ь"(n) coefficients, and
the second section иses the а"(п) and Ь'(n) coefficiel1:ts. 50 there are six dif-
ferent mathematically equivalent ways of combining the sets of соеШ-
cients, Add to this the fact that, for each different combination of
low order sections, there are three factorially distinct ways those three
separate second-order sections сап Ье апаngеd in cascade.
Тhis means that if we want to partition а W-order IIR fi1ter into М dis-
tinct second-order sections, there are М factorial squared, or (М!)2, ways
to do so. As such, there are then (3!)2 = 36 different cascaded filters we
coиld obtain when going from Figure 6-39(а) to 6 39(b).t То fиrther сот-
plicate this fi1ter sectioning problem, the епоrs due to quantization wiII,
in general, Ье different for each possible fiIter combination. Although this
subject is oиtside the scope of this introductory text, ambitioиs readers
сап find further material оп optimizing cascaded filter sections in refer-
ences [19,27], and in part 3 of reference [31].
Тhere's а considerable amount of material in the literature concerning
finite word effects as they relate to digital fi1ters. (References [14,16] and
[19] discuss qиantization noise effects in some detail as well as providing
extensive bibliographies оп the subject.) Although these books and papers
provide useful guidelines in predicting and minimizing quantization
errors, prudent designers eventually resort to computer simulations of ы-
ter designs, This way they сап vary fi1ter hardware characteristics, such as
coefficient word widths, accumulator register sizes, sequencing of cas-
caded sections, and input signal sets. This experimental approach is
attractive because it's both reliable and economical.
6.9 А Brief Comparison of IIR and FIR Filters
The question natиraIly arises as to which filter type, IIR or FIR, is best
suited for а given digital fi1tering application. That's not an easy ques-
tion to answer, but we сап point out а few factors that should Ье kept in
mind. First, we сап assume that the differences in the ease of design
between the two fi1ter types are unimportant, There are usuaIly more
imp()rtant performance and implementation properties to consider than
t These combinations сап get very large, For example, there are (5!)2 = 14,400 ways to сот-
Ыnе five second-order filters to implement а lO'h- or der IIR filter,
.
А Br/ef Coтpor/soп of J/R oпd F/R F//ters 293
design difficu1ty when choosing between an IIR and an FIR filter, One
design consideration that тау Ье significant is the IIR filter's ability to
simulate а predefined prototype analog fi1ter. FIR fi1ters do not have this
design flexibility.
From а hardware standpoint, with so many fundamental differences
between IIR and FIR filters, our choice must Ье based оп those filter char
acteristics that are most and least important to us, For example, if we
needed а filter with exactly linear phase, then an FIR filter is the only way
to go. on the other hand, if our design required that а filter accept very high
data rates and slight phase nonlinearity is tolerable, we might lean toward
IIR filters with their reduced number of necessary mu1tipliers per output
sample. ТаЫе 6--1 presents а brief comparison between IIR and FIR filters
based оп several different perfonnance and implementation properties.
ТаЫе 6-1 IIR and FIR Fllter Characterlstlcs Comparison
Characteristic IIR FIR
Number of nece$sary Lea$t MO$t
multiplications
Sensitivity to filter Сan ье high for Very low
coefficient quantization Direct Form"
Probability of overflow Сan ье high for Very low
error$ Direct Form"
Stability Must Ье designed in Guaranteed
Linear phase No Guaranteed ""
Сan $imulate prototype Yes No
analog filters
Required hardware Lea$t MO$t
memory
Нardware filter control Moderate Simple
complexity
Avai1ability of design Good Very good
$oftware
ЕаБе of design or Moderately Simple
complexity of design complicated
$oftware
Difficulty of quantization Most complicated Lea$t complicated
noise analY$is
Supports adaptive Yes Уе$
filtering
" The$e problems сan Ье minimized though cascade or parallel implementation$,
"" Guaranteed $0 long as the FIR coefficients are $ymmetrical,
294 /пfiпite /тpu/se Respoпse Filters
References
[1] Churchi11, R. V. Moderп Operational Мatheтatics in Engineering, McGraw-НЩ
New York, 1944, рр, 307 34, .
[2] Aseltine, J, А, Transform Method in Linear 5ystem Analysis, McGraw-Нill, New
York, 1958, рр. 287 92.
[3] Nixon, F. Е, Handbook о! Laplace Transformation, Tables and Examples,
Prentice-Hall, Englewood Cliffs, New Jersey, 1960,
[4] Kaiser, J. F. "Digital Filters," in 5ystem Aпalysis ьу Digital Computer, Ed. Ьу F.
F. Кио and J. F. Kaiser, JoOO Wiley and Sons, New York, 1966, рр, 218---77.
[5] Kaiser, J, F. "Design Methods for Samp1ed Data Filters," Chapter 7, in 51963
Proc, 1st Allerton Confereпce, рр. 221-36,
[6] Ragazzini, J, R, and Frank1in, G, F. 5ampled-Data Control 5ystems,
McGraw-НШ, New York, 1958, рр, 52....s3.
[7] Mi1ne-Тhomson, L, М, Тhe Calculus о! Finite Differences, Macmillan, London,
1951, рр, 232 51,
[8] Truxal, J, G, 1955, Automatic Feedback Confrol System Synthesis, McGraw-НШ,
New York, 1955, рр, 283.
[9] Blackman, R, В. Linear Data 5moothing and Prediction in Theory and Practice,
Addison-Wesley, Reading, Mass., 1965, рр, 81....s4.
[10] Gold, В, and Jordan, К. L" Jr. "А Note оп Digital Filter Synthesis," Proceedings
ofthe IЕЕЕ, Vol. 56, October 1968, рр. 1717,
[111 Rabiner, L, R., et аl, "Terminology in Digital Signal Processing," IЕЕЕ Trans,
оп Audio and Electroacoustics, Vo!, AU-20, No. 5, December 1972, рр. 327,
[12] Stearns, 5, О, Digital 5igпal Analysis, Hayden Book Со, Inc., Rochelle Park,
New Jersey, 1975, рр, 114,
[13] Stanley, W. О" et аl" Digital 5ignal Processing, Reston Publishing Со, Inc"
Reston, Virginia, 1984, рр, 191,
[14] Oppenheim, А, V., and Schafer, R, W, Discrete-Time 5ignal Processing,
Prentice-Hall, Englewood Cliffs, New Jersey, 1989, рр, 406,
[15] WiШаms, С. S. Desigпing Digital Filters, Prentice-Hall, EngIewood Cliffs, New
Jersey, 1986, рр, 16б....s6,
[16] Rabiner, L, R., and Gold, В, Theory and Application о! Digital 5igпal Processing,
Prentice Hall, Englewood Cliffs, New Jersey, 1975, рр. 216,
[17] JoOOson, М, "Implement Stable IIR Filters Using Minimal Hardware," EDN,
14 April1983,
[18] Oppenheim, А. V" WШskу, А, 5" and Yoиng, 1, Т, 5ignals and 5ystems,
Prentice Hall, Englewood Cliffs, New Jersey, 1983, рр, 659.
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Refereпces 295
[19] Kaiser, J, F. "Some Practical Considerations in the Realization оЕ Linear
Digital Filters," Proc, Third Аппиа/ A/lerton Conference оп Circuit and System
Theory, 1965, рр, 621 33
[20] Deczky, А. G, "Synthesis оЕ Digital Recursive Filters Using the Мinimиm Р
Епоr Criterion," IEEE Тrans, оп Audio aпd E/ectroacoustics, Vol, AU 20, No, 2,
October 1972, рр, 257.
[21] Steiglitz, К. "Computer-Aided Design оЕ Recursive Digital Filters," IEEE
Trans, оп Audio aпd E/ectroacoustics, Vol, 18, No, 2, 1970, рр, 123,
[22] Richards, М, А. "Application оЕ Deczky's Program for Recursive Filter
Design to the Design оЕ Recursive Decimators," IEEE Trans, оп Acoustics,
Speech, aпd Sigпa/ Processing, Vol, ASSP-30, October 1982, рр, 811,
[23] parks, Т. W" and Burrus, С. 5, Digita/ Fi/ter Desigп, John Wiley and Sons, New
York, 1987, рр, 244,
[24] Rabiner, L" Graham, У" and Helms, Н, "Linear Programming Design оЕ ПR
Digital Filters with Arbitrary Magnitude Functions," IEEE Trans, оп Acoustics,
Speech, and Sigпa/ Processing" Vol, ASSP-22, No, 2, April1974, рр, 117,
[25] Friedlander, В" and Porat, В, "Тhe Modified Yule-Walker Method оЕ ARМA
Spectral Estimation," IEEE Тrans, оп Aerospace E/ectronic Systems, Vol, AES-20,
No, 2, March 1984, рр. 158-73,
[26] Jackson, L, В. "On the Interaction оЕ Roundoff Noise and Dynатс Range and
Dynamic Range in Digital Filters," Веи System Technica/ jourпa/, Vol, 49,
February 1970, рр, 159-84,
[27] Jackson, L, В, "Roundoff Noise Analysis for Fixed-Point Digital Filters
Realized in Cascade or Parallel Form," IEEE Trans, Audio E/ectroacoustics, Vol,
AU 18, June 1970, рр, 107-22,
[28] Sandberg, 1, W, "А Тheorem Conceming Limit Cycles in Digital Filters," Proc,
Seventh Аппиа/ A/lerton Conference оп Circuit aпd System Theory, Monticello,
Illinois, October 1969,
[29] Ebert, Р, М" et аl, "Overflow Oscillations in Digital Filters," Веи Sys, Tech,
jourпa/, Vol, 48, November 1969, рр, 2999 3020,
[30] Oppenheim, А. V. "Realization оЕ Digital Filters Using Block Floating Point
Arithmetic," IEEE Тrans, Audio E/ectroacoustics, Vol, AU-18, June 1970, рр,
130-36,
[31] Rabiner, L, К, and Rader, С. м., Eds" Digita/ Sigпa/ Processing, ШЕЕ Press,
New York, 1972, рр, 361,
!CHAPTER SEVEN
Advanced Sampling
Techniques
Беуопd the low-pass and bandpass sampling schemes discussed in
Chapter 2 there are specialized sampling techniques that we're likely to
encounter in practice. In this chapter we introduce several of those
advanced sampling methods and show why and when they're used,
7.1 Quadrature Sampling
Many digital signal processing applications use the complex data for
mat with its real (in-phase) and imaginary (quadrature phase) parts.
These applications [аН in the general category known as quadrature pro
cessiпg. For example, the phase preservation characteristic of complex
data representation is exploited in high data rate digital communica-
tions systems, radar systems, and time difference of arrival (TDOA)
processing in radio direction finding schemes[1,2]. The enhanced phase
measurement capabilities of quadrature processing are used in coher-
ent pulse measurement systems, antenna beamforming applications,
and single sideband mоdиlаtоrs[З 6], Complex data representation
al10ws us to realize additional processing power through the coherent
measurement of the phase of sinusoids comprising an input signal, То
obtain the time domain representation of а continuous signal using
complex notation, the signal must Ье digitized Ьу а technique cal1ed
quadrature saтpliпg,t
t Quadrature sampling goes Ьу various other names in the literature, such as vector demod-
ulation, complex demodulation, complex down-conversion, quadrature heterodyning, I/Q
sampling, and complex sampling[7 9), Care must Ье exercised here from а semantic view-
point, There's а testing scheme referred to as cohereпt saтpliпg, used to characterize the
dynamic performance of А/О converters, that is unrelated to quadrature sampling[10,1l),
This clever A/D testing technique is discussed in Section 10,9,
297
298 Advoпced Saтpllпg Techп/ques
Quadrature-phase (imaginary) axis
In-phase (real) axis
Figure 7-1 In-phase /(t) aпd quadrature phase q(t) coтpoпeпts of а
continuous signal >«t),
То illustrate this idea, let's consider а continuous time-domain signal
x(t) that has some instantaneous magnitude at а phase angle of 0(t), This
signal сап, equally well, Ье defined Ьу its in phase i(t) and quadrature-
phase q(t) components as shown in Figure 7-1, We could use the low-pass,
or bandpass, sampling schemes in Chapter 2 to obtain discrete samples
representing x(t) and then calculate the i(t) and q(t) components.
Quadrature sampling, оп the other hand, allows us to obtain the discrete
samples representing i(t) and q(t) direct1y, Тhus, quadrature sampling
produces two separate sampled data sequences, one sequence represent
ing the in-phase and one sequence representing the quadrature phase
components of the original continuous signal x(t).
Let's keep two things in mind as we begin our discussion of quadra
ture sampling, First, in studying quadrature processing, we'll encounter
frequency domain spectra that are not symmetrical about the О нz axis,
A1though frequency domain symmetry is inherent in real signals, the
complex nature of quadrature signals does not have this restriction. It's
the lifting of this symmetry restriction that makes quadrature processing
so powerful. Second, recall from Euler's relationships discussed in
Appendix С that
, ej21tft e j21tft jej21tft je j21tft
Sln ( 21t :ft) = +
2j 2 2
(7-1)
and
ej21tft + e j21tft ej21tft e j21tft
cos ( 21t :ft) = = +
2 2 2'
(7-2)
Quadrature Saтpliпg 299
renUnding us that real sinusoids or cosinusoids can Ье represented as coт
plex phasor quantities through the convention оЕ negative frequency, that is,
negative exponentials. With these thoughts in mind, we view the hybrid
processing known as I/Q demodulation (or Weaver demodulation for those
readers with experience in communications theory), shown in Figure 7 2(a),
: -< ContinU0U8 )O- i -< Dilcret. !
,
(а) x(t) } Цп) jq(n)
sin(2xf c t)
t X(f) ...........B
(Ь)
fc О 'с Freq
(с) l' )I
2fc fc 8I2 О +812 'с 2fc Freq
(d) Filtered
8I2 О 812 Freq
(е) j'- )' 2fc
I ......, F:q
2 с fc 'с
" Filtered
(1)
Freq
О 812
(g)
2fs fs
Iex spectrum 01 i(n) jq(n)
8I2 О 812 f s 2' s Freq
Flgure 7-2 Quadrature bandposs sompling: (о) sampiing bIock diagram; (Ь)
contlnuous spectrum of contlnuous input x(t); (с) continuous in-phase
component of x(t); (d) flltered In-phase component of X(t); (е)
contlnuous quadrature-phase component of x(t); (f) flltered quadrature-
phase component of x(t); (g) sampied sum of the flltered in-phase
component аnd the flltered quodrature-phase component of x(t),
300 Advaпced Saтpliпg Techпiqиes
Consider а bandpass signal x(t), centered at carrier frequency fc Hz,
that has the real spectrum Х(Л shown in Figure 7 2(b). When we mix x(t)
with а cosine wave of fc Hz, the positive phasor portion of the cosine (that
is, ei 21r N /2) will translate the spectral components located at fc (dashed
curve) up to baseband centered at О Hz, resulting in what is sometimes
called а zero-frequency IF format[4]. Likewise, the negative phasor por
tion of the cosine (e-- i2 7tf,t /2) will translate the spectral components located
at +f c (solid curve) down to baseband a1so centered at О Hz. The resultant
continuous signal spectrum is shown as Х;(Л in Figure 7-2(с) where we
show the two translated spectra as overlaid, not summed, to see their
individuality. Х;и) is low pass fi1tered Ьу an analog filter with а cutoff
frequency slight1y greater than В/2 Hz, ca11ed the in phase component of
x(t), and shown as the I(Л magnitude spectrum in Figure 7 2(d), (For the
reader who's not at ease with the notion of negative frequencies, it's а
good idea to review Appendix С to see why negative frequencies exist
when we use complex signal notation.)
The continuous heterodyned product оЕ x(t) and the sinewave of fc Hz
is shown in Figure 7 2(e) where the spectra lying below the frequency axis
result from the negative sign of the positive frequency jei21ifcl /2 сотро-
nent in Eq. (7 1). This product is low-pass filtered and represents the
quadrature phase component оЕ x(t), shown as the jQ(f) spectrum in
Figure 7 2(Л. The origina1 real signal x(t) is now represented in а complex
format as i(t) jq(t). (ТО recover the real x(t), we need only multiply i(t) Ьу
cos(21tfi), multiply q(t) Ьу sin(21tfct), and sum the two products.)
Figure 7-2 helps remind us that real signa1s can Ье represented Ьу а sin
gle sequence of sampled values and that their spectra show symmetry
about zero Нz. Complex, or quadrature, signa1s must Ье represented Ьу
two sequences of sampled values, оnе sequence for the in phase part and
оnе for the quadrature part. Again, complex signa1 spectra are not required
to Ье symmetrical about zero Нz. Retuming to Figure 7 2(a), the two sep
arate, continuous signals, i(t) and q(t), are digitized Ьу an A/D coi1verter
at а sample rate of fs samples / s. The resultant spectrum for the complex
i(n) jq(п) is shown in Figure 7 2(g). Again, note the spectral replications
every fs Нz. 50 what's the big deal here? In Figure 7 2(g), notice that the
sample rate need only Ье greater than twice В /2 to avoid aliasing.
Iл quadrature sampling fs need only Ье greater than the signal band
width, as opposed to twice the signal bandwidth previously described in
Chapter 2 for low pass and bandpass sampling. This means that а given
А/О converter can Ье used to sample signal bandwidths twice as wide
with the quadrature sampling technique. Notice, however, that two low
pass filters and two A/D converters are necessary with this scheme,
Quadrature Samp/iпg with Dlgital Mixlng_
З
What we have is two A/D converters operating at half the equival J:'Lt
first order bandpass sampling rate, Are we violating the Nyquist crite.rf!
rion? Not really we stШ have two discrete sample values being gener
ated for every cycle in the bandwidth of our original signal, 50, when we
need to digitize very wide bandwidths, the quadrature sampling вфете
is attractive, particularly, when our goal is spectral analysis using the ррт.
When the two data streams representing the in phase and quadrature
phase components are applied to а complex radix-2 pFТ algorithm, the
pFТ operates more efficient1y than when its inputs are real signаls[З,12],
(Тhat particular topic is discussed in Section 10.5.)
А"";
7.2 Quadrature Sampling with Digital Mixing
То оЫаin quadrature samples in practice, the process shown in Figure
7 2(a) is difficu1t to implement without errors. Over moderately wide
bandwidths, it's not еаву to configure two analog signal paths that have
identical frequency responses (gain and phase) phase differences of а
few degrees are соттоп. Moreover, large DC components are routinely
induced Ьу the mixers in the xi(t) and xq(t) signals before fi1tering, Final1y,
if the low-pass fi1ters are active devices, а kind of continuous noise known
ав 1/f noise from the semiconductor components in the mixers/fi1ters
induces increased noise energy near О Нz.
Тhese disadvantages сап Ье avoided with digital mixers and fi1ters
where the sample rate is four tiIПев the carrier frequency of the bandpass
signаl[13--1б]. Тhis process is il1ustrated in Figure 7 З(а). Тhe input signal
x(t) from Figure 7-2(а), with the spectrum shown in Figure 7 З(Ь), is digi
tized at а sample rate of fs = 4fc providing the discrete samples x(nt s )
whose вресtrшn, Х(т), is shown in Figure 7 З(с). Тhe quadrature mixing
withfc = 1./4, to center the input signal's in phase and quadrature phase
components about О Hz, is performed digitally with the results of the in
phase replicated components depicted in Figure 7-З(d) and the low-pass
fi1tered in-phase spectrum in Figure 7-З(е). Although the A/D converter
must sample at four times the carrier frequency, the difficulties in signal
path phase and amplitude matching are eliminated because only опе
A/D converter is necessary. In addition, the DC Ыав problems associated
with analog signal mixing to zero нz are avoided,
Тhe real advantage of this technique is how еаву the quadrature mix-
ing Ьу fc сап Ье perforrned digitally. For example, the input sampled val
ues x(nt s ) сап Ье multiplied Ьу the repetitive four element sequence
1, О, 1, О, etc" to provide i(n). Likewise, x(nt s ) is mu1tiplied Ьу the four-
element sequence О, 1, О, 1, to generate the q(n) samples, Тhe trick is that
302 Advoпeed Samp/iпg reehпiques
i..... Contlnl.ЮU8 ...: (
I
I'(n)
(а) x(t) cos(21tfcnt o )
q'(n)
10 = 41c=11to
sln(21tf c nt o )
, X(I) B
(Ь) ..
-Ic О 'с Freq
r-;1 , IX(m)1 Sampled
(с) I I rj1 МaQ , I
-41 с - 31 с - 2f c -Ic О 'с = 10/4 21с 31с 41с Freq
II(m)1
--. Mag12
(d)
О 'с =10/4 Freq
(е) l:t1 g12
-10 О 10 Freq
Figure 7-3 Quadrature bandpass sampling wlth digltal mixlng: (а) quadrature
sampling bIoek diagram; (Ь) spectrum of contlnuous bandpass slgnal
X(t); (е) dlscrete spectrum of sampled bandpass slgnal x(nt s ); (d)
frвqueney-trans/ated ,/(т) I d/serete speetrum of /n-phase eomponent
of x(nt s ); (е) low-pass filtered I (т) I speetrum of In-phase component,
these two orthogonal ones sequences translate the input signal's spectrum
Ьу exactly 1.14, and that' s why our origina11. was made equal to 4fc' Тhis
clever technique enables complex sampling without multiplications
because the mixing is implemented merely Ьу sign changes оп x(пt s ).
If we implement this digital mixing process, we'll find that the spectral
replication period in Figure 7 3(d) is half what is was in Figure 7-3(с). Тhis
comes about from the convolution of the real cosine's positive and nega
tive frequency components with ! Х(т)! in Figure 7 3(c), То use this dig
ital mixing scheme, we must ensure that the condition В < Ic = 1./4 holds
true. In addition to the reduced spectral replication periodicity, we also
Dlgital Resaтpliпg 303
need to Ье aware that the DFТ magnitude о! the frequency translated i(n),
I I(m) 1, is related to the DFТ magnitude о! x(nt s )' I Х(m) 1, Ьу
IХ(m)1
I I(m) Il,O, l,O= '
(7 3)
This is indicated Ьу the Mag and Mag/2 factors in Figures 7 3(c) and (d),
Likewise, the DFТ magnitude о! the frequency translated q(п), I Q(m) 1, is
re1ated to the DFТ magnitude I Х(m) I , Ьу
IХ(m)1
I Q(m) 10,1,o, 1 = -----т----
(7-4)
50 what this аП means is that the i(n) and q(n) signals, after translation
down to zero Hz, have been reduced in amplitude Ьу а factor о! 2, We сап
find out why Figure 7 3(d)'s replication period and equations (7 3) and
(7 4) are true Ьу reviewing 5ection 10.1, which discusses several different
digital quadrature mixing sequences in more detail,
Thus far, we've reviewed the fundamentals of, and an efficient tech
nique for, quadrature bandpass samp1ing. Let's introduce three other
advanced sampling topics, all о! which fall in the general category о! dig
ital resampling.
7.3 Digital Resampling
The useful and fascinating process of digital resampling is а scheme for
changing the effective sampling rate of а discrete signal after the signal has
already Ьееn sampled, As such, resampling has таnу applications; it's
used to minimize computations Ьу reducing data rates when signal band
widths are narrowed through low pass filtering. Resampling is manda
tory in real-time processing when а hardware,digital processor must
accept data at а rate that differs from that processor's fundamental clock
rate. (А simple example of this is when а digital processor сап accept
input data at а l MHz rate and the incoming data is arriving at а rate of
1.5 MHz. In this саве, the incoming data rate must Ье converted from а
l.5 MHz rate to а l МНz sample rate with а minimum о! signal distor
tion.) In satellite and medical image processing, digital resampling is nec
essary for image enhancement, image scale change, and image rotation,
Resampling is also used to reduce the computational complexity of cer-
тin narrowband digital filters.
304 Advonced Soтpling Techn/ques
We сап define resampling as follows: Consider the process where а
continuous signal ус has been sampled at а rate off o1d = 1/T o1d ' and the
discrete samples are xo1ct<п) = yc(пT o1d ). Resampling is necessary when we
need xnew(п) = Yc(пT new )' and direct sampling of the continuous ус at the
rate of fnew = l/Т new is not possible, For example, imagine we have an ana
log to digital (A/D) conversion system supplying а sample value every
TOld seconds. But our processor сап accept data only at а rate of one sam
ple every Tnew seconds. How do we obtain xnew(п) directly from xo1d(п)?
One possibility is to digitaHo analog (D/ А) convert the xo1d(п) sequence
to regenerate the continuous Ус and, then, A/D convert Ус at а sampling
rate of fnew to obtain xnew(п). Due to the spectral distortions induced Ьу
D / А followed Ьу A/D conversion, this technique limits our effective
dynamic range and is typically avoided in practice, Fortunately, accurate
all digital resampling schemes have been developed, as we shall see.
7.3.1 Resampling Ьу Decimation
5ampling rate changes соте in two flavors: rate decreases and rate
increases. Decreasing the sampling rate is known as decimation. (Тhe
term decimatioп is somewhat of а misnomer because decimation originally
meant to reduce Ьу а factor of ten, Currently, decimatioп is the term used
for reducing the sample rate Ьу any integral factor,) When the sampling
rate is being increased, the process is known as interpolation, i.e" esti
mating intermediate sample values, Because it's the simplest of the two
rate changing schemes, let's explore decimation first.
We сап decimate, or downsample, а sequence of sampled values Ьу а
factor of D Ьу retaining every Dth sample and discarding the remaining
samples, Relative to the original sample rate,fold' the new sample rate is
J, = fold
new D
(7 5)
For example, to decimate а sequence xo1d(п) Ьу а factor of D = З, we retain
xo1d(0) and discard x o1d (1) and x o1d (2), retain хо1d(З) and discard x o1d (4) and
x o1d (5), retain x o1d (6), and so оп. 50 xnew(п) = Хо1ct<Зn), where п = О, 1, 2, etc.
Тhe result of this decimation process is identical to the resu1t of originally
sam.pling at а rate of fnew = fо1d/З to obtain xnew(п). Тhe spectral implica
tions of decimation are what we should expect, as shown in Figure 7 4,
where the spectrum of an original bandlimited continuous signal is indi
cated Ьу the solid lines. Figure 7 4(a) shows the discrete replicated spec
trum of xo1d(п), Xo1d(m). With xnew(п) = Хо1ct<зп), the discrete spectrum
D/g/to/ Resamp//пg 305
Xnew(m} is shown in Figure 7-4(Ь}, Two important features are illustrated
in Figure 7 4, First, Xnew(m) could have been obtained direct1y Ьу sam
pling the original continuous signal at а rate of f new ' as opposed to deci-
mating xo1in} Ьу а factor of 3. And second, there is, of course, а limit to
the amount of decimation that сап Ье performed relative to the band-
width of the original signa1 В, We must ensure thatfnew > 2В to prevent
aliasing after decimation, When an application requires that f new Ье less
than 2В, then xo1in} must Ье low-pass filtered before the decimation
process is performed. An exam.ple of the need for low pass filtering before
decimation is shown in Figure 7-4(с}. If the original signal has а band
width В, and we're interested in retaining only the band В', the signal
between В' and fnew/2 must Ье low pass filtered before the decimation
м Х (ш !у/1
B
(а) i ! I I I ! I
I I
fold О f 01d Freq
fY1 h1 h1 ( Ш v1 \'у/1 ["т
! 1 I i , I
: i I ' I
(Ь) j . I ....
--Зf new 2fnew fnew О f new 2' new 3' new Freq
= fold
Xo1d(m)
(с)
I
Freq
fold
f new /2
(d)
.
fnew B
r
B' -::: l
о f new /2 В
, ....
f nfNi Freq
Flgure 7-4 Decimotion Ьу о foctor of three: (о) spectrum ond replicotions of
orlginol slgnol; (Ь) spectrum of signol decimoted Ьу о foctor of three;
(с) exomple where only the bondwldth В' from the orlginol bondwidth В
Is to Ье retoined; (d) relationship of о low-poss fllter's cutoff frequency
relotlve to the bondwidth 8',
306 Advoпced SaтplJпg Techпiques
process is performed. Figure 7-4(d) shows this in more detail where the
frequency response of the low-pass fi1ter, shaded, must attenuate the sig
nal amplitudeabove fnew/2. In practice, the direct form of FIR filters (the
structure shown in Figure 5 13) is the prevailing choice for low pass deci-
тatioп filters because of the FIR filter's linear phase response[l7],
When the desired decimation factor D is large, say D > 10, there is an
important feature of the filter/decimation process to keep in mind,
5ignificant computational savings тау Ье had Ьу implementing decima-
tion in multiple stages, Ву way of example, let's assume that we have
input data arriving at а sample frequency of 400 kНz, and we must deci-
mate Ьу а factor of D = 100 because our digital processor сan only accept
data at а rate of 4 kНz. Also, let's assurne that the baseband frequency
range of interest is from О to 1.8 kНz, 50, with fnew = 4 kНz, we must filter
out anу signal above fnew/2 Ьу having our fi1ter transition band between
1.8 kHz andfnew/2 = 2 kНz. It's Ьееn shown that the number of stages 5
in the direct form of а FIR low pass filter is proportional to the ratio of the
original sample frequency over the filter transition band, N in Figure
7-4(d)[17,18]; that is,
S=k. fold k.k&
(/new /2 В' ) N '
(7 6)
where 2 < k < 4 depending оп the amount of filter passband and stop-
band ripple that сап Ье tolerated, 50 for our case, if k is 3, for example,
5 = 6000, Тhink of it, а filter with 6000 stages! Fortunately there's а better
way. Let's do something that appears to compound our problem and par-
tition our decimation example into two stages: decimation Ьу 50 fol-
lowed Ьу decimation Ьу 2, as shown in Figure 7 5(a), We'll assume that
the original input signal spectrum extends from zero Hz to something
greater than 100 kНz, as shown in Figure 7-5(Ь). If the first low pass fil
ter LPF 1 has а cutoff frequency of 1.8 kНz and its stopband is defined to
begin at 6 kНz, the output of the D = 50 decirnator wil1 have а spectrum,
as shown in Figure 7 5(c), where our 1.8-kНz band of interest is shaded.
Notice that there is aliasing of the signal between 2 kHz and 4 kHz not
to worry, LPF 2 will take care of this. When LPF 2 has а cutoff frequency of
1.8 kНz and its stopband is designed to Ье 2 kHz, the output of the D = 2
decimator will have our desired spectrurn, shown in Figure 7-5(d). Тhe
point is that the total number of stages in the two low-pass filters, STotal'
is greatly reduced from 6000, From Eq, (7 6) for the combined LPF 1 and
LPF 2 filters,
Digital Resampliпg 307
Xold(п)
(а)
fOId = 400 kHz
(Ь) 1 х.".,
I
О 200
X ld (т LPF, frequency
/ response
(с)
О 1,8 2 4
Xnew(п)
f new
= 4 kHz
r
I ..
400 Freq (kHz)
Reflection 01 LPF,
frequency response
centered аЬои! 8 kНz
6
8 Freq (kHz)
(d)
1 (тl
v
I ..
о 1,8 2 4 Freq (kHz)
Flgure ]-5 Multlstoge declmatioп: (о) decimation bIock diogram; (Ь) spectrum of
origlnal signal: (с) output of the D = SO decimotor; (d) output of the
D = 2 declmator,
400 8
5Total :: 5 LPF + 5 LPF :: 3. + 3. 665 stages, (7 7)
t 2 (4 1.8) (2 1.8)
This is an irnpressive computational reduction. Reviewing Eq, (7-6) for
each stage, in the first stage we see that, althoughf o1 d = 400 kНz remained
constant, we increased I1f. In the second stage, both t:.f and fold were
reduced. Тhe fact to remember in Eq. (7-6) is that the ratio f o1d / t:.f has а
much more profound effect than k in determining the number оЕ stages
necessary in а low pass filter. Тhis ехатрlе, although somewhat exag-
gerated, shows the kind of computational saving afforded Ьу mu1tistage
decirnation, Isn't it interesting that adding more processing stages to our
original D :: 100 decirnation problem actually decreased the necessary
computational requirement?
308 Advaпced Saтp/lпg Techп/ques
А sensible way to evaluate FIR decimation fi1ter implementations or
any filter implementation, for that matter, is to determine the necessary
nиmber of multip1ications/second, Going back to oиr original D = 3 dec
imation example, let's assume fol d = 60 kНz and our decimation filter need.
have on1y S = 5 mu1tiplier stages, as shown in Figure 7 6(a), with the ш
ter coefficients being ho to h 4 , Because the xo1d(п) samples are arriving at а
rate of fOld' the low pass filter requires C D multiplications/s where
C D = Sfo1d = 5 . 60,000 = 300,000 multip1ications/s. (7 8)
In Figure 7-6(а), our decimation Ьу three is symbolized Ьу а rotary switch
that rotates clockwise 1200 each time an xo1ct<п) sample апivеs and on1y
every third filter output sample is retained for xnew(п), Тhis strиcture is
not 80 desirable becaиse, for every 15 multiplications, we use on1y 5 prod
ucts, an inefficient arrangement even when D is small. Fortunately, the
switching (decim.ation) сап Ье moved ahead of the mu1tip1iers, as illus
trated in Figure 7-6(Ь), and we need perform on1y CD' mu1tip1ications/s
where
C D '= C D = Sfo1d = 100,000 mu1tiplications/s.
D D
(7-9)
Ву moving the decimation process ahead of the multipliers, we give the
multipliers D = 3 times as long to perform а mиltiplication, and xnew(п) is
identical in the two filter structures, (Н our odd order FIR filter coeffi
cients are symmetrical, so that h(4) = h(O), and h(3) = h(l), there's а clever
way to further reduce the number of necessary multiplications, as
described in Section 10.8,)
Let' s step back for а moment to make sure we иnderstand what' s going
оп here, The rotating switches in Figure 7 6 are on1y conceptual, Again,
the 8witch in Figure 7 6(a) means that we retain on1y every third output
sample of x new ' In this case, the filter is operating at the input sample rate
of fold and we discard two out of every three fi1ter output samples, In
Figure 7 6(b), we don't even bother to calculate the two out of every three
samples that are to Ье discarded; here, the fi1ter is operating at the
reduced output sample rate offnew where we calculate on1y those output
samples that we need, Ву the way, ПR filters cannot take advantage of the
reduced computation scheme in Figure 7-6(Ь) because the various imple
mentations of ПR fi1ters mandate that they must always operate at the
input sample rate f01d'
Dlgital Resaтpllпg 3()4
11(0)
Swltch О
(8)
11(0)
(Ь)
FJgure 7-6 Declтotloп fllter lтpleтentatlons: (о) flve-stage FIR dlgltol fllter:
(Ь) deciтotlon toking ploce bвfore the fllter тultlpllcotlons,
Before we leave the subject, it's interesting to rea1ize that decimation
is опе of those rare processes that is not time invariant. From the very
nature of its operation, we know that if we delay the input sequence Ьу
опе sample, а downsampler wi1l generate an entirely different output
sequence. For example, if we apply ап input sequence х(п) = х(й), x(l),
х(2), х(3), х(4), etc., to а downsampler and D = 3, the output у(п) will Ье
the sequence х(й), х(3), х(6), etc, Should we delay the input sequence Ьу
опе sample, our delayed х'(п) input would Ье x(l), х(2), х(3), х(4), х(5),
etc, In this case, the de1ayed output sequence у'(п) would Ье x(l), х(4),
х(7), etc., which is not а delayed version of у(п), Тhus, а decimator
(downsampler) is not time invariant.
7.3.2 Resampling Ьу Interpolation
As we said before, decimation is only part of the digital resampling
story let's now consider interpolation, Sample rate increase Ьу interpo-
lation is а bit more involved than decimation because, with interpolation,
new sample values need to Ье ca1culated, Conceptually, interpolation
.
31 О Advaпced Saтpliпg Techп/ques
comprises generating а continuous curve that passes through our Xo1in)
sampled values, followed Ьу resampling that curve at the new sample
rate fnew to obtain the sequence xnew(n), Of course, continuous curves can't
exist inside а digital machine, so we'll just have to ()btain xnew(n) directly
from xo1in), То increase а given sample rate or upsample Ьу а factor of и,
we have to calculate U 1 intermediate values between each sample in
x o1d (11). Тhe process is beautifully straightforward and best understood Ьу
way of an example.
Let's assume that we have the sequence xo1d(n), part of which is
shown in Figure 7 7(a), and we want to increase its sample rate Ьу а fac
tor of 4, so, и::;: 4. The sequence's spectrum is provided in Figure 7 7(b)
where only the signal spectrum between О Hz and 4fo 1 d is shown. То
upsample xo1d(n), Ьу а factor of four, we must insert three zeros between
each sample of xo1d(n) as shown in Figure 7-7(с), to create the new
sequence x'new(n'), (This insertion of zeros is called zero padding,)
Notice that
X'new(n') = xo1d(n) , when n' = 4n ;
(7 10)
that is, the old sequence is now embedded in the new sequence. Тhe inser
tion of the zeros establishes the sample index for the new sequence xnew(n')
where the interpolated values will Ье assigned, Тhe spectrum of x'new(n'),
X'new(m), is shown in Figure 7 7(d) where fnew = 4fold' In Figure 7-7(d),
notice that xo1in)'s spectrum is replicated и = 4 times in the frequency
range of zero Hz to f new ' Тhis shouldn't surprise us; xo1in)'s spectrum was
already replicated 4 times between О Hz and fnew in Figиre 7 7(b), What
we've done Ьу adding the zeros is merely increase the effective sample fre
quency to fs = fnew in Figure 7-7(d). Likewise, we've also increased the new
folding frequency Ьу а factor of 4 to 2fold'
ТЬе final step in interpolation is to fi1ter the x'new(n') sequence with а
low pass digital filter whose frequency response about zero Hz and fnew
Hz is shown as the dashed lines in Figure 7 7(d), Тhis low-pass filter is
called Щl interpolation filter, and its output sequence is the desired xnew(n'),
shown in Figure 7-7(е) while its output spectrum is given in Figure 7-7(f).
Is that аН there is to interpolation? Well, not quite because we can't
implement an ideallow-pass fi1ter, xnew(n') will not Ье an exact interpola
tion of xo1d(n). Тhe error manifests itself as the unwanted images indicated
Ьу the low level replications in Figure 7 7и), With an ideal fi1ter, these
images would not exist, We сап only approximate an ideal, low pass fi1-
ter, and the accuracy of our entire interpolation process depends оп the
frequency response of our filter approximation! As with decimation,
O/g/ta/ Resaтpllпg 311
(а) t XOld(п "
I
т.,'d
Xo1d(т)
"
(Ь)
Time
о f o1d
(f s )
2fold
3f o ld 4f o1d Freq
1 X ew (rt)
(с) ""
'..I 1'
Tnew Told
"
!
(d)
f o1d
2f o1d
3f o1d
f new Freq
(f s )
... ...
Time
(е)
X new (rt)
...I!I.... .T
.. i' ::
" t x... ' '
о f o1d
" .. ,
2f o1d
3f d'
f new Freq
(fэ)
т,;еw Id Time
Figure 7-7 Interpolation Ьу а faetor of foШ: (а) origina/ sampled sequenee;
(Ь) speetrum of orlglnal sequenee; (е) zeros inserted in orlglnai
sequenee; (d) speetrum of sequenee with Inserted zeros; (е) output
sequenee of interpo/ation filter; (f) output speetrum of interpolation
interpolation сап Ье thought of as, fundamentally, а low-pass filter
design exercise, and the choice of interpolation filter structure deserves
attention,
If the FIR interpolation filter design requires S filter stages, we need
not perform S multiplications to get each of the Xnew(n') samples in Figure
7 7(e). Ву way of an example, retuming to our U == 4 interpolation case,
let's assume we've decided that we need а low-pass filter with S == 19
stages. Our example uses an FlR filter with an odd number of stages
because this is the орНmиm structure used for interpolation filters[19],
The job of our low-pass filter is to convolve its impulse response with the
x'new(n') sequence, Figure 7-8(а) shows the low-pass filter's impulse
response being applied to а portion of the x'new(n') samples to calculate
the ninth sample of xnew(n'), x new (9), The 19 filter coefficients are indi-
cated Ьу the Xs in Figure 7-8(а), With the dots in Figure 7-8(а) represent-
ing the x'new(n') sequence, we see that, although there are 19 Xs, only the
bold Xs generate nonzero products that contribute to the convolution surn
x new (9), The issue here is that we need not perform the multiplications
associated with the zero-valued samples in x'new(n'), In our example, we
need only perform five multiplications to get x new (9) , For the general case,
312 Advaпced Saтpliпg Techпiques
X new (9)
j
1
х....х'><
/ \
х х
I \
i /1. .,
b.Jt 1 J. I Jt ' Jc . .L .."
....X'-Х...Jt' \ 'X x"'X
.
(а)
R ....
Time
п'=9
X new (9)
(Ь)
.". "Х....А...о
/ \
,;0 0\ .
.. : /' .\ i ...o
c.:>A......O...O .JC....6 6.....0 0.... /6
Time
Flgure 7-8 Nlneteen-stoge Interpolotion filter coefflclents: (о)
Interpolotion filter's iтpulse response shlft posltlon to
colculote x new (9); (ь) Xs тark the only filter coefficlents thot
need Ье used to colculote x new (9},
the number of multiplications necessitated Ьу an odd order symmetrical
interpolation FIR filter, for each sample of Xnew(п'), сaлnоt Ье expressed in
а closed {оrrn for all possible values of 5 and и, but it сan Ье shown that
the minimum number of nonzero multiplications C Umin is
C Umin = r 1 multiplications,
U
(7-11)
where r 1 means "the largest integer contained in." C Umin is the number of
x'new(п,) values that will Ье used in the interpolation to deterrnine each
xnew(п,) sample. Тhe number of necessary multiplications/s is Cuminfnew'
50 the good news is that the number of interpolation ffiter multiplica
tions necessary is approximately equal to S/U, and that's quite а reduc
tion in computational complexity, Тhe bad news is that we still require
that 5 = 19 filter coefficients Ье stored, and we can't take advantage of anу
filter coefficient symmetry to reduce the number of multiplications Ьу а
further factor of two as described in 5ection 10,8. Both of these minor
Digital Resaтpliпg 313
problems are the price we must рау to take advantage оЕ Eq, (7-11), and
they can Ье understood Ьу examining Figure 7 8(b). То illustrate our U = 4
example, in Figure 7-8(Ь), we use the filter coefficients indicated Ьу the Xs
to calculate x new (9) , When we slide the filter's impulse response to the
right one sample, we use the coefficients indicated Ьу the circles to calcu
late x new (10) because the nonzero values of x'new(n') wi111ine ир under the
circled coefficients. Likewise, when we slide the impulse response to the
right one more sample to calculate x new (l1), we use the coefficients indi
cated Ьу the diamonds. Finally, when we slide the impulse response to the
right once more, we use the coefficients indicated Ьу the triangles to cal
culate x new (12). 5liding the filter's impulse response once more to the
right, we retum to using the coefficients indicated Ьу the Xs to calculate
x new (13). Уои can see the pattem her there are и, where U =' 4 in our
example, different sets of coefficients that are used to calculate xnew(n')
from the x'new(n') samples. Each time а new xnew(n') sample value is to Ье
calculated, we rotate one step through the sets of coefficients and use
the 11 coefficient set, when п' = О, 4, 8, 12, . . ., 4п,
the Х coefficient set, when п' = 1, 5, 9, 13, . , ., 4п+ 1,
the о coefficient set, when п' = 2, 6, 10, 14, . . ., 4п+2, and
the О coefficient set, when п' = 3, 7, 11, 15, . . ., 4п+3.
This rotation through U sets of coefficients, where each set contains
approximately 5 I U coefficients, is why interpolation fi1ters are sometimes
called time varying, or po/yphase, filters. At different times, we use differ-
ent filter coefficients.
50, even though we don't have to perform 5 = 19 multiplications for
еаф xnew(n') value, we do have to have 5 = 19 fi1ter coefficients stored and
available to calculate еаф xnew(n'). (During the calculation of any given
xnew(n') sample, because the coefficients are not necessarily symmetrical,
we can't take advantage of any symmetrical ffiter strncture to reduce the
number of multiplications Ьу an additional factor of two as discussed in
5ection 10.8.) There's one finа1 important feature of the FIR filter's соеШ-
cients indicated Ьу the triangles. For our example, when the filter's
impulse response peak is aligned above one of the nonzero values of
x'new(n'), notice that the other coefficients indicated Ьу the remaining tri
angles have va1ues of zero. Тhis results in the agreeable characteristic that
xnew(n') = x'new(n') = x'пew(4п), so that the output of the filter at the original
sampling times is equal to the original samples.
314 Advaпced Saтpllпg Techпiques
7.3.3 Gomblnlng Decimation and Interpolation
While changing sampling rates through decimation or interpolation Ьу
integer factors can Ье шеЫ, what can we do if we need а sample rate
change that is not an integer? Тhe good news is that we can implement
sample rate changes Ьу any rational fraction и/о with interpolation Ьу а
factor о! и followed Ьу decimation Ьу а factor о! О. Весаше the fraction
и/о can Ье obtained as accurately as we want with the сопесt choice о!
integers и and О, we can change sample rates Ьу almost any factor in
practice, For example, а sample rate increase Ьу а factor о! 7.125 сап Ье
performed Ьу an interpolation о! и = 57 followed Ьу а decimation о!
0= 8, Ьесаше 7.125 = 57/8.
This и/о sample rate change is illustrated Ьу the processes shown
in Figure 7-9(а). The really neat part here is that the computational bur-
den о! changing the sample rate Ьу the combination о! и/о is less than
the sum о! an individual interpolation followed Ьу an individual deci
mation because we сап сотЬinе the interpolation filter LPF u and the
decimation filter LPF D into а single filter, shown as LPFU/D in Figure 7-
9(Ь)_ The process in Figure 7-9(Ь) is normally called а sample rate con-
verter because, if и > о we have interpolation, and, when О > и, we
have decimation. (The filter LPFU/D is sometimes referred to as an
interpolation filter, regardless о! the ratio и/о, and sometimes it's
called а тultirate filter,) LPFU/D serves two purposes for us now, First, it
must sufficiently attenuate the interpolation images illustrated in
Figure 7-7и), so that, after decimation, they don't contaminate our
desired signal beyond acceptable limits. Second, it must have а cutoff
frequency that prevents aliasing о! our desired signal images resulting
from the final decimation,
То illustrate these filter requirements, let's assume we need to resam
ple an input sequence Ьу 4/3 or 1.333, so, и = 4, and О = 3, Тhe original
sequence xo1d(n) has the spectrum shown in Figure 7 9(c). Тhe spectrum о!
the intermediate sequence x'new(n') after upsampling Ьу 4 and low pass
filtering Ьу LPFU/D is shown in Figure 7-9(d), Тhe spectral result, after the
0= з decimation to obtain,the desired output sequence xnew(n"), is shown
in Figure 7-9(е). Notice that the frequency response о! LPFU/D must Ье
designed so that the stopband frequency, !stop' is less than !new/2 to avoid
aliasing after the decimation. О! course LPFU/D is а time varying filter as
described above for Figure 7-8, Тhe attenuation о! LPFU/D beyond !stop
must Ье great enough, so that the images (shown in Figure 7 9(d) at mul-
tiples о! !old) do not induce intolerable levels о! noise when they're aliased
Ьу decimation into the final band о! О to !MW/2 Нz. If the number о! mul
Interpolation
Digito/ Resoтpliпg 315
Dвclmation
(а)
xold(n)
Xnew(n")
Sample Rate Converter
(Ь)
xOld(n)
(с)
(d)
'Old
('s)
1] ne W(m) LPF
U/D
. . t
О 'stop 'Old
(в)
В\ ,тl
о !
'stop
I
\
'new
-т
Lr i
'new
('s)
/1
! ) 1
2'old
3'Old
4'Old Freq
' 1 . .
, ,, 1 . . ,
Iб '"
..
Freq
"new
(I S )
r T /!
1"''>
, !
..
Frвq
2, new
Flgure 7-9 Samp/e rate eonversion Ьу а faetor of 4/3: (а) eombinatlon of
Interpolatlon and deelmatlon; (Ь) sample rate eonverslon with а slngle
low pass fi/ter; (е) orlglnal dlserete Input sequenee speetrum;
(d) speetrum of the low-pass filter output; (е) final diserete speetrum
after deelmation Ьу а faetor of 3,
tiplication stages in LPFU/D is 5, the computational complexity оЕ our
interpolator/decimator implementation is given Ьу
С и / о = multiplications/output sample,
UD
(7-12)
316 Advoпced Saтpllпg Techп/ques
Because the input to LPFU/D is nonzero only for every uth filter input
sample, so, for each filter output sample we need only perform S/U mul
tiplications, Moreover, our decimator discards аll but every Dth filter out-
put sample, so the actual number ofmultiplications necessary is (S/U)/D
multiplications per decimator output sample,
Again, our interpolator / decimator problem is an exercise in low-pass
filter design, and аll the knowledge and tools we have to design low-pass
filters сап Ье applied to this task. In software interpolator / decimator
design, we want our low pass filter algorithm to prevent aliasing images
and Ье fast in execution time, For hardware interpolator / decimators, we
strive to implement а design that optimizes the conflicting goals of high
performance (minimum aliasing), simple architecture, high data through-
put speed, and low power and parts COunt requirements,
This introduction to digital resampling has, Ьу necessity, only touched
the surface of this important signal processing technique, Fortunately for
us, ilie excellent work of the first engineers and mathematicians to explore
this subject is well documented in the literature. Тhe inquisitive reader
сап probe further to learn how to choose the number of stages in а multi
stage process[17,20], the interrelated considerations of designing opti
mum FIR filters[17,21], the benefits of half band FIR filters[18,22], when
ПR filter structures тау Ье advantageous[21], what special considerations
are applicable to resampling in image processing[23 25], guidance in
developing the controllogic necessary for hardware implementations of
resarnpling algorithms[21], how resampling improves the usefulness of
commercial test equipment[26,27], and software development tools for
designing multirate filters[28],
References
[1] Floyd, Р., and Taylor, J. "Dual-Channel Space Quadrature-Interferometer
System," Microwave Systeт Desigпer's Haпdbook, Fifth Edition, Microwave
Systems News, 1987.
[2] Hack, Т, "IQ Sampling Yields Flexible Demodulators," RF Desigп Мagaziпe,
Aprill991,
[3] Mirage Systems, "Automated-Radar-Measurement System for Pulsed I/Q
Data Collection," Microwave Systeт News aпd Coттuпicatioпs Techпology,
December 1987.
[4] Chester, О, В" and РhiШрs, G, "Use DSP Filter Concepts in IF System
Design," Electroпic Desigп, July 11, 1994,
[5] Steyskal, Н, "Digital Beamforming Antennas," Microwave Jourпal, January
1987,
Refereпces 317
[6] Maruta, R., and Tomozawa, А. "An Improved Method for Digital SSB FDM
Modulation and Demodulation," IЕЕЕ Trans, оп Communications, Vol,
СОМ-26, No, 5, Мау 1978,
[7] Albaugh, N, "New ADCs for RF Signal Processing," RF Desigп, November 1987,
[8] Oxaal, J, "DSP Hardware Improves Multiband Filters," EDN, March 1983.
[9] Sigman, Е, "Simplified Digital Down-Converters," NASA Tech Briefs, July 1995,
[10] Coleman, В" Meehan, Р" Reidy, J" and Weeks, Р, "Coherent Sampling Helps
When Specifying DSP А/О Converters," EDN, October 1987,
[11] Kester, W, "DSP Test Techniques Кеер Flash ADCs in Check," EDN Magazine,
January 18, 1990,
[12] Morris, G., Jr" and Wilck, Н, "JPL 220 Channel 330 MНz Bandwidth Digital
Spectrum Analyzer," IЕЕЕ Proc" Int, Conference, Acoust" Speech, Signa/
Processing, 1978,
[13] Considine, V, "Digital Соmрlех Sampling," Electronics Letters, 19,4 August,
1983,
[14] Rader, С. М, "А Simple Method for Sampling In Phase and Quadrature
Components," IЕЕЕ Trans, оп Aerospace and Electronic Systems, Vol. AES-20,
No, 6, November 1984,
[15] Юсе, О" and Wu, К. "Quadrature Sampling with Нigh Dynarnic Range," IЕЕЕ
Trans, оп Aerospace and Electronic Systems, Vol, АЕ5-18, No. 4, November 1982,
[16] Pellon, L, Е, "А ОоиЫе Nyquist Digital Product Detector for Quadrature
Sampling," IЕЕЕ Trans, оп Signal Processing, Vol, 40, No, 7, July 1992.
[17] Crochiere, R. Е., and Rabiner, L, R. "Optimum FlR Digital Implementations
for Decimation, Interpolation, and Narrow-band Filtering," IЕЕЕ Trans, оп
Acoust, Speech, and Signa/ Proc" Vol, ASSP-23, No. 5, October 1975,
[18] Bal1anger, М, G, "Computation Rate and Storage Estimation in Multirate
Digital Fi1tering with Half-Band Filters," IЕЕЕ Trans, оп Acoust, Speech, and
Sigпal Proc" Vol. ASSP-25, No. 4, August 1977,
[19] Schafer, R, W" and Rabiner, L. R, "А Digital Signal Processing Approach to
Interpolation," Proceedings ofthe IЕЕЕ, Vol, 61, No, 6, June 1973,
[20] Crochiere, R, Е" and Rabiner, L. R. "Decimation and Interpolation оЕ Digital
Signals-----A Tutorial Review," Proceedings ofthe IЕЕЕ, Vol, 69, No, 3, Мзrch 1981,
[21] Crochiere, R, Е" and Rabiner, L, R. "Further Considerations in the Design оЕ
Decimators and Interpolators," IЕЕЕ Traпs, оп Acoust, Speech, and Signal Proc"
Vol, ASSP-24, No, 4, August 1976,
[22] Bal1anger, М, G" Daguet, J. L., and Lepagnol, G, Р, "Interpolation,
Extrapolation, and Reduction оЕ Computational Speed in Digital Filters," IЕЕЕ
Trans, оп Acoust, Speech, and Sigпal Proc" Vol, ASSP-22, No, 4, August 1974.
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[23] Нои, Н, 5" and Andrews, Н, С. "СиЫс Splines for rmage Interpolation and
Digital Filtering," IБЕЕ Trans, оп Acoust, Speech, and Signal Proc" Vol, ASSP-26,
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[24] Keys, R, G, "СиЫс Convolиtion Interpolation for Digital Image Processing,"
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[25] Parker, J, А, Кеnуоn, R. V" and Troxel, О, Е, "Comparison оЕ Interpolating
Methods for rmage Resampling," IЕЕЕ Тrans, оп Medical lтaging, Vol, МИ,
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[26] Вlие, К. J" et аl, "Vector Signal Analyzers for Difficu1t Measиrements ол
Тime-Varying and Соmрlех Modиlated Signals," Hewlett Packard Journal,
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[27] Bartz, М" et аl" "Baseband Vector Signal Analyzer Hardware Design,"
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[28] Mitchell, J, А "Mиltirate Fi1ters Alter Sampling Rates Even After Yoи've
Captured the Data," EDN, Aиgиst 20, 1992,
ICHAPTER EIGHTI
Signal Averaging
How do we determine the typical amount, а valid estimate, or the true
value of some measured parameter? In the physical world, it's not so easy
to do because unwanted random disturbances contaminate our measure-
ments. Тhese disturbances are due to both the nature of the variable being
measured and the fa11ibility of our measuring devices, Each time we try to
accurately measure some physical quantity, we'll get а slight1y different
value. Тhose unwanted f1uctuations in а measured value are ca1led пoise,
and digital signal processing practitioners have learned to minimize noise
through the process of averaging, In the literature, we сan see not only
how averaging is used to improve measurement accuracy, but that aver
aging also shows ир in signal detection algorithms as we1l as in low pass
fi1ter schemes, Тhis chapter introduces the mathematics of averaging and
describes how and when this important process is used, Accordingly, as
we proceed to quantify the benefits of averaging, we're compe1led to
make use of the statistical measures known as the mеan, variance, and
standard deviation,
In digital signal processing, averaging often takes the {олn of summing
а series of time domain signal samples and then dividing that sum Ьу the
number of individual samples, Mathematica1ly, the average of N samples
of sequence х(п), denoted xave' is expressed as
1 N
xave = N Lx(п)
п=l
х(l) + х(2) + х(3) +.,. + x(N)
N
(8-1)
(What we саН the average, statisticians call the mеап.) In studying averag
ing, а key definition that we must keep in mind is the variance of the
sequence а 2 defined as
319
320 Sigпal Averogiпg
2 1 N 2
(j == N :rx(n) Xave] ,
п=l
(8 2)
== [х(l) Xave]2 + [х(2) Xave]2 + [х(3) Xave]2 + .., + [x(N) Xave]2 (8 2')
N '
As explained in Appendix D, the а2 variance in Eqs, (8 2) and (8 2')
gives us а well-defined quantitative measure of how much the values in
а sequence fluctuate about the sequence's average. Тhat's because the
x(l) xave value in the bracket, for example, is the difference between the
x(l) value and the sequence average x ave ' Тhe other important quantity
that we'll use is the standard deviation defined as the positive square root
of the variance, or
1 N
(1= N )X(n) Xave]2 ,
п=l
(8-3)
То reiterate our thoughts, the average value xave is the constant level
about which the individual sequence values тау vary. Тhe variance а2
indicates the sum of the magnitudes squared of the noise fluctuations of
the individual sequence values about the xave average value. If the
sequence х(п) represents а time series of signal samples, we сап say that
xave specifies the constant, or DC, value of the signal, the standard devia
tion (j reflects the amount of the fluctuatin or АС, component of the sig-
nal, and the variance а2 is an indication of the power in the fluctuating
component. (Appendix D explains and demonstrates the nature of these
statistical concepts for those readers who don't use them оп а dai1y basis.)
We're now ready to investigate two kinds of averaging, coherent and
incoherent, to leam how they're different from each other and to see under
what conditions they should Ье used.
8.1 Coherent Averaging
In the coherent averaging process (also known as linear, predetection, or
vector averaging), the key feature is the timing used to sample the origi
nal signal; that is, we collect mu1tiple sets of signal plus noise samples,
and we need the time phase of the signal in each set to Ье identical. For
example, when averaging а sinewave embedded in noise, coherent aver-
aging requires that the phase of the sinewave Ье the same at the beginning
of each measured sample set. When this requirement is met, the sinewave
Cohereпt Averagiпg 321
will average to its true sinewave amplitude value, Тhe noise, however, is
different in each sample set and wil1 average toward zero,t The point is
that coherent averaging reduces the variance of the noise, while preserv
ing the amplitude of signals that are synchronous, or coherent, with the
beginning of the sampling interval, With coherent averaging, we сап actu
ally improve the signal to noise ratio of а noisy signal, Ву way of ехаm-
ple, consider the sequence of 128 data points plotted in Figure 8 l(a),
Тhose data points represent the time-domain sampling of а single pulse
contaminated with random noise. (For illustrative purposes the pulse,
whose peak amplitude is 2.5, is shown in the background of Figure 8-1,)
It's very difficu1t to see а pulse in the bold pulse-plus-noise waveforrn in
the foreground of Figure 8 1(a). Let's say we collect 32 sets of 128 pulse
plus noise samples of the forrn
Sample Set 1 = х 1 (1), х 1 (2), Х 1 (3) , .." х 1 (128) ,
Sample Set 2 = Х/1), х 2 (2), Х 2 (3) , ..., х 2 (128) ,
Sample Se = Хз(l), х з (2), Х з (3) , ,." х з (128) ,
Sample Sеt З2 = х з2 (1), х з2 (2), Х з2 (3) , ..., х з1 (128) .
(8-4)
Here's where the coherent part comes in; the signal measurement times
must Ье synchronized, in some manner, with the beginning of the pulse,
so that the pulse is in а constant time relationship with the first sample of
each sample set, Coherent averaging of the 32 sets of samples, adding ир
the columns of Eq, (8 4), takes the form of
1 З2
xave(k) = Lxn(k) = [Х 1 (k) + x2(k) + хз(k) +... + ХЗ2(k)] / 32 ,
32 n=1
or
Хауе (1) = [Х 1 (1) + Х 2 (1) + Хз (1) +... + Х З2 (1)] / 32
Х ауе (2) = [Х 1 (2) + х 2 (2) + х з (2) + .., + х з2 (2)]/ 32
Х ауе (3) = [х 1 (3) + Х 2 (З) + хз(З) + ." + Х З 2(3)]/ 32
Х ауе (128) = [Х 1 (128) + Х2(128) + х з (128) +... + х з2 (128)] / 32 (8-5)
t Noise samples are assumed to Ье uncorrelated with each other and uncorrelated with the
sample rate, If some component of the noise is correlated with the sample rate, that noise
component will ье preserved after averagirlg,
322 S/gпa/ Averag/пg
. ...
(а)
i
i
.......... ...j..
о
1
20
40
, N=1
60
80
100
120
Time
Amplitude
4
j ! i )
3 . .,. ..". ... . ....ш.l, -l ,. i J... ..
J' .оМ f\f'I л J rI\ "".;; .Л i
2 V iш YV V; VVT V '-' ! ' Yi'-'/Y' , "
(Ь)
1
1
20
40
: N=32 :
60 80
100
120
Тiтe
Amplltude
4
, ' ,
! Pulse falling !
. L
Pulse rlsJng
3 / ' 9 iшшшшшJш
, J +" .
(с)
1
1
20
40
iN=256:
60 80
100
120
Тiтe
Figure 8.1 Slgnal pulse plus noise: (а) one sample set; (Ь) average of 32
sample sets; (е) average of 256 sample sets,
If we perform 32 averages indicated Ьу Eq. (8 5) оп а noisy pulse like
that in Figure 8 1(a), we'd get the 128 point xave(k) sequence plotted in
Figure 8 1(b), Here, we've reduced the noise fluctuations riding оп the
pulse, and the pulse shape is beginning to Ьесоmе apparent, The coher
ent average of 256 sets of pulse measurement sequences results in the
plot shown in Figure 8 1(c), where the pulse shape is clearly visible
now_ We've reduced the noise fluctuations while preserving the pulse
amplitude, (An important concept to keep in mind is that summation
Cohereпf Averagiпg 323
and averaging both reduce noise variance. Summation is merely imple-
menting Eq, (8 5) without dividing the sum Ьу N = 32, If we perform
summations and don't divide Ьу N, we merely change the vertical
scales for the graphs in Figure 8-1(Ь) and (с), However, the noise Лис-
tuations will remain unchanged relative to true pulse amplitude оп the
new scale.)
The mathematics of this averaging process in Eq. (8-5) is both straight-
forward and important, What we' d like to know is the signal-to-noise
improvement gained Ьу coherent averaging as а function of N, the num-
ber of sample sets averaged, Let's say that we want to measure some con
stant time signal with amplitude А, and еат time we actually make а
measurement we get а slightly different value for А. We realize that our
measurements are contaminated with noise such that the nth measure-
ment result r(n) is
У(n) =А + noise(n) ,
(8-6)
where noise(n) is the noise contribution, our goal is to determine А when
the r(n) sequence of noisy measurements is all we have to work with, For
а more accurate estimate of А, we average N separate r(n) measurement
samples and calculate а single average va1ue 'ауе' То get а feeling for the
accuracy of rave' we decide to take а series of averages rave(k), to see how
that series fluctuates with each new average; that is,
rave(l) = [r(l) + r(2) + r(3) + ... + r(N)]/N , lstN-роiпt average
r ave (2) = [r(N+1) + r(N+2) + r(N+З) +.., + r(2N)]/N, 2ndN.pointaverage
rave(3) = [у(т+1) + r(2N+2) + r(2N+3) +... + r(ЗN)]/N, ЭrdN-роiпtаvеrаgе
rave(k) = [r([k l] .N+1) + r([k l] .N+2) +r([k 11.N+З) + ... +r(k. N)l/N , (8-7)
or, more concisely,
1 N
rave(k) = N Lr([k l]'N+n)
п=l
(8-8)
То see how averaging reduces our measurement uncertainty, we need to
compare the standard deviation of our rave(k) sequence of averages with
the standard deviation of the original r(n) sequence,
324 Sigпal Averaglпg
If the standard deviation of our original series of measurements r(п) is
0in' it has been shown [1-5] that the standard deviation of our rave(k)
sequence of N-point averages, Оауе' is given Ьу
(Т,
(Тауе == Ш
(8-9)
Equation (8-9) is significant because it tells us that the r ave(k) series of aver-
ages will not fluctuate ав much about А as the original r(п) measurement
values did; that is, the rave(k) sequence will Ье less noisy than any r(п)
sequence, and the more we average Ьу increasing N, the more closely an
individual rave(k) estimate will approach the true value of л. t
In а different way, we сап quantify the noise reduction afforded Ьу
averaging, If the quantity А represents the amplitude of а signal and Оin
represents the standard deviation of the noise riding оп that signal атрН-
tude, we сап state that the original signal-amplitude--to-noise ratio is
А
SNR in ==
(Т.
m
(8-10)
Likewise, the signal-amplitude--to-noise ratio at the output of ал averag-
ing process, SNR ave is defined ав
SNR == rave ==
ауе
(J ауе (J ауе
(8-11)
Continuing, the signal-to-noise ratio gaiп, SNR coh gain, that we've realized
through coherent averaging is the ratio of SNR ave over SNR in , or
SNR ' SNR ave
g am
coh SNR.
m
А / (J ауе = (J in
А / (Тin (J аУе
(8-12)
Substituting (Тауе from Eq. (8-9) in Eq, (8-12), the SNR gain becomes
SNR coh gain == (Jin.JN ==.JN '
(Тin / N
(8-13)
t Equation (8-9) is based оп the assumptions that the average of the origtna! nоше is zero and
that neither А nor а in change during the time we're perfonning our averages.
Cohereпt Averagiпg 325
Тhrough averaging, we Сan realize а signal to noise ratio improvement
proportional to the square root of the number of signal samples averaged,
In terms of signaHo noise ratio measured in decibe1s, we have а coherent
averaging, or iпtegratioп, gain of
SNR coh gain(dB) = 20.1og lO (SNR coh ) = 20.10g lO (#J) = 10.1og lO (N) , (8 14)
Again, Eqs, (8 lЗ) and (8 14) are valid if А represents the aтplitude of а sig
nаl and O'in represents the original noise standard deviation,
Another way to view the integration gain afforded Ьу coherent aver
aging is to consider the standard deviation of the input noise, O'in and the
probability of measuring а particular value for the Figиre 8-1 pulse ampli
tude. Assume that we made тапу individual measurements of the pulse
amplitude and created а fine grained histogram of those measиred values
to get the dashed curve in Figиre 8 2(a), Тhe vertical axis of Figure 8 2(a)
represents the probability of measuring а pulse amplitude value corr
sponding to the values оп the horizontal axis, If the noise fluctuations fol
low the well known пorтal, or Gaussian distribution, that dashed
probability distribution curve is described Ьу
р(х) = 1 e--(х Jl)2 12(12 = Ke (x Jl)2/2(12
0'$
(8 15)
where о' = o'in and the true pulse amplitude is represented Ьу 11 = 2.5, We
see from that dashed curve that аnу given measured value wil1 most
likely (with highest probability) Ье near the actual pulse amplitude value
of 2.5, Notice, however, that there's а nonzero probability that the тea
sured value could Ье as low as 1.0 or as high as 4.0. Let's say that the
dashed curve represents the probability curve of the pulse-plus noise sig-
nаl in Figure 8 1(a), If we averaged а series of 32 pulse-amplitude values
and plotted а probability curve of our averaged pulse-amplitude теа-
surements, we' d get the solid curve in Figure 8-2, Тhis curve characterizes
the pulse plus noise values in Figиre 8-1(Ь), From this solid curve, we see
that there's а very low likelihood (probability) that а measured value,
after 32 point averaging, will Ье less than 2.0 or greater than 3.0.
From Eq, (8 9), we know that the standard deviation of the result of
averaging 32 signal sample sets
0" 0"
о' =.......J!L=.......J!L
ауе .J32 5.65
(8-16)
326 Sigпol Averoglпg
Р(Х)
к
О,2К ш
'"
I
/
О,ВК
О,БК
О.4К ш
"-
...
о
о
1,0
2,0
3,0
4,0
Measured
amplitude (Х)
Flgure 8-2 Probobility density curves of meosured pulse omplitudes with по
overaglng (N = 1) and wlth N = 32 overaglng,
In Figure 8-2, we can see а statistical view of how an averager's output
standard deviation is reduced from the averager's input standard devia-
tion. Taking larger averages Ьу increasing Nbeyond 32 would squeeze the
solid curve in Figure 8 2 even more toward its center value of 2.5, the true
pulse amplitude,t
Returning to the noisy pulse signal in Figure 8 1, and performing
coherent averaging for various numbers of sample sets N, we see in
Figure 8 3(a) that as N increases, the averaged pulse amplitude
approaches the true amplitude of 2.5. Figure 8 3(b) shows how rapidly
the variance of the noise riding оп the pulse falls off as N is increased, An
alternate way to see how the noise variance decreases with increasing N
is the noise power plotted оп а logarithmic scale as in Figure 8-3(с). In
this plot, the noise variance is normalized to that noise variance when по
averaging is performed, i.e., when N = 1. Notice that the slope of the
curve in Figure 8 3(c) closely approximates that predicted Ьу Eqs. (8 13)
and (8 14); that is, as N increases Ьу а factor of 10, we reduce the average
noise power Ьу 10 dB. Although the test signal in this discussion was а
pulse signal, had the signal been sinusoidal, Eqs. (8-13) and (8 14) would
still apply,
t The curves in Figure 8-2 are normaJized for convenient iJJustration, From Eq, (8-15) and
assUlnmg that cr = 1 when N = 1, then К = 0,3989, When N = 32, the new standard deviation
is cr' = crl ffi = 11 -J3i and К = 0,3989' -J3i = 2,23,
/пcohereпt Averagiпg 327
2.60
Averaged pulse amplllude
2,55
.
.... . '
(а) 2,50 ...\/ ..
.
2,45 . ........ ... .....;..... .
.
2.40
О
100
200
зоо
400
500
N
Averaged pulse-amplitude variance, а;уе
0,9
: . ,
0.6
0,5 ... ......... ................
(Ь) 0.4 f..
о.зl .
. .
0,2 \ .
0,1 ....
О ...,,_.,. . .
"'{},1
О 100 200 ЗОО 400
.
500
N
10
(с) 15
20
25
30
1 10 100 500 N
Flgure 8-3 Results of overaging signol pulses plus nolse: (а) meosured pulse
omplltude vs, N; (Ь) meosured vorlonee of pulse omplitude vs, N; (е)
meosured pulse-omplitude noise power vs, N оп а logorithmlc seole.
8.2 Incoherent Averaging
The process of incoherent averaging (also known as rms, postdetection,
scalar, or video averaging) is the averaging of signal samples where по
sample timing constraints are used; that is, signal measиrement time
328 Sigпcl Averagiпg
intervals are not synchronized in any way with the phase of the signal
being measured. Think for а moment what the average would Ье of the
noisy pulse signal in Figure 8-1(а) if we didn't in some way synchronize
the beginning of the collection оЕ the individual signal sample sets with
the Ъеginning of the pulse. The resu1t would Ье pulses that begin at а dif-
fereI1t time index in each sample set. The averaging of multiple sample
sets would then smear the pulse acrDSS the sample set, or just "average
the pulse signal away." (For those readers fam.iliar with using oscillo-
scopes, incoherent averaging would Ье like trying to view the pulse
when the beginning of the scope sweep was not triggered Ьу the signal.)
As such, incoherent averaging is not so useful in the time domain. t In the
frequency domain, however, it's а different story because incoherent
averaging can provide increased accuracy in measuring relative signal
powers. lndeed, incoherent averaging is used in many test instruments,
such as spectrum, network, and signal analyzers.
In some analog test equipment, time domain signals are represented in
the frequency domain using а narrowband sweeping filter followed Ьу а
power detector, These devices measure signal power as а function of fre
quency. The power detector is necessary because the sweeping measure-
ment is not synchronized, in time, with the signal being measured. Thus
the frequency-domain data represents power only and contains по signal
phase information. Although it's too late to improve the input's signal-
amplitude to noise ratio, incoherent averaging can improve the accuracy
of signal power measurements in the presence of noise; that is, if the
signal power spectrum is very noisy, we can reduce the power estimation
fluctuations and improve the accuracy оЕ signal power and noise-power
measurements. Figure 4(a) illustrates this idea where we see the power
(magnitude squared) output of an FFТ of а fundamental tone and several
tone harmonics buried in background noise. Notice that the noise power
levels in Figure 8 4(a) fluctuate Ьу almost 20 dB about the true average
noise power indicated Ьу the dashed line at 19 dB.
lf we take 10 FFТs, average the square of their output magnitudes, and
normalize those squared values, we get the power spectrum shown in
Figure 8 4(b). Here, we've reduced the vаriaщ:е оЕ the noise in the power
spectrum but have not improved the tones' signal power to-noise power
ratios; that is, the average noise power level remains unchanged.
Averaging the output magnitudes squared оЕ 100 FFТs results in the spec-
t Тhe term incoherent averaging is а bit of а misnomer, Averaging а set of data is just that, aver-
aging we add ир а set of data values and divide Ьу the number of samples in the set,
lncoherent averaging should probably Ье called averaging data that's obtained incoherently,
/пeohereпt Averagiпg 329
Power (dB)
о
N=1
5
(8)
True average
noise power
10
20
зо
40
50
60
Freq
Power (dB)
о
(Ь)
5
10 20 30 40 50 60 Freq
Power (dB)
О
....
10
(с)
15...
20
25
О 10 20 30 40 50 60 Freq
Figure 8-4 Results of averaging signal tones plus noise-power speetra: (а) по
averaging, N= 1; (Ь) N= 10; (е) N= 100,
trurn in Figure 8-4(с), which provides а more accurate measure of the rel
ative power levels of the fundamental tone's harmonics.
Just as we arrived at а coherent integration SNR gain expression in Eq,
(8-14), we сап express аn incoherent integration gain, SNR incoh gain, in
terms of SNR measured in dB as
330 Sigпo/ Averagiпg
Processing gain in dB
60
10
106 N
50
40
зо
20
Flgure 8-5 Tlme-domain ampiltude SNR processing gain from Eq, (8-14), and the
frequency-domaln power SNR processlng galn from Eq, (8-17), as
functlons of N,
SNR incoh gain(dB) = 10 .10g10(-J.N) .
(8-17)
Equation (8-17) applies when the quantity being averaged represents the
power of а signal. That's why we used the factor of 10 in Eq. (8-17) as
opposed to the factor of 20 used in Eq, (8-14),t We сап relate the process
ing gain effects of Eqs. (8-14) and (8-17) Ьу plotting those expressions in
Figure 8-5.
8.3 Averaging Multiple Fast Fourier Transforms
We discussed the processing gain associated with а single DFТ in Section
3,12 and stated that we сап realize further processing gain Ьу increasing
the point size of аnу given N-point DFТ, Let's discuss this issue when the
DFТ is implemented using the PFТ algoritlun. The problem is that large
FFТs require а lot of number crunching. Because addition is easier and
faster to perform than multiplication, we сап average the outputs of mul-
tiple FFТs to obtain further PFТ signa1 detection sensitivity; that is, it's
t Section Е,l of Appendix Е explains why the multiplymg factor is 10 for signal-power
measurements and 20 when dealing with signal-amplitude values,
Averaglпg Multlple Fast Fourier Traпsforms 331
easier and typically faster to average the outputs of four 128 point FFТs
than it is to ca1culate one 512 point FFТ. The increased FFТ sensitivity, or
noise variance reduction, due to multiple FFТ averaging is also called
integration gain, 50 the random noise fluctuations in а FFT's output Ьins
will decrease, while the magnitude of the FFT's signal Ып output
remains constant when mu1tiple FFТ outputs are averaged. (Inherent in
this argument is the assumption that the signal is present throughout
the observation intervals for all of the FFТs that are being averaged and
that the noise sample values are independent of the original sample
rate,) There are two types оЕ FFТ averaging integration gain, incoherent
and coherent.
Incoherent integration, relative to FFТs, is averaging the corresponding
bin magnitudes of multiple FFТs; that is, to incoherently average k FFТs,
the zeroth Ьin of the incoherent FFf average Fincoh(O) is given Ьу
1Ft (0)1 + IF 2 (0)1 +IF з (о)1 + ... + IFk(O)1
F incoh (О) ::; k '
( 18)
where I Fп(O) I is the magnitude of the zeroth Ып from the пth FFT,
Likewise, the first Ьin of the incoherent FFТ average, F incoh (l), is given Ьу
I F t(1}1 + IF 2 (l)j +IF 3 (l)j + ,., + IF k (l)1
F incoh (1) ::; k '
(8-18')
and, so оп, out to the last bin of the FFТ average, Fincoh(N l), which is
IFt(N 1)1+IF2(N l)l+IFз(N l)I+...+IFk(N 1)I "
Fincoh(N 1)::; k (8 18 )
Incoherent integration provides additional reduction in background
noise variation to augment а single FFТ's inherent processing gain, We
сап demonstrate this in Figure 8 6(a) where the shaded curve is а single
FFТ output of random noise added to а tone centered in the 16th bin of а
64-point FFf. Тhe solid curve in Figure 8 6(a) is the incoherent integration
of 10 individual 64--point FFТ magnitudes. Both curves are normalized to
their peak values, so that the vertical scales are referenced to О dB. Notice
how the variations in the noise power in the solid curve have been
reduced Ьу the averaging of the 10 FFТs, Тhe noise power values in the
solid curve don't fluctuate as much as the shaded noise power values. Ву
332 Sigпol Averagiпg
Power (dB)
...."...."' ...':...... .
I
...}\I
f \ i I
Ш( П е ;=;ТШ ш
50 60
True
average
nolse power
10
(а) 15
20
25
--30
О
10
20
30 40
m
Power (dB)
о
--30
О
10
20 30
Ш\:/,
Ili
... .v..........шШ.....ШIШШШШ...
t Топе is Ьеlwее"
i FFТ Ып centers
50 60
True
average
noise power
10
(Ь) 15
20
25
40
m
Flgure 8-6 Single FFТ output magnltudes (shaded) and the average of 1 О FFТ output
magnitudes (solld): (а) tone at Ыn center; (Ь) tone between Ыn centers,
averaging, we haven't raised the power оЕ the tone in the 16th Ьin, but we
have reduced the peaks оЕ the noise power values. Тhe larger the number
оЕ FFrs averaged, the closer the individual noise power Ьin values will
approach the true average noise power indicated Ьу the dashed horizon
talline in Figure 8 6(a).
When the signal tone is not at а Ьin center, incoherent integration still
reduces fluctuations in the FFr's noise power bins, Тhe shaded curve in
Figure 8 6(b) is а single FFТ output оЕ random noise added to а tone
whose frequency is halfway between the 16th and 17th bins оЕ the 64-
point FFТ, Likewise, the solid curve in Figure 8 6(b) is the magnitude
average оЕ 10 FFrs, Тhe variations in the noise power in the solid curve
have again been reduced Ьу the integration оЕ the 10 FFrs. 50 incoherent
integration gain reduces noise-power fluctuations regardless оЕ the fre
Averagiпg Mu/t/p/e Fast Fourier Traпsforтs 333
quency location of апу signals of interest. As we would expect, the signal
peaks are wider, and the true average noise power is larger in Figure
8-6(Ь) relative to Figure 8 6{a) because leakage raises the average noise
power level and scalloping loss reduces the FFТ bin's output power level
in Figure 8-6(Ь). Тhe thing to remember is that incoherent averaging of
FFТ output magnitudes reduces the variatioпs in the background noise
power but does not reduce the average background noise power.
Equivalent to the incoherent averaging results in Section 8.2, the reduc-
поп in the output noise variance[6] of the incoherent average of k FFТs rel-
ative to the output noise variance of а single FFТ is expressed as
2
cr kFFfs
2
cr single FFf
1
k
(8 19)
Accordingly, if we average the magnitudes of k separate FFTs, we reduce
the noise variance Ьу а factor of k,
In practice, when mu1tiple FFТs are averaged and the FFТ inputs are
windowed, an overlap in the time domain sampling process is commonly
used, Figure 8 7 Шustrаtеs this concept with 5.5Nt s seconds worth of time
series data samples, and we wish to average ten separate N-point FFТs
where t s is the sample period {l/fJ Because the FFТs have а 50 percent
overlap in the time domain, some of the input noise in the N time samples
for the first FFТ wШ also Ье contained in the second FFТ, Тhe question is
Time -<
series I
5.5Nt s seconds
'-.'-. I
1st FFТ I
2nd FFТ I
Зrd FFТ I
4th FFТ
Time
,
N samples
I .
10th FFТ I
Flgure 8-7 Tlme relatlonshlp of multlple FFТs wlth 50 percent overlap,
334 Signal Averagiпg
"VVhat's the noise variance reduction when some оЕ the noise is соттоn
to two FFТs in this averaging scheme?" Well, the answer depends оп the
wir1.dow function used оп the data before the FFTs are performed. It's
Ьееn shown that for the most соттоn window functions using an over
lap оЕ 50 percent or less, Eq. (8 19) still applies as the level оЕ noise vari
ancereduction[7].
Coherent integration gain is possible when we average the real parts of
multiple PFТ bin outputs separately from the average оЕ the imaginary
parts and then combine the single real average and the single imaginary
average into а single complex Ьin output average value; that is, to coher
ently average k separate FFТs, the zeroth Ьin оЕ the complex coherent PFТ
average, Fcoh(O), is given Ьу
F ( 0 ) = Fl(O)real +F 2 (0)real +...+Fk(O)real
coh k
, 11 (O)imag + [ 2 (O);mag +... + Fk (O)imag
+ J k
(8 20)
'Пlе first bin оЕ the complex coherent РРТ average, F coo (l), is
F ( 1 ) = F 1 (l)real + F 2 (l)real +... + Fk (1}real
coh k
, F 1 (1);mag + F 2 (1)imag + ...+ F k (l);mag
+ J k '
(8 20')
and so оп out to the last Ьin of the complex PFТ average, Fcoh(N l), which is
F ( N 1 ) = FJ (N l)real + [ 2 (N 1)real +... + Fk (N l)real
coh k
, F 1 (N l)imag + Р 2 (N l)Unag +... + Fk (N l)imag ( 8 20" )
+ J k .
Let's consider why the integration gain afforded Ьу coherent averaging
is more useful than the integration gain from incoherent averaging
Where incoherent integration does not reduce the background noise'
Averag/пg Mu/t/p/e Fast Four/er Traпsforтs 335
power average 1eve1, it does reduce the variations in the averaged FFT's
background noise power because we're dea1ing with magnitudes only,
and аН FFr noise Ьin va1ues are positiv they're magnitudes. 50 their
averaged noise Ьin magnitude va1ues will never Ье zero. Оп the other
hand, when we average PFТ comp1ex bin outputs, those comp1ex noise
Ьin values сап Ье positive о, negative. Тhose positive and negative values,
in the rea1 or imaginary parts of multip1e РРТ Ьin outputs, will tend to
cance1 each other. This means that noise Ьin averages can approach zero
before we take the magnitude, while а sigJ;;la1 Ьin average will approach
its Пuе nonzero magnitude. If we say that the coherent1y averaged PFТ
signa1 to noise ratio is expressed Ьу
SNR = 20.10 ( signa1 Ьin magnitude )
coh g10, Ь ' . d '
nOlse m magrutu е
(8 21)
we сап see that, should the denominator of Eq. (8 21) approach zero, then
SNR coh сап Ье increased, Let' s 100k at an examp1e of coherent integration
gain in Figure 8 8(a) where the shaded curve is а sing1e FFr output of ran
dom noise added to а tone centered in the 16th Ьin of а 64 point PFТ. Тhe
solid curve in Figure 8-8(а) is the coherent integration of ten 64 point
FFTs, Notice how the new noise power average has actuaHy been
reduced.+ Тhat's coherent integration, Тhe 1arger the number of FFrs aver-
aged, the greater the reduction in the new noise power average.
When the signa1 tone is not at bin center, coherent integration stiIl
reduces the new noise power average, but not as much as when the tone
is at Ьin center, The shaded curve in Figure 8-8(Ь) is а sing1e PFТ output
of random noise added to а tone whose frequency is ha1fway between the
16th and 17th bins of the 64 point FFr, Likewise, the solid curve in Figure
8 8(b) is the coherent integration of 10 FFТs, Тhe new noise power aver
age has, again, been reduced Ьу the integration of the 10 FFТs. Coherent
integration provides its maximum gain when the origina1 samp1e rate fs is
an integra1 multip1e of the tone frequency we're trying to detect. Тhat way
the tone will a1ways have the same phase ang1e at the beginning of the
samp1e interval for each РРТ. 50, if possible, the fs samp1e rate should Ье
t The sharp-eyed reader might say, "Wait а minute, Coherent integration seems to reduce the
noise-power average, but it doesn't look like it reduces the noise-power variations," Well,
the noise-power variations realiy are reduced in magnitude----remember, we're using а log-
arithmic scale, The magnitude of а half-division power variation between 20 dB and зо dB
is smalier than the magnitude of а half-division power variation between 10 dB and 20 dB,
.
336 Sigпol Averaglпg
о
10 ..............шш.ш.
...! ................ ........ .....
(а) 20
o
10
20
30
40
50
60
Power
(dB)
о
10
(Ь) 20
o
--40
О
10
30
20
40
50
60
о
10
(с) 20
--(J()
--40
о
10
20
30
40
50
60
Original
average
noise
power
New
average
noise power
m
Original
average
noise
power
New
average
noise power
m
New
average
noise power
Original
average
noise
power
m
Flgure 8-8 Single FFТ output magnltudes (shaded) and the eoherent Integratlon of
10 FFТ outputs (solld): (а) tone at bIn eenter; (Ь) tone between bIn
eenters; (е) tone at bIn eenter wlth random phase shift Indueed prlor to
the FFТ proeess,
chosen to ensure that the tone of interest, assuming it's not drifting in fre-
quency, will Ье ехасНу at а FFТ bin center,
Кеер two thoughts in mind while using coherent integration of multi-
ple FFТs, First, if the random noise contaminating the time signal does not
have an average value of zero, there will Ье а nonzero DC term in the
Averag/пg Mu/tip/e Fast Four;er Traпsforms 337
averaged FFI"s Рсоь(О) Ьin, Averaging, of course, will not reduce this DC
level in the averaged FFТ РсоЬ(О) output. Second, coherent integration сап
actually reduce the averaged FFI"s SNR if the tone being detected has а
random phase at the beginning of each FFI' sample interval, Аn example
of this situation is as follows: the shaded curve in Figure 8 8(c) is а single
FFI' output of random noise added to а tone centered in the 16th Ьin of а
64 point FFТ, Тhe solid curve in Figure 8 8(c) is the coherent integration
of 10 individual 64 point FFТs, However, for each of the 10 FFТs, а ran
dom phase shift was induced in the fixed frequency test tone prior to the
FFI'. Notice how the averaged FFI"s new average noise power is larger
than the original average noise power for the single FFI', When the tone
has а random phase, we've actually lost rather than gained SNR because
the input tone was по longer phase coherent with the analysis frequency
of the FFТ Ыn.
Тhere's а good way to understand why coherent integration behaves
the way it does. If we look at the phasors representing the successive out
puts of а FFI' Ьin, we сап understand the behavior of а single phasor rep
resenting the sum of those outputs. But, first, let's refresh our memory for
а moment concerning the vector addition of two phasors,t Because аn FFТ
Ьin output is а complex quantity with а real and аn imaginary part, we
сап depict а single FFТ Ьin output as а phasor, like phasor А shown in
Figure 8 9(a). We сan depict а subsequent output from the same FFI' Ьin
as phasor В in Figure 8-9(Ь). Тhere ме two ways to coherent1y add the two
FFI' Ьin outputs, i.e" add phasors А and В, As shown in Figure 8 9(c), we
сап add the two real parts and add the two imaginary parts to get the sum
phasor С. А graphical method of summing phasors А and В, shown in
Figure 8 9(d), is to position the beginning о! phasor В at the end of pha
sor А. Тhen the sum phasor С is the new phasor пот the beginning of
phasor А to the end of phasor В, Notice how the two С phasors in Figures
9(c) and 8 9(d) are identical. We'll use the graphical phasor summing
technique to help us understand coherent integration о! FFI' Ьin outputs,
Now we're ready to look at the phasor outputs of аn FFI' signal Ыn to
see how а single phasor representing the sum of multiple signal Ьin pha
sors behaves. Consider the three phasor combinations in Figure 8 10(a),
Each phasor is а separate output о! the PFТ Ыn containing the signal tone
we're trying to detect with coherent integration, Тhe dark arrows are the
phasor components due to the tone, and the small shaded arrows are
the phasor components due to random noise in the FFI' Ыn containing
the tone, In this case, the original signal sample rate is а multiple о! the tone
t Following the argиments put forth in Section А,2 of Appendix А, we'll use the term phasor,
as opposed to the term vector, to describe а single, complex DFf output value,
338 Sigпol Averaglпg
Z haso'A l '
maglnary
partol А
Аеаl part 01 А
Real part 01 В
Imaginary
Phaso, В part 01 В
(а)
(Ь)
(с)
Imaginary part of А . '
Imaglnary
partolB
Phaso' С
Аеаl part 01 А
Аеаl part 01 В
Phaso'B
Phaso, А ...............
(d) / ...............
Phaso, С
Figure 8-9 Two ways to add phaso,s А aпd В, where phasor С = А + В,
frequency, so that the tone phasor phase angle I?J is the same at the begin
ning of the three FFТs, sample intervals, Тhus, the three dark tone pha-
sors have а zero phase angle shift relative to one another, Тhe phase
angles of the random noise components are random, If we add the three
phasor combinations, as а first step in coherent integration, we get the
actual phasor sum shown in Figure 8-10(Ь), The thick shaded vector in
Figure 8-10(Ь) is the ideal phasor sum that would result had there been
по noise components in the original three phasor combinations, Because
the noise components are reasonably small, the actual phasor sum is not
too different from the ideal phasor sum,
Now, had the original signal samples been such that the input tone's
phase angles were random at the beginning of the three FFТs, sample
intervals, the three phasor combinations in Figure 8 10(c) could resu1t,
Summing those three phasor combinations results in the actual phasor
sum shown in Figure 8-10(d), Notice that the random tone phases, in
Figиre 8-10(с), have resulted in an actual phasor sum magnitude (length)
that's shorter than the dark lines of the phasor component due to the tone
in Figure 8-10(с). What we've done here is degrade, rather than improve,
Averag/пg Mu/t/p/e Fast Four/er Traпsforтs 339
(а)
""'W"""_
(с)
t
(Ь)
,,"
(d)
Actual
phasor sum
Figure 8-10 FFТ bin outputs represented 05 phosors: (о) three outputs fюm the
some bin when the input tone is ot bin center; (Ь) coherent integration
(phasor additlon) of the three bin outputs; (с) three outputs fюm the
some bin when input tone hos о random phose at the beglnning о'
eoch FFТ; (d) coherent integration о' the three bin outputs when input
tone hos random phose,
. 340 Sigпa/ Averagiпg
the averaged FFТ's output SNR when the tone has а random phase at the
beginning of each FFТ sample interval,
The thought to remember is that, although coherent averaging of FFТ
outputs is the preferred technique for realizing integration gain, we're
rarely lucky enough to work real world signals with а constant phase at the
beginning of every time--domain sample interval, 50, in practice, we're usu
аНу better off using incoherent averaging, Of course, with either integration
teclmique, the price we рау for improved FFТ sensitivity is additional coт
putational complexity and slower system response times because of the
additional surnmation ca1culations, (А slick way to reduce the computa-
tional burden of averaging multiple FFТs is discussed in Section 10,7.)
8.4 Filtering Aspects of Time-Domain Averaging
In Section 5,2 we introduced FIR fi1ters with an averaging example, and
that's where we first leamed that the process of time-domain averaging
performs low pass filtering, In fact, successive time domain outputs of an
N point averager are identical to the output of аn (N lHap FIR fi1ter
whose coefficients are аН equal to l/N, as shown in Figure 8 11,
The question we'H answer here is "What is the frequency magnitude
response of а generic N point averager?" We could evaluate Eq, (6 42),
from Section 6.3, that describes the frequency response of а generic N
stage FIR filter, In those expressions, we'd have to set аН the Ь(О) through
b(N) coefficient values equal to l/N and ca1culate HFIR(ro)'s magnitude
over the normalized radian frequency range of О ro п, Тhat range cor
responds to an actual frequency range of О f fs/2 (where fs is the equiv
alent data sample rate in Hz), А simpler approach is to recaH, from Section
5.2, that we сап ca1culate the frequency response of an FIR filter Ьу taking
the DFТ of the fi1ter's coefficients. In doing so, we'd use аn M point FFТ
1st
2nd
(N--1)th
1/N
Averager output
Averager input
Figure 8.11 An N-polnt averager deplcted as an FIR fllter,
Expoпeпtial Averaglпg 341
Averagar frequency magnitude response
0,4
fJ8
fJ4
fJ2
0,8
0,6
0,2
Flgure 8.12 N-point overoger's frequency magnitude response os а functlon of N,
software routine to transform а sequence of N coefficients whose values
are al1 equal to l/N. Of course, М should Ье larger than N so that the
вin(Х)/Х shape of the frequency response is noticeable. Following through
оп this Ьу using а 128 point PFТ routine, our N-point averager's fre-
quency magnitude responses, for various values of N, are plotted in
Figure 8--12. То make these curves more meaningfиl, the frequency axis is
defined in terms of the sample ratefs in samples/s,
8.5 Exponential Averaging
Тhere is а kind of time-domain averaging used in soте power measure-
ment equipment it's cal1ed exponential averaging[8---l1], This technique
provides noise reduction Ьу multiplying аn input sample Ьу а constant
and adding that product to the constant's ones complement multiplied Ьу
the most recent averager output, Sounds complicated in words, but the
equation for the exponential averager is the simple expression
у(n) = ш(n) + (1 а)у(n 1),
(8-22)
where у(n) is the cuпеnt averager output sample, y(n 1) is the previous
averager output sample, and а is the weighting factor constant. The
342 Sigпol Averagiпg.
х(п)
Averager output
у(п)
а
Output
Storage
Register
l a
y(п 1)
Figure 8-13 Exponential averager,
process described Ьу Eq. (8-22) is implemented as shown in Figure 8 lЗ.
The advantage of exponential averaging is that only one storage register
is needed to hold the value y(п l) while waiting for the next input data
sample х(п),
ТЬе exponential averager's name stems from its time domain impulse
response. Let's assume that the input to the averager is а long string of
zeros and we apply а single sample of value 1 at time t = О. Тhen the input
returns again to а string of zero-valued samples. Now, if the weighting fac
tor is а = 0.4, the averager's output is shown as the curve in Figure 8-14,
When t = О, the input sample is multiplied Ьу а, so the output is 0.4, on
the next clock cycle, the input is zero, and the old value of 0,4 is multiplied
Ьу (1 0.4), or 0.6, to provide an output of 0.24, on the following clock
cycle, the input is zero, and the old value of 0.24 is multip1ied Ьу 0,6 to pro
vide an output of 0.144, Тhis continues with the averager's output, or
impulse response, falling off exponentially because of successive multipli
cations Ьу 0.6,
А useful feature of the exponential averager is its capability to vary the
amount of noise reduction Ьу changing the value of the а weighting fac-
tor, If а equals one, input samples are not attenuated, past averager out-
puts are ignored, and по averaging takes place. In this case, the averager
output responds immediately to changes at the input. As а is decreased
in value, input samples are attenuated, and past averager outputs begin
to affect the present output. Тhese past values represent an exponentially
weighted sum of recent inputs, and that summation tends to smooth out
noisy signals. Тhe smaller а gets, the more noise reduction is realized,
However, with smaller values for а, the slower the averager is in respond
ing to changes in the input. We сап demonstrate this behavior Ьу looking
Expoпeпtial Averagiпg З<t
4 Averager output
::"\
0,25 .
0,20 !
0.15 ! .
0,10 ..
0.05 i' .__________.
. . .
о
о
2
3
4
5
6
7
8
.,.............
9 Time
Flgure 8-14 Exponential averager impulse response with <х = 0,4 ,
at the exponential averager's time domain step response for various val-
ues of а as shown in Figure 8 15,t
50 we have а trade off, The more the noise reduction, the more slug-
gish the averager will Ье in responding to changes at the input. We сап
see in Figure 8 15 that, as а gets smaller, affording better noise reduc-
tion, the averager's output takes longer to respond and stabilize, 50те
test instrumentation manufacturers use а clever scheme to resolve this
noise reduction versus response time trade off. Тhey use а large value
for а at the beginning of а measurement so that the averager's output
responds immediately with а nonzero value, Then, as the measurement
proceeds, the value of а is decreased to reduce the noise fluctuations at
the input.
Тhe exponential averager's noise variance reduction as а fиnction of
the weighting factor а has been shown to Ье[9,lО),
output noise variance = SNR а
input noise variance ехр 2 а
(8 23)
t The step response is the averager's output when а string of аН zeros foHowed Ьу а string
of аН ones is applied to the input.
344 Sigпa/ Averog/пg
0,7
,.......................
........---
..............
.....
.. ."""
......
........
........
0,8
0,6
а=О,1
.'
---.....;7"..-.
./
O / Oo OOoo _ _. _ .. _oo _oo
.
-- -----.-
.
.L.o_. _...0....._0..0
/
0,1 ,"' 1 1
2
0,5
0,4
0,3
0,2
о
4
1
6
1
8
1
10
12
1
14
1
16
1
18
1'-
Time
Figure 8.15 Exponentlal averager output vs, а when а step Input Is appllвd at tlme
t= О,
Тhus, the exponential averager's noise power reduction in dB is given Ьу
SNR ex p = 10 'log10 ( ) ,
dB 2 a
(8 24)
Equation (8-24) is plotted in Еgше 8 16 to illustrate the trade--off between
noise reduction and averager response times.
То demonstrate the exponential averager's noise power reduction
сараЬШtiеs, Figure 8-17 shows the averager's output with а cosine wave
plus noise as an input. Тhe weighting factor <х starts out with а value of
1.0 and decreases linearly to а final value оЕ 0.1 at the 180th data input
sample, Notice that the noise is reduced as а decreases. However, the
cosine wave's peak amplitude шо decreases due to the smaller а value.
The reader тау recognize the implementation of the exponential aver-
ager in Еgше 8 13 as а one--tap infinite impu1se response (ПR) digital fil-
ter[12]. 1ndeed it is, and, as such, we сап determine its frequency
response. We do this Ьу remembering the general expression in Chapter 6
for the frequency response of an lIR filter, or Eq. (6 28) repeated here as
Expoпeпtlal Averaglпg з45
N N
Ib(k). cos(kO») j Ib(k). sin(kO»)
Н ( О» ) = k=O k=O
ехр М М
1 Ia(k).cos(kO») + jL,a(k).sin(kO»
k=l k=l
(8-25)
Noise power
reduction in dB
16
14
12
i
Slower response .
/.
..... ../..
../
.,;::;-/.
.......
......117.......... ........
................<.
..................
2 .. ,- . .
...............
О . .....Т I I I
.8
Faster response
10
8
6
4
,9
,7
.6
.5
.4
,3
,2
I
,1
'
CJ.
Flgure 8-16 Exponential averager nolse--power reductlon аз а functlon of the
weighting factor а,
; а=1
4(
а = .45
.}
а=.1
.}
О
2
2
--4
1
45
90
135
180
225
Samples
(Time)
Flgure 8-17 Exponentlal averager output nolsв reduction аз а decreases.
'346 Slgпol Averagiпg'
From Figure 6-18, we modify Eq, (8 25) to set N = О, and М = 1, so that
Н (О) = Ь(О). cos(OO) jb(O). sin(OO)
ехр 1 а(l)' cos(lO) + ja(l) . sin(lO)
(8-26)
Now with Ь(О) = а and а(l) = 1 а, our exponential averager' s frequency
resp<>nse is the complex expression
а
Нехр(О)= l (l a).cos(O)+ j(l a).sin(O)
(8 26')
For now, we're interested only in the magnitиde response о! our filter, so
we сап express it as
1Hex (0)1 = / I
р 1 (1 а)' COS(O) + j(l а). sin(O)
а
= [1 (1 а). cOS(0)]2 + [(1 а). sin(0)]2
а
= 1 2.(1 a)'cos(0)+(1 a)2
(8 27)
Evaluating Eq, (8-27) over the normalized angular range о! О::; 0)::; 1t (cor-
responding to а frequency range о! О ::; f::; 1./2 Нz), the frequency mаgni
tиde responses о! our exponential averaging filter for various values о! а
are shown in Figure 8 18(a) using а normalized decibel scale. Notice that,
as а decreases, the exponential averager behaves more and more like а
low pass filter.
We сап get some practice manipulating complex nиmbers Ьу deriving
the e:xpression for the phase о! an exponential averager, We know that the
phase angle, as а function о! frequency, is the arctangent о! the ratio о! the
imaginary part о! Нехр(О) over the real part о! Нехр(О), or
l ( imaginary part о! Нехр(О) ]
" х (О) = tan ,
е р real part о! Нехр(О)
(8 27')
То find those real and imaginary parts, knowing that Нехр(О) is itself а
ratio, we determine what the variables are in Eq. (8--26') сопеsропding to
the form о! Eq, (A 20), from Appendix А, Doing that, we get
Expoпeпtlol Averog/пg 347
Normalized magnitude response in dB
о
0,9
--5
0,7
0,5
10
(а) 15
20
25
--30
О 's/8
Phase angle 'п degrees
О
10
20
(Ь)
--30
--40
--50
--50
70
0,3
..,..........
а=О,1
.
's/4
f /2 Freq
s
's/4
's/2
Freq
Flgure 8-18 Exponentiol overager frequency response vs, CI. : (а) normolized
mognitude response in dB; (Ь) phose response in degrees,
R 1 = а, [1 = О, R 2 = 1 (1 а). cos(i), and [2 = (1 а). sin(i) ,
Substituting those variables in Eq. (А-20), yields
Н (Ф) = а[l (1 а). cos(i)] Ла(1 а). sin(i)]
ехр [1 (1 а). cos(i)]2 + [(1 а). sin(i)]2
(8 28)
Representing the denominator of this messy Eq. (8 28) with the term
Deп, we use Eq, (8--27) to express the phase angle of Нехр(Ф) as
348 S/gпQI AveragJпg
( ) I ( a(l a)'sin(ro)/Deп )
0ех ro tan
р а(1 (1 а). cos(ro)]/ Deп
I ( (1 a),sin(ro) )
=tan ,
1 (1 а). cos(ro)
(8-29)
Тhe very nonlinear phase of 0 ехр (ro) from Eq, (8-29) is calculated over the
norrnalized angular range of О ::; ro ::; п, corresponding to а frequency
range of О ::; 1::; 1/2 Нz, and plotted in Figure 8 18(Ь).
References
[1] Mil1er, 1" and Freund, ], Probabi1ity and Statistics /or Engineers, 2nd Ed.,
Prentice-Hall, Englewood Cliffs, New]ersey, 1977, рр, 118,
[2] Bel1er, J" and РlеSБ, W, "А Modular All-Haul Optical Time-Domain
Reflectometer for Characterizing Fiber Links," Hewlett-Packard Jourпal,
February 1993,
[3] Spiegel, М, R, Theory and ProbIems o/Statistics, Shaum's Outline Series,
McGraw-Нi1l Book Со" New York, 1961, рр, 142,
[4] Papou1is, А. ProbabiIity, Random VariabIes, aпd Stochastic Processes, McGraw-
нill Book Со" New York, 1984, рр, 245,
[5] Davenport, W, В" ]r., and Root, W, L. Random Signals aпd Noise, McGraw-Нill
Бооk Со" New York 1958, рр, 81....s4,
[6] Welch, Р. О, "Тhe Use оЕ Fast Fourier Transform for the Estimation оЕ Power
Spectra: А Method Based оп Time Averaging over Short, Modified
Periodograms," IЕЕЕ Transactions оп Audio and Electroacoust" Vol, AU-15, No,
2, June 1967
[7] Напis, F. J, "On the Use оЕ Windows for Нarmonic Analysis with the Discrete
Fourier Transform," Proceedings о/ the lБЕЕ, Vol, 66, No, 1, January 1978,
[8] Eooster, О, Н, et аl, "Design оЕ а Precision Optical Low Coherence
Ref1ectometer," Hewlett-packard Jourпal, February 1993,
[9] Witte, R. А. "Averaging Techniques Reduce Test Noise, Improve Accuracy,"
Microwaves & RF, February 1988,
[10] Охааl, J, "Temporal Averaging Techniques Reduce Image Noise," EDN,
March 17,1983,
[11] Lymer, А. "Digital-Modulation Scheme Processes RF Broadcast Signals,"
Microwaves & RF,April1994,
[12] Hayden, О, "limer Controls DSP-Filter Frequency Resolution," EDN, April13,
1995,
ICHAPTER NINEI
Digital Data Formats
and Their Effects
In digital signal processing, there are many ways to represent numerical
data in computing hardware. These representations, known as data for-
mats, have а profound effect оп the accuracy and ease of implementation
of any given signal processing algorithm. The simpler data formats enable
uncomplicated hardware designs to Ье used at the ехреnsе of а restricted
range of number representation and susceptibility to arithmetic епоrs,
The more elaborate data formats are somewhat difficult to implement in
hardware, but they allow us to manipulate very large and very small
numbers while providing immunity to тапу problems associated with
digital arithmetic, The data format chosen for any given application can
mean the difference between processing success and failur it's where
our algorithmic rubber meets the road.
ln this chapter, we'll introduce the most comm.on types of fixed-poiпt
digital data formats and show why and when they're used. Next, we'll
use analog to-digital (A/D) converter operation to establish the precision
and dynamic range afforded Ьу these fixed point formats along with the
inherent errors encountered with their use. Рinаllу, we'll cover the inter-
esting subject ofjloatiпg poiпt binary formats,
9.1 Fixed-Point Binary Formats
Within digital hardware, numbers are represented Ьу binary digits
known as bits in fact, the term bit originated from the words Biпary
349
350 Digital Data Formats aпd Тheir Effects
digIТ, А single bit сап Ье in only оnе о! two possible states: either а оnе
or а zero, t А six bit binary number could, for example, take the form
101101, with the leftmost bit known as the тost sigпificaпt bit (msb), while
the rightmost bit is called the least sigпificaпt bit (lsb), Тhe number о! bits
in а binary number is known as the word length hence 101101 has а
word length о! six. Like the decimal number system so familiar to us, the
binary number system assumes а weight associated with each digit in the
number, Тhat weight is the base оЕ the system (two for binary numbers
and ten for decimal numbers) raised to аn integral power. То illustrate
this with а simple example, the decimal number 4631 is
(4 . 103) + (6 . 1()2) + (3 . 101) + (1 . l()О)
= 4000 + 600 + 30 + 1 = 4631 ,
(9-1)
The factors, 103, 102, 101, and 100 are the digit weights in Eq, (9-1),
Similarly, the six bit binary number 101101 is equal to decimal 45 as
shown Ьу
(1 . 25) + (О . 24) + (1 . 23) + (1 . 22) + (О . 21) + (1 . 20)
= 32 + 8 + 4 + 1 = 45 ,
(9 2)
Using subscripts to signify the base о! а number, we сап write Eq. (9 2) as
1011012 =: 4510' Equation (9-2) shows us that, like decimal numbers, binary
numbers use the place value system where the position о! а digit signifies
its weight. If we use В to denote а number system's base, the place value
representation о! the four digit number аз а1ао is
(а з . В3) + (а 2 . В2) + (а 1 . В1) + (а о . ВО) ,
(9 3)
In Eq_ (9 3), Вп is the weight multiplier for the digit а п , where о:::; а п :5: B l,
(Тhis place value system о! representing numbers is very old-----so old, in
fact, that it' s origin is obscure, However, with its inherent positioning о! the
deciInal or Ьinту point, this number system is so convenient aпd powerfи1
that its importaпce has Ьееn compared to that о! the alphabet[l].)
t Binary numbers are used because early eIectronic computer pioneers quickIy reaIized that it
was much more practical and reliable to use electrical devices (relays, vacuum tubes, transis-
tors, etc,) that had on1y two states, оп or off. Thus, the on/off state of а device couJd represent
а single binary digit,
F/xed-Po/пt B/пary Forтats 351
9.1 .1. Octal Numbers
As the use of minicomputers and microprocessors rapidly expanded in the
1960s, реорlе grew tired of manipulating 10ng strings of ones and zeros оп
paper and began to use more convenient ways to represent binary nurn-
bers, Onе way to express а binary nurnber is аn octal format with its base
of eight, Converting from binary to octal is as simple as separating the
binary nurnber into three bit groups starting from the right, For ехатрlе,
the binary nurnber 101010012 сan Ье converted to octal format as
101010012..... 10 11011 001 = 2518 .
Each of the three groups of bits above are easi1y converted from their
binary formats to а single octal digit because, for three bit words, the octal
and decimal formats are the same. Тhat is, starting with the left group of
bits, 102 = 210 = 28' 1012 = 510 = 58' and 0012 = 110 = 18' Тhe octal format also
uses the рlасе value system meaning that 2518 = (2 . 82 + 5 . 81 + 1 . 80),
Octal format enables us to represent the eight digit 101010012 with the
three digit 2518' of course, the only valid digits in the octal format are О
to 7 the digits 8 and 9 have по meaning in octal representation.
9.1.2. Hexadecimal Numbers
Another popular binary format is the hexadecimal number format
using 16 as its base, Converting from binary to hexadecimal is done,
this time, Ьу separating the binary number into four bit groups start
ing from the right, Тhe binary number 101010012 is converted to hexa
decimal format as
101010012..... 1010 11001 = А9 16 .
If уои haven't seen the hexadecimal format used before, don't let the А9
digits confuse you,ln this format, the characters А, В, С, D, Е, and F rep-
resent the digits whose decimal values are 10, 11, 12, 13, 14, and 15
respectively, We convert the two groups of bits above to two hexadeci
mal digits Ьу starting with the left group of bits, 10102 = 1010 = Ан>, and
10012 = 910 = 916' Hexadecimal format numbers also use the рlасе value
system, meaning that А9 16 = (А . 161 + 9 . 160). For convenience, then, we
сап represent the eight digit 101010012 with the two digit number А916'
ТаЫе 9 llists the permissible digit representations in the number sys
tems discussed thus far,
352 Dig/to/ Doto Forтots oпd Their Effects
ТаЫе 9-1 AllowabIe Dlgit Representotions VS, Number System Вase
Decimal
Binary Octal Decimal Hexadecimal Equivalent
о о о о о
1 1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
5 5 5 5
6 6 6 6
7 7 7 7
8 8 8
9 9 9
А 10
В 11
С 12
D 13
Е 14
F 15
9.1.3. Fractional Binary Numbers
Fractions (numbers whose magnitudes are greater than zero and less than
one) can al80 Ье represented Ьу binary numbers if we use а binary point
identical in function to our familiar decimal point. In the binary numbers
we've discussed so far, the binary point is assumed to Ье ftxed just to the
right of the rightmost digit, Using the symbol О to denote the binary point,
the six bit binary fraction 11 0101 is equal to decima13,3125 as shown Ьу
(1.21)+ (1. 20) + (О, т1) +(1. т2) +(0' ТЗ) +(1. т4)
1 1 1 1
= (1. 2) +(1.1)+(0. )+(1. )+(O. )+(1. )
2 4 8 16
= 2 + 1 +0 +0.25+ 0+0.0625 = 3.3125 ,
(9 4)
F/xed-Poiпt B/пary Forтats 353
For the example in Eq. (9 4), the binary point is set between the second
and third most significant bits, Having а stationary position for the binary
point is why binary numbers are often cal1edfixed-poiпt binary,
For some binary number formats (like the floating-point formats that
we'll cover short1y), the binary point is fixed jиst to the left о! the most sig
nificant bit. тhis forces the number valиes to Ье restricted to the range
between zero and оnе. In this format, the largest and smallest valиes pos-
sibIe for а b-bit fractional word are 1 2--b and 2--Ь, respectively, Taking а six-
bit binary fraction, for example, the largest valиe is 01111112' or
0(1. т1) + (1. т2) +(1. ТЗ) +(1. 2--4) + (1. т5) + (1. 2--6)
1 1 1 1 1 1
=0 (1' )+(1' ) +(1. )+(1. ) +(1. ) +(1. )
2 4 8 16 32 64
= 0.5 + 0.25 + 0,125 + 0,0625 + 0.03125 + 0.015625 = 0,984375 , (9-5)
which is 1 2--6 = 1 1/64 in decimal. The smallest nonzero valиe is 00000012'
eqиaling а decima1 1 / 64 = 0,01562510'
9.1.4. Sign-Magnltude Blnary Format
For binary numbers to Ье at all usefи1 in practice, they mиst Ье аЫе to rep-
resent negative valиes. Binary numbers do this Ьу dedicating оnе о! the
bits in а binary word to indicate the sign о! а number. Let's consider а
popиlar binary format known as sign-magnitиde. Here, we assume that а
binary word's leftmost bit is а sign bit and the remaining bits represent
the magnitиde о! а number which is always positive, For example, we сап
say that the four-bit number 00112 is +310 and the binary number 10112 is
eqиal to 310' or
magnitude Ыtэ
Н.!.
00112 = 310'
i
and
magnitude Ыtэ
Н.!.
10112 = 31O '
i
sign Ы! of zero
signiiies positive
sign Ы! of one
signiiies negative
354 Digltol Data Forтats aпd Thelr Effects
Of course, using one of the bits as а sign bit reduces the magnitude of the
numbers we can represent, If an unsigned binary number's word length is
Ь bits, the number of different values that can Ье represented is 2 Ь , An
eight bit word, for example, can represent 28 = 256 different integral val
ues. With zero being one of the values we have to express, а b bit unsigned
bincuy word can represent integers пот О to 2b l, The largest value repre-
sented Ьу an unsigned eight bit word is 28 1 = 25510 = 111111112' In the
sign magnitude binary format а b bit word can represent only а таgni
tude of :t2b-- C 1, so the largest positive or negative value we can represent
Ьу al1 eight-bit sign-magnitude word, then, is :t28--1 1 = :t127,
9.1.5 Two's Complement Format
Another common binary number scheme, known as the two's complemeпt
format, also uses the leftmost bit as а sign bit, The two's complement for
mat is the most convenient numbering scheme пот а hardware design
standpoint and has been used for decades, It enables computers to per-
form. both addition and subtraction using the same hardware adder logic.
То get the negative version of а positive two's complement number, we
merely complement (change а one to а zero and change а zero to а one)
each bit and add а one to the complemented word, For example, with
00112 representing а decimal 3 in two's complement format, we obtain а
nega tive decimal 3 through the fol1owing steps:
+3 in two's complement ..... 0011
complement оЕ +3 ..... 1100
add оnе ..... + QJ)JU
in two's complement ..... 1101.
In the two's complement format, а b bit word сап represent positive
amplitudes as great as 2b--1 1, and negative amplitudes as large as 2b--l,
ТаЫе 9 2 shows four bit word examples of sign magnitude and two's
complement binary formats.
While using two's complement numbers, we have to Ье careful when
adding two numbers of different word lengths. Consider the case where
а four bit number is added to а eight-bit number:
+15 in two's complement .....
add +3 in two's complement .....
+18 in two's complement .....
00001111
+ QJLil
00010010,
Fixed Poiпt Biпary Forтats 355
ТаЫе 9-2 Exaтples of Blпary Nuтber Forтats
Sigп- Тwo's Offset Deciтal
Мagпitude Complemeпt Binary Equivaleпt
0111 0111 1111 7
0110 0110 1110 6
0101 0101 1101 5
0100 0100 1100 4
0011 0011 1011 3
0010 0010 1010 2
0001 0001 1001 1
0000 0000 1000 О
1001 1111 0111 1
1010 1110 0110 2
1011 1101 0101
1100 1100 0100 4
1101 1011 0011 ...s
1110 1010 0010 --6
1111 1001 0001 7
1000 0000 --8
No problem so [М, The trouble occurs when our four-bit number is nega
tive. Instead о! adding а +3 to the +15, let's try to add а 3 to the +15:
+15 in мо'. complement ....
add а 3 in мо'. complement ....
+28 in мо'. complement ....
00001111
+ llJU
00011100
, .... Wrong an.wer
The аЬоуе arithmetic error can Ье avoided Ьу performing what's cal1ed а
sign-exteпd operation оп the four bit number. This process, typically per
formed automatically in hardware, extends the sign bit о! the four-bit neg
ative number to the left, making it an eight bit negative number. If we
sign-extend the 3 and, then perform the addition, we'l1 get the correct
answer:
+15 in мо'. complement.... О О О О 1 1 1 1
.dd а .ign-extended --З in мо'. complement .... + 1 1 1 1 1 1 О 1
+12 in мо'. complement....1 О О О О 11 О О
i
overflow Ы! is ignored
.... ТЬаУ. better
356 D/gita/ Data Forтats aпd Their Effects
9.1.6. Otfset Binary Format
Another useful binary number scheme is known as the offset biпary format,
While this format is not as common as two's complement, it still shows ир
in some hardware devices. Table 9-2 shows offset binary fопnаt examples
for four-bit words, Offset binary represents negative numbers Ьу subtract-
ing 2Ь--1 from ап unsigned binary value, For example, in the second row of
Table 9 2, the offset binary number is 11102' When this number is treated as
ап unsigned binary number it's equivalent to 1410' For four-bit words Ь = 4
and 2Ь--1 = 8, so 1410 810 = 610 which is the decimal equivalent of 11102 in
offset binary. Тhe difference between the unsigned binary equivalent and
the actual decimal equivalent of the offset binary numbers in Table 9 2 is
always .....g, Тhis kind of offset is sometimes referred to as а bIOS when the
offset binary fопnаt is used, (Н тау interest the reader that we сan convert
back and forth between the two's complement and offset binary fопnаts
merely Ьу complementing а word's most significant bit,)
УЬе history, arithmetic, and utility of the тапу available number for-
mats is а very broad НеЫ of study. А thorough and very readable discus
sion of the subject is given Ьу Кnuth in reference [2],
9.2 Binary Number Precision and Dynamic Range
As we implied before, for апу binary number format, the number of bits
in а data word is а key consideration, The more bits used in the word, the
better the resolution of the number, and the larger the maximum value
that сап Ье represented,+ Assuming that а binary word represents the
amplitude of а signal, digital signal processing practitioners find it useful
to qиantify the dynamic range of various binary number schemes, For а
signed integer binary word length of Ь+1 bits (опе sign bit and Ь таgni-
tude bits), the dynamic range in decibels is defmed Ьу
d ' 20 1 ( largest possible word value )
ynатlC rangedB = . oglO
smallest possible word value
= 20 .10glO( 2 Ь 1 1 ) = 20 .loglO(2 b 1) , (9 6)
t Some compиters иse 64-bit words, Now 264 is approximate!y eqиa! to 1,8 . 1019 that's а
pretty !arge nиmber, 50 !arge, in fact, that if we started incrementing а 64-bit coиnter оnсе
per se<:ond at the beginning of the иniverse (=20 Ыlliоn years ago), the most significant foиr
bits of this coиnter woи!d still Ье аН zeros today,
Effects of F/п/te F/xed Po/пt B/пary Word Leпgth 357
When 2 Ь is much larger than 1, we can ignore the 1 in Eq, (9-6) and state
that
dynamic rangedB = 20.1og 10 (2 b )
= 20 .log10(2). Ь = 6.02. Ь dB ,
(9-6 ')
Equation (9-6'), dimensioned in dB, te11s us that the dynamic range of our
number system is direct1y proportional to the word length, Thus, an
eight-bit two's complement word, with seven bits available to represent
signal magnitude, has а dynamic range of 6,02 . 7 = 42,14 dB. Most people
simplify Eq. (9-6') Ьу using the rule of thumb that the dynamic range is
equal to "six dB per bit,"
9.3 Effects 01 Finite Fixed-Point Binary Word Length
The effects of fiлitе binary word lengths touches a1l aspects of d.igital signa1
processing, Using fiлitе word lengths prevents иs from representing values
with infinite precision, increases the background noise in our spectral esti-
mation techniques, creates nonideal d.igital filter responses, induces noise in
analog-to--d.igital (A/D) converter outputs, and can (ifwe're not careful) lead
to wildly inaccurate arithmetic results. The sma11er the word lengths, the
greater these problems will Ье. Fortunately, these fiлitе, word.-length effects
are rather well understood, We can predict their consequences and take
steps to minimize any unpleasant surprises, The first fiлitе, word-length
effect we'll cover is the errors that occur during the A/D conversion process,
9.3.1 A/D Converter Quantization Errors
Practical A/D converters are constrained to have binary output words of
finite length. Commercial A/D converters are categorized Ьу their output
word lengths which are norma11y in the range from 8 to 16 bits. А typical
A/D converter input analog voltage range is from 1 to + 1 volt, If we used
such an A/D converter having eight-bit output words, the least signifi-
cant bit would represent
1 Ь 1 fu11 vo1tage range 2 vo1ts 7 81 . 11 " 1
s уа иe dl gth 8 , nu lVO ts,
2 wor еn 2
(9-7)
What this means is that we can represent continuoиs (analog) voltages
perfect1y as long as they're integral multiples of 7,81 mil1ivo1ts any
358 D/glto/ Dato Forтots oпd Тhe/r Effects
intermediate input voltage will cause the A/D converter to output а best
estirnate digital data value. Тhe inaccuracies in this process are called quaп-
tizatioп errors because an A/D output least significant bit is an indivisible
quалtity. We illиstrate this situation in Figиre 9-1(а) where the continuous
waveform is being digitized Ьу an eight-bit A/D converter whose output
is in the sign-rnagnitude format. When we start sampling at time t = О, the
continuous waveform happens to have а value of 31.25 millivolts (mv),
and our A/D output data word wШ Ье exactly correct for sample х(О). At
Нте Twhen we get the second A/D output word for sample х(I), the con-
tinuous voltage is between О and 7.81 тУ. In this case, the A/D converter
outputs а sample value of 10000001 representing 7.81 ту, even though
the continuous input was not quite as negative as 7,81mv. The 10000001
Slgnal amplitude in millivolts
31,25 х(О)
7,81
23,43 -
15,62 ---
(а)
О
-7,81
-+
-15,62
" ,Х(6)
tr t j +
I , : r
-23.43
. .., . ". .J . _ "_
-31.25
О
т
5Т
3Т
4Т
2Т
(Ь)
A/D quantization error in millivolts
7,811
r "
О " 1 _ - _ _" -- ----- - '--- "-- ----- - -,-,,""""- --
-7,81
О Т ппп я п
ND ou1put valu..
00000100
00000011
00000010
00000001
00000000
10000001
10000010
10000011
10000100
6Т Time
00000001
} Quantization
errOr range
10000001
Time
Figure 9.1 QuonHzat/on errors: (а) digitized х(п) volues of а contlnuous sigпol;
(Ь) quontizatlon error between the octuol onolog signol volues
ond the digltized signol volues,
i
Effects о( F/п/te F/xed-Po/пt B/пary Word Leпgth 359
A/D output word contains some quantization error. Each successive
sample contains quantization error because the A/D's digitized output
va1ues must lie оп а horizonta1 dashed line in Figure 9 1(a). Тhe differ
ence between the actua1 continuous input voltage and the A/D con-
verter's representation of the input is shown as the quantization error in
Figure 9-1(Ь). For an ideal A/D converter, the quantization error, а kind
of rouпdoff noise, сап never Ье greater than ::1:1/2 an lsb, or ::1:3.905 ту,
While Figure 9 1(b) shows A/D quantization noise in the time domain,
we сап a1so illustrate this noise in the frequency domain. Figure 9 2(a)
depicts а continuous sinewave of one cycle over the sample interval shown
as the dotted 1ine and а quantized version of the tim domain samples of
that wave as the dots, Notice how the quantized version of the wave is
constrained to have only integra1 values, giving it а stair step effect oscil
1ating аЬоуе and below the true unquantized sinewave. The quantization
here is 4 bits, meaning that we have а sign bit and three bits to represent
the magnitude of the wave, With three bits, the maximum peak values for
the wave are ::1:7, Figure 9 2(b) shows the discrete Fourier transfопn (DFТ)
of а discrete version of the sinewave whose tim domain sample values
are not forced to Ье integers, but have high precision, Notice in this case
that the DFТ has а nonzero va1ue only at m = 1, on the other hand, Figure
9 2(c) shows the spectrum of the 4-bit quantized samp1es in Figure 9-2(а)
where quantization effects have induced noise components across the
entire spectra1 band, If the quantization noise depictions in Figures 9 1(b)
and 9 2(c) 100k random, that's because they are, As it tums out, even
though A/D qиantization noise is random, we can still quantify its effects
in а useful way,
In the field of communications, peop1e often use the notion of output sig-
naHo noise ratio, or SNR = (signa1 power)/(noise power), to judge the use-
fulness of а process or device. We сап do likewise and оЫаin an important
expression for the output SNR of an ideal A/D converter, SNR л / D , account-
ing for finite word-1ength qиantization effects. Because quantization noise
is random, we can't explicitly represent its power 1evel, but we can use its
statistical equiva1ent of variance to define SNR AlD measured in decibels as
SNR = 1 О .10 ( input signa1 variance )
А D gl ""
/ О A/D quantization n01se varlance
( 2 )
(1 signaJ
= 10 .10g10 2 '
(1 А/О noise
(9 8)
I
360 D/g/ta/ Data Forтats aпd Тhe/r Effects
Slnewave amplltude
8
(а)
..........,.,.", continuous aI....wav.
8 ....' j 1.. '... "'-' /
/, 1 ' 1 1111'1' /"
/ . I : I 1 1 1 1 1 1; 11 1 1 1 1. 1 .'
4 .. 1 " 1 \ \ 1 t 1 ..
. ! 1 i 11 11 1 ' 1,: '1 II .
2 /.11:ilj' 11 11 !llllli" j .
". i 1 11 : 1, 1 11 1 11 ' 1 i , 1 1 ' 1 ,1 ' I ! I (:' зз 36 з9 42 45 48 51 54 57 80 6з
0,++1 111111 11111111111111111 .
О 3 8 9 12 15 18 21 24 27 за ..1 I " ,1 , 11 [1 I (! .. n
-2 ! ШШ J!, Ч t . !tITt,r' 1 11 ii/
-4 . .ji \11 11 1 [ i 11.1'
.6 1... 1+ Ш" .
'-11 "
.......
.8 .
DFТ 01 unquantlzed slnewave (dB)
О
-20
(Ь)
.40
-60
.60 ...L...,_. , . ,_. , . , . , ._, ...., . , ..., . , . , . , . ,... ,
О 2 4 6 8 10 12 14 18 18 20 22 24 26 28 за m
DFТ 01 quan zed sinewave (dB)
О ,
I
-20 +
(с)
:: .... .... ! . i f . r " ; i j l ij! 1
-80 ...L...L.............. . ...L, . .....I...................J......_. . .........,..L.......L..
О 2 4 8 8 10 12 1 1'6 1 2 2 2'6 28 m
Figure 9.2 Quantlzatlon noise effects: (а) input sinewave applled to а 64-polnt DFТ;
(Ь) theoretlcal DFТ magnltude of high-preclslon slnewave samples; (с)
DFТ magnitude of а slnewave quantlzed to 4 bits,
Next, we'll determine an A/D converter's quantization noise variance rel-
ative to the converter's тaxiшшn input peak voltage V p ' If the full scale
( Vp to +V p volts) continuous input range of а b-bit A/D converter is 2Vp,
а single quantization level q is that voltage range divided Ьу the number
of possible A/D output binary values, or q = 2V p /2 b , (In Figure 9 1, for
example, the quantization level q is the lsb value of 7.81 ту.) А depiction
of the likelihood of encountering any given quantization error value,
called the probability density function р(е) of the quantization error, is
shown in Figure 9 З,
Effects of Fiпite Fixed-Poiпt Biпary Word Leпgth 361
А/О quantization error probability
1 density function р(е)
Т...
...
q/2 О q/2 е (Error value)
Flgure 9-3 Probability denslty functlon of A/D converslon roundoff error (noise).
This simple rectangular function has much to tell иs, It indicates that
there's an equal chance that any error value between ---q/2 and +q/2 сап
OCcur. Ву definition, because probability density functions have an area of
unity (i,e" the probability is 100 percent that the error wi1l Ье somewhere
under the curve), the amplitude of the р(е) density function must Ье the
area divided Ьу the width, or р(е) = l/q. From Figure D 4 and Eq, (D 12)
in Appendix D, the variance of our uniform р(е) is
q/2 ql2
2 . J 2 ( )d 1 J 2 d q2
о' А/О nOlБe е р е е . е е
q 12
-ql2 q/2
(9-9)
We can now express the A/D noise error variance in terms of A/D para-
meters Ьу replacing q in Eq. (9 9) with q = 2Vpl2 b to get
2 (2V p )2
о' А/О поШе = 12. (2 Ь )2
v 2
Р
3.2 2Ь
(9-1 О)
ОК, we're halfway to our goal with Eq. (9 10) giving us the denomina-
tor of Eq, (9 8), we need the numerator. То arrive at а general resu1t, let's
express the input signal in terms of its root mean square (rms), the A/D
converter's peak vo1tage, and а loading factor LF defined as
LF = rms of the input signal O'signal . t
V p V p
(9-11)
t Recall from Section О,2 that, although the variance 0'2 is associated with the power оЕ а
signal, the standard deviation а is associated with the rms value of а signal.
362 Digitol Doto Formots oпd Тhelr Effects
With the loading factor defined as the input rms voltage over the A/D
converter's peak input vo1tage, we square and rearrange Eq, (9 1l) to
show the signal variance cr gna1 as
2 2 2
crsignal = (LF) V p
(9-12)
Substituting Eqs. (9-10) and (9 12) in Eq, (9 8),
[ (LF)2V; J 2 2Ь
SNR A / D =10.1og 1o 2 2Ь =10.1og lO [(LF) .(3.2 )]
V p /(3.2 )
= 6.02. Ь + 4.77 + 20 .lOg10(LF) ,
(9-13)
Eq. (9 13) gives us the SNR A / D of an ideal b bit A/D converter in terms
of the loading factor and the number of bits Ь, Figure 9 4 plots Eq. (9 13)
for various A/D word lengths as а function of the load.ing factor. Notice
that the loading factor in Figure 9 4 is never greater than 3dB, Ьесаше
the maximum continuous A/D input peak value must not Ье greater V p
vo1ts, Тhus, for а sinusoid input, its rms value must not Ье greater than
V p / fi vo1ts (3 dB below V p )'
SNR ND
90
80
70
40
30
20
10
21
..,
i 14-Ы!
.. I
j12-bit
ш '11 О-Ы!
18-Ы!
60
50
6-Ы!
18
15
12
9
4)
Loading
lactor (dB)
Figure 9-4 SNRA/D 01 ideol A/D converters 05 а function 01 looding factor In dB,
Effects of Fiпite Fixed-Poiпt Biпary Word Leпgth З6З
When the input sinewave's peak amplitude is equal to the A/D con
verter's full scale voltage V Р' the ful1 scale LF is
V p / {i 1
LFfull scale = = {i
(9-14)
Under this condition, the maximum A/D output 5NR from Eq, (9 13) is
SNR A / D _ max = 6,02.Ь+ 4,77 + 20.10g lO (l/ {i)
= 6,02. Ь + 4.77 зm = 6,02. Ь + 1,76 dB . (9 15)
Тhis discussion of 5NR relative to A/D converters теans three important
things to us:
· An ideal A/D converter will have an SNR A / D defined Ьу Eq, (9 13), so
апу continuous signal we're trying to digitize with а b bit A/D con
verter wil1 never have ап SNRA/D greater than Eq. (9 13) after A/D
conversion, For example, let's say we want to digitize а continuous
signal whose 5NR is 55 dB, Using an ideal eight-bit A/D converter
with its full scale SNR A / D of 6,02. 8 + 1,76 = 49,9 dB from Eq, (9 15),
the quantization noise wil1 contaminate the digitized values, and the
resultant digital signal's SNR сап Ье по better than 49.9 dB. We'H have
lost signal 5NR through the A/D conversion process, (А ten bit A/D,
with its ideal SNR A / D '" 62 dB, could Ье used to digitize а 55 dB 5NR
continuous signal to reduce the 5NR degradation caused Ьу quanti
zation noise.) Equations (9 13) and (9 15) apply to ideal A/D convert
ers and don't take into account such additional A/D noise sources as
aperture jitter error, missing output bit pattems, and other nonlinear-
ities, 50 actual A/D converters are likely to have 5NRs that are lower
than that indicated Ьу theoretical Eq, (9 13). То Ье safe in practice, it's
sensible to assume that SNRA/D_max is 3 to 6 dB lower than that indi-
cated Ьу Eq, (9 15),
· Equation (9 15) is often expressed in the literature, but it сan Ье а lit
tle misleading because it's imprudent to force ап A/D converter's
input to full scale, It's wise to drive an A/D converter to some level
below fuH scale because inadvertent overdriving wil11ead to signal
clipping and wil1 induce distortion in the A/D's output, 50 Eq. (9-15)
is overly optimistic, and, in practice, A/D converter 5NRs wil1 Ье less
364 DlgltQI Data Forтats aпd Thelr Effects
than that indicated Ьу Eq. (9 15), The best approximation for an
A/D's SNR is to determine the input signal's rms value that will
never (or rarely) overdrive the converter input, and plug that value in
Eq, (9-11) to get the loading factor value for use in Eq, (9-13).t Again,
using an A/D converter with а wider word length will alleviate this
problem Ьу increasing the available SNRA/D'
. Remember now, real-world continuous signals always have their
own inherent continuous SNR, so using an A/D converter, whose
SNR A / D is а great deal larger than the continuous signal's SNR
serves по purpose. In this case, we'd Ье using the A/D converter's
extra bits to digitize the continuous signal's noise to а greater degree
о! accuracy.
А word о! caution is appropriate here conceming our analysis о! A/D
converter quantization епоrs, The derivations о! Eqs. (9 13) and (9 15) are
based upon three assumptions:
· The cause о! A/D quantization errors is а stationary random
process; that is, the performance о! the A/D converter does not
change over time, Given the same continuous input vo1tage, we
always expect an A/D converter to provide exact1y the same output
binary code,
· Тhe probability density function о! the A/D quantization error is
uniform. We're assuming that the A/D converter is ideal in its oper-
ation and аН possible errors between q/2 and +q/2 are equaHy
likely. An A/D converter having stuck bits or missing output codes
would violate this assumption. High quality A/D converters being
driven Ьу continuous signals that cross many quantization levels
wi1l resu1t in our desired uniform quantization noise probabi1ity
density function,
· The A/D quantization errors are uncorrelated with the continuous
input signal. If we were to digitize а single continuous sinewave
whose frequency was harmonically related to the A/D sample rate,
we' d end ир sampling the same input vo1tage repeatedly and the
quantization error sequence would not Ье random, The quantization
t Ву the way, some folks use the term crest factor to describe how hard аn А/О converter's
input is being driven, The crest factor is the reciprocal of the loading factor, or CF = V PI (nns
of the input signal),
Effects of Flпlte F/xвd-Polпt Blпary Word Leпgth 365
error would Ье predictable and repetitive, and our quantization noise
variance derivation would Ье invalid, In practice, complicated con-
tinuous signals such as music or speech, with their rich spectral con-
tent, avoid this problem.
То conclude our discussion of A/D converters, let's consider one last
topic. In the literature the reader is likely to encounter the expression
Ь SNR 1,76
eff 6,02
(9 16)
Equation (9 16) is used Ьу test equipment manufacturers to specify the
sensitivity of test instruments using а b eff parameter known as the num-
ber of effective bits, or effective number of bits (ENOB) [3 8], Equation
(9-16) is merely Eq. (9-15) solved for Ь. Test equipment manufacturers
measure the actual SNR of their product indicating its ability to capture
continuous input signals relative to the instrument's inherent noise char-
acteristics. Given this true SNR, they use Eq, (9 16) to determine the b eff
value for advertisement in their product 1iterature. Тhe larger b eff , the
greater the continuous voltage that can Ье accurately digitized relative to
the equipment's intrinsic quantization noise.
9.3.2 Data Overflow
The next finite, word-length effect we'll consider is called overflow,
Overflow is what happens when the resu1t of an arithmetic operation has
too many bits, or digits, to Ье represented in the hardware registers
designed to contain that result. We can demonstrate this situation to our-
selves rather easily using а simple four function, eight digit pocket calcu-
lator, Тhe sum of а decimal 9.9999999 plus 1,0 is 10.9999999, but оп an
eight digit calculator the sum is 10.999999 as
9,9999999
+ 1,0000000
10 , 9 9 9 9 9 9 9 ,
l'
this digit gets discarded
Тhe hardware registers, which contain the arithmetic result, and drive the
calcu1ator's display, сап hold only eight decimal digits; so the least sig
nificant (of course) digit is discarded. Although the аЬоуе error is less
366 D/gita/ Data Forтats aпd Тhe;r Effects
than one part in ten million, overflow effects сan Ье striking when we
work with large numbers. If we use our ca1culator to add 99,999,999 plus
1, iru;tead оЕ getting the correct result of 100 rnillion, we'll get а resu1t of
1. Now that's an authentic overflow error!
Let's illustrate overflow effects with examples more closely related to
our discussion of binary number formats. First, adding two unsigned
binary numbers is as straightforward as adding two decimal numbers.
Тhe sum of 42 plus 39 is 81, or
+81 in unSigned binary
1 1 1 ..... carry ы..
101010
.... + 100111
.... 1010001.
+42 in unsigned binary ....
+39 in unsigned binary
In this case, two 6 bit binary numbers required 7 bits to represent the
resu1ts. Тhe general rule is the sum о/ m iпdividual b-bit biпary пumbers сап
reqиire as тапу as [ь + logim)] bits to represeпt the results. 50, for example,
а 24-bit result register (accumulator) is needed to accumulate the sum of
sixteen 20 bit binary numbers, or 20 + 10g2(16) = 24. Тhe sum of 256 eight
bit words requires an accumulator whose word length is [8 + 10g2(256)],
or 16 bits, to ensure that по overflow errors occur.
In the preceding example, if our accumulator word length was 6 bits,
an overflow error occurs as
+42 in unSigned binary ....
+39 in unsigned binary ....
+17 in unsigned binary ....
111 .....canyЫ..
101010
+ 100111
1 О 1 О О О 1. ..... Overflowerror
i
an overflow out о! the sign Ы! is ignored, causing an overflow error
Here, the most significant bit оЕ the result overflowed the 6 bit accи
mulator, and an error occurred.
With regard to overflow errors, the two's complement binary format
has two interesting characteristics. First, under certain conditions, over-
flow during the summation оЕ two numbers causes по error. 5econd, with
multiple summations, intermediate overflow errors cause по problems if
the final magnitude of the sum of the b-bit two's complement numbers is
less than 2b--l. Let's illustrate these properties Ьу considering the four bit
two's complement format in Figure 9 5, whose binary values are taken
from ТаЫе 9 2.
Effects о( Fiпite Fixed-Poiпt Biпary Word Leпgth 367
1000
1000
Flgure 9-5 Four-bIt two's complement bInary numbers,
decimal
equivalent
........... +7
.... ш +6
+5
+4
1010
1001
+3
+2
........... +1
О
1
2
--з
....... .. --4
5
ш. --6
7
ш. --6
Тhe first property оЕ two's complement overf1ow, which sometimes
causes по errors, сап Ье shown Ьу the following examples:
о 1 О .... сапу Ыь
5 in two's complement..... 1 О 1 1
+2 in two's complement +.Q..Q..l.Q.
3 in two's complement ..... О 1 1 О 1 .... valid negative resu1t
i
zero overflow out of the sign Ы!
1 1 О .... сапу ЫЬ
2 in two's complement..... 1 1 1 О
+6 in two's complement + Q...Ll.Q
+4 in two's complement ..... 1 О 1 О О .... va1id positive resu1t
i
overflow out of the sign Ы! ignored, по Ьаnn done
Тhen again, the following examples show how two's complement over-
f10w sometimes does cause errors:
368 Digital Data Forтats aпd The/r Effects
о о о +-- carry ЫЬ
7 in мо'" complement.... 1 О О 1
--6 in мо'" complement .... + 1Jl..1..Q
+3 in мо'" complement .... 1 О О 1 1 +-- invalid po"itive rl!Sult
t
overflow out of the "Ign Ы! ignored, cau"ing overflow епоr
1 1 1 +-- carry Ыь
+7 in мо'" complement.... О 1 1 1
+7 in мо'" complement.... Q..LU
2 in мо'" complement .... О 1 1 1 О +-- invaJid negative re"ult
t
zero overflow out of the "ign Ы!
Тhe rule with two's complement addition is ifthe carry bit into the sigп bit
is t'Ш same as the overflow bit out о/ the sigп bit, the overflow bit сап Ье igпored
causing по errors; if the carry bit into the sigп bit is different from the overflow
bit out о/ the sigп bit, the result is invalid, An even more interesting property
оЕ two's complement numbers is that а series of b bit word sununations
сап Ье perfonned where intennediate SUInS are invalid, but the final sшn
will Ье сопесt if its magnitude is less than 2/J.-l. We show this Ьу the fol-
lowing example. If we add а +6 to а +7, and then add а 7, we'l1 encoиnter
an intermediate overflow error but ощ finаl sum wil1 Ье correct as
+7 in мо'" complement .... 0111
+6 in мо'" complement .... + Q.J...l.Q
3 in мо'" complement .... 11 О 1 +-- overflow error here
7 in мо'" complement .... + l.QJU
+6 in мо'" complement .... 10110 +-- valid po"itive resuJt
t
overflow ignored, with по harm done
Тhe magnitude of the sum of the three four bit nиmbers was less than 24--1
«8), so ощ resu1t was valid. If we add а +6 to а +7, and next add а 5,
we'll encoиnter an intennediate overflow error, and ощ final sиm will
also Ье in error because its magnitude is not less than 8.
+7 in мо'" complement .... 0111
+6 in мо'" complement .... + Q...UJl
3 in мо'" complement .... 11 О 1 +--- overf]ow error here
s in мо'" complement .... + 1.Q..U
in мо'" complement .... 11000 +-- invaJid negative resuJt
.
Effects of Fiп/te Fixed-Poiпt B/пary Word Leпgth 369
х
х'
k
у....... W N
l'
Flgure 9-6 Slngle decimatlon-In-tlme FFТ butterfly.
Another situation where overflow problems are conspicuous is dur
ing the calculation о! the fast Fourier transform (РРТ) , It's difficult at
first to imagine that mu1tiplying complex numbers Ьу sines and cosines
сап lead to excessive data word growth particularly, because sines and
cosines are never greater than unity. Well, we сап show how РРТ data
word growth occurs Ьу considering а decimation in time РРТ butterfly
from Figиre 4 14(c) repeated here as Figure 9 6, and grinding through а
1itt1e algebra. Тhe expression for the х' output о! this PFТ butterfly, from
Eq. (4 26), is
x'=x+wt.Y ,
(9 17)
Breaking ир the butterfly's х and У inputs into their real and imaginary
parts and remembering that wt=c-j27rk!N, we сan express Eq, (9 17) as
, ' ( j27rk!N ) . (" ' )
х x real + JX irnag + r:: \!freal + JYirnag .
(9 18)
If we let а Ье the twiddle factor angle о! 21tk/N, and recall that
c-ia = cos(a) jsin(a), we сan simp1ify Eq, (9 18) as
х' = XreaJ + jXimag + [cos(a) jsin(a)]'(Yreal + jYimag)
= XreaJ +cos(a)YreaJ +sin(a)Yimag + j(Ximag +cOS(a)Yimag sin(a)Yrea1)'
(9 19)
If we look, for example, at just the real part о! the х' output, x'real' it coт
prises the three terms
x'real = x real + cos( a)Yreal + sin( a)Yirnag
(9 20)
.
370 O/g/la/ Oata Forтats aпd Тhe/r Effects
If x real ' Yreal' and Yimag are of unity value when they enter the butterfly and
the twiddle factor angle а = 2rck/N happens to Ье 'ТС/4 = 450, then, x'real сan
Ье greater than 2 as
х' real = 1 + cos( 450) '1 + sin( 450) '1
= 1 + 0,707 + 0,707 = 2.414 .
(9 21)
So we see that the real part of а complex nurnber сan more than double in
magnitude in а single stage of an PFТ. Тhe imaginary part of а complex
number is equally likely to more than double in magnitude in а single PFТ
stage. Without mitigating this word growth problem, overflow errors
could render an FFТ algorithrn useless,
ок, overflow problems are handled in one of two ways: Ьу truncation
or rounding.........each inducing its own individual kind of quantization
errors as we shall see.
9.3.3 Truncation
Trullcatioп is the process where а data value is represented Ьу the largest
quantization level that is less than or equal to that data value. If we're
quantizing to integral values, for example, the real value 1,2 would Ье
quantized to 1, An exam.ple of truncation to integral values is shown in
Figure 9-7(а), where all values of х in the range of О х < 1 are set equal
to О, values of х in the range of 1 х < 2 are set equal to 1, х values in the
range of 2 х < 3 are set equal to 2, and so оп,
As we did with A/D converter quantization errors, we сап call upon the
concept of probability density functions to characterize the errors induced
Ьу truncation, Тhe probabllity density function of truncation errors, in
terms of the quantization level, is shown in Figure 9-7(Ь). In Figure 9 7(a)
the quantization level q is 1, so, in this case, we сan have truncation errors
as great as 1, Drawing upon our results from Eqs. (D-l1) and (D 12) in
Appendix D, the теan and variance of our unllorm truncation probabl1ity
density function are expressed as
q
.иtrunсаtiоn = 2
(9-22)
and
Effects of F/п/te F/xed-Po/пt B/пary Word Leпgth 371
Quantized х
/
// ..
. 3
I
,;:..!.J
.
/,
Truncation
...з 2 1
2 3 4 Х
(а)
. 2:
л...............
/ !
.
.
.
--q
Truncation error
] p,<>I>o>Ы11Iy ''''''''' т""""
о Error value
1
q
т I
(Ь)
Figure 9-7 Truncatlon: (а) quantiZatlon nonllnearlties; (ь) error probabillty
denslty functlon,
2
,..2 , q
v truncation
12
(9 23)
In а sense, truncation error is the price we рау for the privilege of
using integer binary arithmetic. Оnе aspect of this is the error introduced
when we use truncation to implement division Ьу some integral power
of 2, We often speak of а quick way of dividing а binary value Ьу 2 Т is to
shift that binary word Tbits to the right; that is, we're truncating the data
value (not the data word) Ьу lopping off the rightmost Т bits after the
shift, For example, let's say we have the value 31 represented Ьу the five-
bit binary number 111112' and we want to divide it Ьу 16 through shift
ing the bits Т = 4 places to the right and ignoring (truncating) those
shifted bits. After the right shift and truncation, we' d have а binary quo-
tient of 31/16 = 000012' Well, we see the significance of the problem
because our quick division gave us аn answer of оnе instead of the
372 Digita/ Data Forтats aпd The/r Effects
correct 31/16 = 1.9375. Our division-by truncation error here is almost 50
percent of the correct answer, Had our original dividend been а 63 rep
resented Ьу the six bit binary number 1111112' dividing it Ьу 16 through
а four bit shift would give us an answer of binary 0000112' or decimal
three. The correct answer, of course, is 63/16 = 3.9375, In this case the
percentage error is 0,9375/3.9375, or about 23.8 percent. 50, the larger the
dividend, the lower the truncation епоr,
If we study these kinds of errors, we'l1 find that the resulting trunca
tion error depends оп three things: the number of value bits shifted and
truncated, the values of the truncated bits (were those dropped bits ones
or zeros), and the magnitude of the binary number left over after trunca
tion_ Although а complete analysis of these truncation errors is beyond
the scope of this book, we сап quantify the maximum error that сап occur
with our division-by-truncation scheme when using binary integer arith
metic. The worst case scenario is when аН of the Т bits to Ье truncated are
ones. For binary integral numbers the value of Т bits, аН ones, to the right
of the binary point is 1 2Т, If the resu1ting quotient N is smal1, the sig-
nificance of those truncated ones aggravates the error. We сап normalize
the maximum division error Ьу comparing it to the correct quotient using
а percentage. 50 the maximum percentage error of the binary quotient N
after truncation for а T bit right shift, when the truncated bits are аН
ones, is
О, ' 100 ' Correct quotient Quotient after truncation
,оtrunсаtiоnепоrmах
Correct quotient
= 100. Truncation епоr 100' 1 тТ
Correct quotient N + (1 Т Т)
(9 24)
Let' s plug а few numbers into Eq, (9 24) to understand its significance, In the
first example above, 31/16, where Т = 4 and the resulting binary quotient
was N = 1, the percentage епоr is
100. (1 0.0625)/(1 + 1 0.0625) = 100 . (0.9375/1.9375) = 48.4%,
Plotting Eq, (9-24) for three different shift values as functions of the quotient,
N, after truncation results in Figure 9-8,
50, to minimize this type of division епоr, we need to keep the result
ing binary quotient N, after the division, as large as possible. Remember,
Eq, (9-24) was а worst case condition where аН of the truncated bits
Effects о( F/п/te F/xed-Po/пt Biпary Word Leпgth 373
Maximum % error
50
45
40
35
за
25
20
15
10
5
О
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 N
Figure 9-8 Maximum error when data shifting and truncatlon are used to
Implement divlslon Ьу 2Т' Т = 1 is dlvislon Ьу 2; Т = 2 Is division Ьу 4;
and Т = 8 is dlvision Ьу 256,
were ones. Оп the other hand, if аН of the truncated bits happened to
Ье zeros, the error wi11 Ье zero. In practice, the errors will Ье some
where between these two extremes, А practical example of how divi
sion by-truncation сап cause serious numerical errors is given in
reference [9],
9.3.4 Data Rounding
Rouпdiпg is an altemate process of dealing with overflow errors where а
data value is represented Ьу, or rounded ОН to, its nearest quantization
level. If we're quantizing to integral values, the number 1,2 would Ье
quantized to 1, and the number 1.6 would Ье quantized to 2, This is
shown in Figure 9-9(а) where аН values of х in the range of 0.5 :;; х < 0.5
are set equal to О, values of х in the range of 0.5 :;; х < 1,5 are set equal to
1, х values in the range of 1.5 :;; х < 2.5 are set equal to 2, etc. The proba-
bility density function of the error induced Ьу rounding, in terms of the
quantization level, is shown in Figure 9-9(Ь), In Figure 9 9(a), the quanti-
zation level q is 1, so, in this case, we can have truncation error mаgni
tudes по greater than q/2, or 1/2. Again, using ош Eqs, (D-l1) and (D 12)
resu1ts from Appendix D, the mеап and variance of our unifопn roundoff
probabi1ity density function are expressed as
Ilroundoff = о ,
(9-25)
and
374 Digffa/ Data Formats aпd Тheir Effects
.
.
Quantized х
з
.
2
1.... , Roundoff
(а)
1
3/2 5/2 7/2
х
.
.
.
1
q
Roundoff error probability
density function
(Ь)
о q/2 Error value
Figure 9-9 Rouпdoff: (а) quaпtizatioп пoпliпearlties; (Ь) error probability
deпsity fuпctioп,
2 q2
(j roundofl = 12
(9 26)
ВесаиБе the mean (average) and maximиm error possibly indиced Ьу
roundoff is less than that of trиncation, roиnding is generally preferred, in
practice, to minimize overflow errors.
In digital signal processing, statistical analysis of qиantization error
effects is exceedingly complicated, Analytical resиlts depend оп the types
of qиantization errors, the magnitude of the data being represented, the
numerical format иsed, and which of the many PFТ or digital filter strиc-
tиres we happen to иse. Ве that as it тау, digital signal processing experts
have developed simplified error models whose analysis has proved иse--
fиl. Althoиgh discиssion of these analysis techniqиes and their resиlts is
beyond the scope of this introdиctory text, many references are available
for the energetic reader[1O 18]. (Reference [11] has an extensive reference
list of its own оп the topic of qиantization error analysis.)
F/oatiпg-Po/пt Biпary Formats 37
Again, the overflow problems using fixed point binary formats that
we try to alleviate with truncation or roundoff arise because so many
digital signal processing algorithms comprise large numbers of additions
or multiplications, This obstacle, particularly in hardware implementa
tions of digital filters and the FFT, is avoided Ьу hardware designers
through the use of floating point binary formats.
9.4 Floating-Point Binary Formats
Floating point binary formats allow us to overcome most of the limita
tions of precision and dynamic range mandated Ьу fixed-point binary for
mats, particularly in reducing the ill effects of overflow [19],
Floating point formats segment а data word into two parts: а mantissa т
and an exponent е, Using these parts, the value of а binary floating point
number п is evaluated as
п = т . 2 е ,
(9 27)
that is, the number's value is the product of the mantissa and 2 raised
to the power of the exponent, (Maпtissa is а somewhat unfortunate
choice of terms because it has а meaning here very different from that
in the mathematics of logarithms maпtissa originally meant the deci
mal fraction of а logarithm,+ However, due to its abundance in the lit-
erature we'll continue using the term maпtissa here,) Of course, both the
mantissa and the exponent in Eq. (9-27) сап Ье either positive or nega
tive numbers.
Let's assume that а b-bit floating point number will use Ь е bits for the
fixed point signed exponent and Ь т bits for the fixed-point signed man-
tissa. The greater the number of Ь е bits used, the larger the dynamic
range of the number, The more bits used for Ь т , the better the resolution,
or precision, of the number, Early computer simulations conducted Ьу
the developers of b-bit floating point formats indicated that the best
trade off occurred with Ь е "" Ь! 4 and Ь т "" 3Ь! 4. We'll see that, for typical
32-bit floating point formats used today, Ь е "" 8 bits and Ь т "" 24 bits. То
take advantage of а mantissa's full dynamic range, most implementa-
tions of floating point numbers treat the mantissa as а fractional fixed-
t For example, the соттоn Iogarithm (log to the base 10) of 256 is 2.4082, Тhe 2 to the Ieft
of the decimal point is caJled the characteristic of the Iogarithm and the 4082 digits are caJled
the mantissa, Тhe 2 in 2.4082 does not теаn that we multiply ,4082 Ьу 102, Тhe 2 means that
we take the antilog of ,4082 to get 2.56 and multiply that Ьу 102 to get 256,
376 Digitol Doto Forтots aпd Тhe/r Effects
pomt binary number, shift the mantissa bits to the right or left, so that
its rnost significant bit is а one, and adjust the exponent accordingly.
This convention is called пorтalizatioп, When normalized, the mantissa
bits are typically called the fractioп of the floating-point number, instead
of the mantissa, For example, the decimal value 3,687510 сап Ье repre
sented Ьу the fractional binary number 11,10112' If we use а two-bit
ехр onent with а six-bit fraction floating-point word, we сап just as well
represent 11.10112 Ьу shifting it to the right two places and setting the
exponent to two as
11.10112
exponent fraction
.j, .j,
о 111 0111 .
t
binary point
(9-28)
The floating-point word above can Ье evaluated to retrieve our decimal
number again as
(0(1' 2-1) + (1. 2-2) + (1, ТЗ) + (0'2"'-4) +(1' т5) + (1' 2--6)], 22
1111 1 12
= [0(1' )+ (1' ) +(1, )+ (O. )+(l, )+ (1' H' 2
2 4 8 16 32 64
= [0.5 + 0.25 + 0.125 + 0,0625 + 0,03125 + 0,015625].22
= 0.921875.4 = 3,6875 .
(9-29)
After some experience using floating-point normalization, users soon
realized that always having а one in the most significant bit of the fraction
was wastefu1. That redundant one was taking ир а single bit position in
аll data words and serving по purpose. 50 practical irnplementations of
floating-point formats discard that one, assume its existence, and increase
the useful number of fraction bits Ьу one, Тhis is why the term hiddeп bit
is used to describe some floating-point formats. While increasing the frac-
tion's precision, this scheme uses less тетоту because the hidden bit is
merely accounted for in the hardware arithrnetic logic, Using а hidden bit,
the fraction in Eq, (9-28)'s floating point number is shifted to the left one
place and would now Ье
Floatlпg Poiпt Biпary Forтats 377
11.10112
exponent fraction
J. J.
[II о 11 О 1 1 01 .
t
binary point
(9-30)
RecaH that the exponent and mantissa bits were fixed-point signed
binary numbers, and we've discussed several formats for representing
signed binary numbers, i.e., sign magnitude, two's complement, and off
set binary. As it tums out, аН three signed binary formats are used in
industry-standard f1oating point formats. Тhe most соттоn f1oating-
point formats, аН using 32 bit words, are listed in ТаЫе 9 3.
ТаЫе 9-3 Floating Point Number Formats
IЕЕЕ Staпdard Р754 Format
Bit 31 30 29 28 27 26 25 24 23 22 21 20 '., 2 1 О
15 27 26 25 24 23 22 21 20 2 1 2-2 2 3 ,.. 2 21 2 22 2 231
Sign(5) .. Exponent (е) .... .. Fraction (п ....
IВМ Forтat
Bit 31 30 29 28 27 26 25 24 23 22 21 20 .. . 2 1 О
r 5 26 25 24 23 22 21 20 2 1 2 2 2--3 2 4 '" 2 22 ;<23 2 241
Sign(5) .. Exponent (е) .... .. Fraction и) ....
DEC (Digital Equipтeпt Corp.) Forтat
Bit 31 30 29 28 27 26 25 24 23 22 21 20 ... 2 1 О
r 5 27 26 25 24 23 22 21 20 2 2 2--3 2 4 .., 2 22 2 23 2 241
Sign (5) .. Exponent (е) .... .. Fraction (Л ....
MIL STD 1750А Forтat
Bit 31 30 29 . .. 11 10 9 8 7 6 5 4 3 2 1 О
r 20 2-1 2-2 .. . 2 20 2 21 2 22 2 23 27 26 25 24 23 22 21 20 I
.. Fraction (п .... .. Exponent (е) ....
378 Dlgital Data Forтats aпd Тhelr Effects
The IEEE Р754 floating point fonnat is the most popu1ar because so many
manufacturers of floating-point integrated circuits comply with this stan
dard [8, 20 22]. Its exponent е is offset Ьinary (biased exponent), and its frac
tion is а sign magnitude binary number with а hidden bit that's assumed to
Ье 20. The decimal value of а nonnalized IEEE Р754 floating point number
is evaluated as
valuе ШЕЕ = ( l)S . 101. 2e 127.
i
hidden Ы!
(9 31)
The IВM floating point fonnat differs somewhat from the other float-
ing point formats because it uses а base of 16 rather than 2, Its exponent
is offset binary, and its fraction is sign magnitude with по hidden bit,
Тhe decimal value of а norma1ized IВM floating point number is eval-
uated as
value lBM = ( l)S . 001' 16e 64.
(9 32)
The DEC floating point format uses an offset binary exponent, and its
fraction is sign magnitude with а hidden bit that's assumed to Ье 2 1, The
decimal value of а normalized DEC floating point number is evaluated as
value DEc = ( l)S . 0011' 2e 128.
i
hidden Ы!
(9 33)
MIL STD 1750А is а United States Мilitary Airborne floating point
standard, Its exponent е is а two's complement binary number residing in
the least significant eight bits, МIL-SТD 1750A's fraction is also а two's
complement number (with по hidden bit), and that's why по sign bit is
specificalIy indicated in Table 9 3, The decimal value of а MIL STD 1750А
floating point number is evaluated as
value 1750A = 1. 2 е .
(9 34)
Notice how the floating point formats in Table 9 3 аН have word
lengths оЕ 32 bits this was not accidental. Using 32 bit words makes
these formats easier to handle using 8 , 16-, and 32 bit hardware proces
sors, That fact not withstanding and given the advantages afforded Ьу
floating point number formats, these formats do require а significant
F/oatiпg-Poiпt B/пory Forтots 379
amount of logical comparisons and branching to correct1y perform arith-
metic operations. Reference [23] provides useful f10w charts showing
what procedural steps must Ье taken when f1oating-point numbers are
added and multip1ied.
9.4.1 Floating-Point Dynamic Range
Attempting to determine the dynamic range of an arbitrary f1oating-point
number format is а challenging exercise. We start Ьу repeating the expres-
sion for а number system's dynamic range from Eq. (9 6) as
d . 20 1 ( largest p ossible word value )
ynanuc rangedB = . oglO .
smallest possibIe word value
(9 35)
When we attempt to determine the largest and sma1lest possible va1ues
for а floating point number format, we quickly see that they depend оп
such factors as
· the position of the binary point
· whether а hidden bit is used or not (If used, its position relative to the
binary point is important.)
. the base value of the floating point number format
· the signed binary format used for the exponent and the fraction (For
example, recall from ТаЫе 9 2 that the binary two's complement for-
mat сап represent larger negative numbers than the sign magnitude
format.)
. how unnormalized fractions are handled, if at аН, (Unnormalized,
also called gradual underf1ow, means а nonzero number that's less
than the minimum normalized format but сап still Ье represented
when the exponent and hidden bit are both zero.)
. how exponents are handled when they're either аll ones or аll zeros,
(For example, the ШЕЕ Р754 format treats а number having an аН
ones exponent and а nonzero fraction as an inva1id number, whereas
the DEC format handles а number having а sign = 1 and а zero expo
nent as а special instruction instead of а valid number.)
Trying to develop а dynamic range expression that accounts for аll the
possible combinations of the above factors is impractical. What we can do
380 O/gita/ Oata Forтats aпd Тheir Effects
is derive а rule of thumb expression for dynamic range that's often used
in practice[8,22,24],
Let's assume the foHowing for our derivation: the exponent is а ь. bit
offset binary number, the fraction is а normalized sign magnitude num
ber having а sign bit and Ь т magnitude bits, and а hidden bit is used just
left of the bmary point. Our hypothetical floating point word takes the
following form:
Bit Ь т +Ь, l Ь т +Ь, 2 Ь т +2 Ь т Ьт l Ьт 2 О
r 5 2ь....1 2 ь ,..2 21 20 2-1 2-2 2--bm+ 1 2--bm I
Sign (5) .... Ехроnеn! (е) --+ .... Fraction (f) --+
First we'H determine what the largest value сап Ье for our floating poin.t
word. The largest fraction is а оnе in the hidden bit, and the remaining Ь т
fraction bits are аН ones, тhis would make fraction f = [1 + (1 2--Ь т )]. The
first 1 in this expression is the hidden bit to the left of the binary point,
and the value in parenthesis is аН Ь т bits equal to ones to the right of the
binary point. The greatest positive value we сап have for the Ь. bit offset
binary exponent is 2(2ь, 1 1). 50 the largest value that сап Ье represented
wi th the floating point number is the largest fraction raised to the largest
positive exponent or
largest possible word value = [1 + (1 2 bт )]. 2(2 Ь '-' 1)
(9 36)
тпе smaHest value we сап represent with our floating point word is а оnе
in the hidden bit times two raised to the exponent's most negative value,
2 2Ь. I), or
(2 ь ,-1 )
smallest possible word value = 1 . 2
(9-37)
Plиgging Eqs, (9 36) and (9 37) into Eq, (9 З5),
, [ [1 + (1 ТЬт)]. 2(2b,-1 1) J
d yn amlc rangedB = 20 'IO g 10 ь ]
1.т(2' )
(9 38)
Now here's where the thumb comes in when Ь т is large, say over seven,
the 2--Ь т value approaches zero; that is, as Ь т increases, the аН ones fraction
B/ock F/oat/пg-Poiпt B/пary Forтat 381
(1 2 bт) value in the nurnerator approaches 1. Assuming this, Eq. (9-38)
becomes
( [1 + 1]' 2(2" 1 1» )
dynamic rangedB '" 20 '10g1o , 1
1,т(2' )
( 2' 2(2,,-1 1) J ( 2(2,,-1) J
= 20 '10g1o т(2,,-I) = 20 '10g1o т(2,,-I)
= 20 '10g10(2' 2(2,,-1» = 20 '10g10(2(2"» = 6.02' 2 Ь ' , (9 39)
Using Eq. (9 39) we сап estimate, for example, the dynamic range of the
sing1e-precision IEEE Р754 standard floating-point format with its eight
bit exponent:
dynamic rangelEEEP754 = 6.02.28 = 1529 dБ.
(9 40)
Although we've introduced the major features of the most comтon
floating-point formats, there are stil1 more details to leam about floating
point nurnbers, For the interested reader the references given in this sec
tion provide а good place to start.
9.5 Block Floating-Point Binary Format
А marriage of fixed point and floating-point binary formats is known as
block floatiпg poiпt. Тhis scheme is used, particularly in dedicated FFТ inte-
grated cirruits, when large arrays, or blocks, of associated data are to Ье
manipulated mathematicaHy. Бlосk floating point schemes begin Ьу
examining all the words in а block of data, normalizing the largest valued
word's fraction, and establishing the correct exponent. This normalization
takes advantage of the fraction's fuH dynamic range. Next, the fractions of
the remaining data words are shifted appropriately, so that they сап use
the exponent of the largest word. In this way, аН of the data words use the
same exponent va1ue to conserve hardware memory,
In FFТ implementations, the arithmetic is performed treating the block
normalized data values as fixed point binary. However, when an addi-
tion causes an overflow condition, аН of the data words are shifted one
bit to the right (division Ьу two), and the exponent is incremented Ьу
382 Digital Data Formats aпd Тheir Effects
one. As the reader тау have guessed, block floating point formats have
increased dynamic range and avoid the overflow problems inherent in
fixed point formats but do not reach the performance level of true float
ing point formats[8,25,26].
References
[1] Neugebauer, О, "The History оЕ Ancient Astronomy," Jourпal ofNear Eastern
Studies, Vol, 4, 1945, рр, 12,
[2] Кnuth, D, Е, тhe Art о! Cornputer Prograrnrniпg: Serniпurnerical Methods, Vol, 2,
Section 4.1, Addison-Wesley Publishing, Reading, Massachusetts, 1981, рр, 179,
[3] Kester, W, "Peripheral Circuits Сап Make or Break Sampling-ADC Systems,"
EDN Мagaziпe, October 1,1992,
[4] Grove, М, "Measuring Frequency Response and Effective Bits Using Digital
Signal Processing Techniques," Hewlett-Packard Jourпal, February 1992,
[5] Tektronix, "Effective Вits Testing Evaluates Dynamic Range Performance оЕ
Digitizing Instruments," Tektroпix App/icatioп Note, No, 45W-7527, December
1989,
[6] Ushani, R, "Subranging ADCs Operate at High Speed with Нigh Resolution,"
EDN Magaziпe, Aprill1, 1991,
[7] Demler, М, "Time domain Techniques Enhance Testing оЕ High-speed
ADCs," EDN Magaziпe, March 30, 1992,
[8] Hilton, Н. "А 10-МНz Analog-to-Digital Converter with 110-dВ Linearity,"
Hewlett-Packard Jourпal, October 1993,
[9] Lyons, R. G, "Providing Software Flexibility for Optical Processor Noise
Analysis," Cornputer Desigп, July 1978, рр, 95,
[10] Кnuth, D, Е. тhe Art о! Cornputer Prograrnrniпg: Serniпurпerical Methods, Vol, 2,
Section 4,2, Addison-Wesley Publishing, Reading, Massachusetts, 1981, рр. 198,
[11] Rabiner, L, R., and Gold, В. Тheoтy aпd App/icatioп ofDigital Sigпal Processiпg,
Chapter 5, Prentice-Hall, Englewood Cliffs, New Jersey, 1975, рр, 353,
[12] Jackson, L, В, "An Analysis оЕ Limit Cycles Оие to Multiplicative Rounding
in Recursive Digital Filters," Ртос, 7th Allertoп Соп! Circuit Systern Theory,
1969, рр, 69 78,
[13] Кап, Е, Р, F, and Aggarwal, J, к. "Error Analysis оЕ Digital Filters Employing
Floating Point Arithmetic," IЕЕЕ Traпs, Circuit Тheoтy, Vol, СТ-18, November
1971, рр, 67в....s6,
[14] Crochiere, R, Е. "Digital Ladder Structures and Coefficient Sensitivity," IЕЕЕ
Traпs, Audio Electroacoustics, Vol, AU 20, October 1972, рр, 240 46,
Refereпces 383
[15] Jackson, L, В, "On the Interaction оЕ Roundoff Noise and Dynamic Range in
Digital Filters," Веи System Techпica/ Jourпa/, Vol, 49, February 1970, рр, 159 84,
[16] Roberts, R. А., and МuШs, С. Т. Digita/ Sigпa/ Processiпg, Addison-Wesley
Publishing, Reading, Massachusetts, 1987, рр. 277.
[17] Jackson, L, В. "Roundoff Noise Analysis for Fixed-Point Digital Filters
Realized in Cascade or Parallel Form," IЕЕЕ Traпs, Audio E/ectroacoustics, Vol,
AU-18, June 1970, рр, 107 22.
[18] Oppenheim, А. V" and Schafer, R. W, Discrete-Time Sigпa/ Processiпg, Sections
6,8 and 9,8, Prentice-Hall, Englewood Cliffs, New Jersey, 1989, рр, 335,
[19] Larimer, J., and D. Chen, "Fixed or Floating? А Pointed Question in DSPs,"
EDN Magaziпe, August 3, 1995,
[20] Ashton, С. "Floating Point Math Handles Iterative and Recursive
Algorithms," EDN Magaziпe, January 9,1986,
[21] Windsor, В" and Wilson, J, "Arithmetic Dио Excels in Computing Floating
Point Products," E/ectroпic Desigп, Мау 17,1984,
[22] Windsor, W, А. "IEEE Floating Point Chips Implement DSP Architectures,"
Computer Desigп, January 1985,
[23] Texas Instruments Inc" Digita/ Sigпa/ Processiпg Applicatioпs with the TMS320
Family: ThEory, A/gorithms, aпd Imp/emeпtatioпs, SPRA012A, Texas Instruments,
Dallas, ТХ, 1986,
[24] Straиss, W, 1, "Integer or Floating Point? Making the Choice," Computer Desigп
Magazine, April1, 1990, рр. 85,
[25] Oppenheim and Weinstein. "Effects оЕ Finite Register Length in Digital
Filtering and the Fast Fourier Transform," Proc, IЕЕЕ, August 1972, рр, 957 76,
[26] Woods, R. Е. "Transform-based Processing: "How Миф Precision Is Needed,"
ESD: ThE E/ectroпic System Desigп Magaziпe, February 1987,
ICHAPTER TENI
Digital Signal
Processing Tricks
As we study the literature of digital signal processing, we'l1 encounter
some creative techniques that prolessioпals use to m.ake their algorithms
more efficient, These techniques are straightforward examples of the phi-
losophy "don't work hard, work smart" and studying them will give us а
deeper understanding of the underlying rnathernatical subt1eties of digital
signa1 processing, In this chapter, we present а col1ection of these clever
tricks of the trade and explore several of them in detail, because doing so
reinforces some of the lessons we've learned in previous chapters.
10.1 Frequency Translation without Multiplication
Frequency translation is often cal1ed for in digital signal processing algo
rithm.s. А filtering scheme cal1ed traпsтultiplexiпg (using the FFТ to eт
cient1y implement а bank ofbandpass filters) requires spectral shifting Ьу
half the sample rate, or 1/2[1]. Inverting bandpass sampled spectra and
converting low pass FIR filters to highpass fi1ters both саН for frequency
translation Ьу half the sample rate. Conventional quadrature bandpass
sampling uses spectral translation Ьу one quarter of the sample rate, or
1./4, to reduce unnecessary соmрutаtiоns[2,З], There are а couple of tricks
used to perform discrete frequency translation, or тixiпg, Ьу 1./ 2 and 1./ 4
without actua1ly having to perform any multiplications. Let's take а look
at these mixing schemes in detail.
First, we'l1 consider а slick technique for frequency translating an input
sequence Ьу 1./2 Ьу merely multiplying that sequence Ьу ( 1)n, or ( l)O,
( 1)1, н)2, ( l)З, etc. Better yet, this requires only changing the sign of
every other input sample value because ( 1)n = 1, 1, 1, 1, etc. This
process тау seem а bit mysterious at first, but it can Ье explained in а
straightforward way if we review Figure 10-1. The figure shows us that
385
386 Dlgltol Sigпal Processiпg Tricks
.
]
о
1 1
"-.
"
/
"
, (.
\ Tlme
"'о
-"8
/
/
I
-'о,.
Flgure 10-1 Mixing sequeпce comprising Н)n; 1, 1, 1, 1, etc,
multiplying а time domain signal sequence Ьу the ( l)п mixing sequence
is equivalent to mu1tiplying the signal sequence Ьу а sampled cosinusoid
where the mixing sequence values ахе shown as the dots in Figше 1O 1.
Because the mixing sequence' s cosine repeats every two sample values, its
frequency isf/2. Let's look at this situation in detail, not only to under
stand mixing sequences, but to il1ustrate the DFТ equation's analysis
capabilities, to reiterate the nature of complex signals, and to reconfirm
the important equivalence оЕ shifting in the time domain and phase shift-
ing in the frequency domain,
We can verify the ( l)п mixing sequence's frequency translation off/2
Ьу taking the DFT of the mixing sequence expressed as Fl, l,1, l_.. (т)
where
N l
Fl, l,l, l,_Jm) = L (1, 1, 1, 1,...)e j211nm/N
п=О
(1O 1)
Using а 4 point DFТ, we expand the sum in Eq. (10-1), with N = 4, to
F (т) = e j21tOт/4 e j21iJ.т/4 + /:j2112т/4 /:j21t3 т /4 .
1, 1,l, 1
(1O 2)
Notice that the mixing sequence is embedded in the signs of the terms оЕ
Eq. (10-2) that we eva1uate from т = О to т = 3 to get
т = о: F 1, 1,1, 1 (О) = /:1'0 e--iO + /:jO e jO = 1 1 + 1 1 = О ,
(1O 3)
т = 1: F ( 1 ) = /:jO /:j211/4 + /:j411/4 /:jб11/4 = 1 + J 'l 1 J '1 = О ( 10-4 )
l, l,l, l '
т = 2: Fl ll l (2) = /:jO /:1'411/4 + /:1'811/4 e j1211/4 = 1 + 1 + 1 + 1 = 4, (10-5)
Frequeпcy Тюпs/оt/оп W/thout Mu/t1p//COtloп 387
N t MagnitudeofH)" SBq B::
о . ....'...'. ..'" .,.....j.",.. .'".'.....",.
о 2 4 6 6 10 12 14 16 18 20 22 24 26 28 30
п
t Phase of ( 1) sвquence
100
50
о, .IJI"., ,...,., ,.,., ,",., '"' .... .,.,.,., ,",., ."..,.,
о 2 4 6 6 10 12 14 16 18 20 22 24 28 26 30
Flgure 10-2 Frequency-domain magnitude oпd phase of ап N-point ( 1)n sвquence,
and
m = 3: Fl, l,1, l (3) = e jO e--j61t/4 + rj127t/4 rj187t/4 = 1 --11 1 + j1 = О . (1O 6)
See how the 1, 1, 1, 1 mixing sequence has а nonzero frequency compo
nent only when m = 2 corresponding to а frequency of mfs/ N =2//4 = //2,
50, in the frequency domain the four sample 1, 1, 1, 1 mixing sequence
is an fs/2 sinusoid with а magnitude of 4 and а phase angle of 00. Had our
mixing sequence contained eight separate values, the resu1ts of ап 8-point
DFТ would муе Ьееп аН zeros with the exception of the m = 4 frequency
sample with its magnitude of 8 and phase angle of 00. In fact, the DFТ of
N (1)n has а magnitude of N at а frequency 1./2, When N = 32, for eXaт
ple, the magnitude and phase of а 32-point ( 1)n sequence is shown in
Figиre 10-2.
Let's demonstrate this ( 1)n mixing with an example. Consider the 32
discrete samples of the sum of 3 sinusoids comprising а real time domain
sequence х(п) shown in Figure 10-3(а) where
2п10п 2п11п 1t 2п12п 3п
х(п) = cos('"32) + 0,5. cos( 4')+ 0.25.cos('"32 8) , (10 7)
Тhe frequencies of the three sinusoids are 10/32 Hz, 11/32 Hz, and 12/32
Hz, and they each have an integra1 number of cycles (10, 11, and 12) over
32 samples of х(п). То show the magnitude and phase shifting effect of
using the 1, 1, 1, 1 mixing sequence, we've added arbitrary phase shifts
of п/4 ( 450) and 3п/8 ( 7.50) to the second and third tones, Using а
32 point DFT results in the magnitude and phase of Х(m) shown in
Figure 10 3(b),
Let's notice something about Figure 10 3(b) before we proceed, Тhe
magnitude of the m = 10 frequency sample I Х(10) I is 16. Remember why
this is so? RecaH Eq. (3 17) from Chapter 3 where we leamed that, if а real
input signal contains а sinewave component of peak amplitude Ао with
388 Digita/ Sigпa/ Processiпg Tricks
(а)
2 Т x(n)
1,' Ш .', .' . . 1(1) = C/OI) + O,5cos(2x11t-1tI4) + O::\5COS(2X12t-зItl8)
i , . . ')
0511 .8 1\.\ 8\ j\
. , '.' I I
О I I 1 I I I I .. I ]1 1: I J i I 1\ I .. 1;1 1,' I } I
.05 f O. : ' , " ;; ' .; 'J \/ J ,31 Т1m.
1 : I j i
, '.
I ,; 1. \:
15 · \8
.2
Negc ::': nCY Phase of Х(т) il') degrees. 0(m)
! Magnitude 01 X(m) 100 t
16 . .
50 '.
, 2 4 в 8 10 12 !
(Ь) 8 : . ' . . , . о ......,IIJI..IIJI.. .,+t . , , , +i- , , , ,
4 ; .. f / i 14 16 18 20 22 24 26 28 за
О ....IIJI........+++........I!+++...IIJI"......, -50 о /8.
О 2 4 6 В 10 12 14 16 18 20 22 24 26 28 за -100 ....5 l'
.67.5
Figure 10-3 Discrete sigпal sequence х(п): (а) time-domain representation of х(п);
(ь) frequency-domaln magnitude and phase of Х(т),
ап integral number of cycles over N input samples, the output magnitude
of the DFT for that particular sinewave is AoN 12, In our case, then,
I Х(10) I = 1.3212 = 16, I Х(Щ I = 8, and I Х(12) I = 4,
If we multiply х(п), sample Ьу sample, Ьу the ( 1)" mixing sequence, our
new time domain sequence Х 1 l(п) is shown in Figure 10 4(a), and the
DFT of the frequency transl ted Х 1 1 is provided in Figure 1O 4(b),
(Remember now, we didn't really pe;form anу explicit multiplications
the whole idea here is to avoid multiplications we merely changed the
sign of altemating х(п) samples to get Xl. l(п),) Notice that the magnitude
and phase of Xl. l(т) are the magnitude and phase of Х(т) shifted Ьу f./2,
or 16 sample shifts, from Figиre 10-3(Ь) to Figure 10 4(b), Тhe negative fre
quency components of Х(т) are shifted downward in frequency, and the
positive frequency components of Х(т) are shifted upward in frequency
resulting in Х 1 jm), It's а good idea for the reader to Ье energetic and
prove that th magnitude of Xl l(т) is the convolution of the ( 1)"
sequence's spectral magnitude in Figure 1O 2 and the magnitude of Х(т)
in Figure 10-3(Ь), Another way to look at the Xl, l(т) magnitudes in Figиre
10 4(b) is to see that multiplication Ьу the ( 1)" mixing sequence f1ips the
positive frequency band of Х(т) from zero to +//2 Hz about 1/4 нz and
flips the negative frequency band of Х(т) from f./2 to zero Hz, about
Freqиeпcy Traпs/atioп Withoиt Mи/tip//cat/oп 389
(а)
' Т,':": " " ,О,
05 .
О 01 I 1 :
i O ; , 1
, .
-'5
'.
:8
.
:.
.: i ! i : :
1 11 1 i 1 I ! . i I ! ' 11 ! 1 , ",.
. .'
.
oi
oi .
1 Magnitud. 01 х,,_,(т) 100 t Phase 01 x,, ,(т) In degrees, "1,_, (т)
16. .
(Ь) 1 i i 7 , : .,..::..,..,..,..,.....,..,..,..,..:: .,..
о .....+++........................ .... ---50 1 2 .. 6 8 10 12 14 16 18 20 22 24 ., эо
0246810121416182022 ииэо
1OO
Flgure 10-4 Frequency tronslotion Ьу './2: (о) mlxed seqиence х, l(п) = ( 1)". х(п);
(Ь) mognltude ond phose of frequency-tronsloted х;,_,(т),
f/ 4 Нz. Тhis process сап Ье used to invert spectra when bandpass sam-
pling is used, as described in Section 2.4.
Another useful mixing sequence is 1, 1, 1,1, etc. It's used to translate
spectra Ьу f/4 in quadrature sampling schemes and is illustrated in
Figure 10 5(a). In digital quadrature mixing, we сап multiply an input
data sequence х(n) Ьу the cosine mixing sequence 1, 1, 1,1 to get the in-
phase component of x(n) what we'll саН i(n). То get the quadrature-
phase product sequence q(n), we multiply the original input data
sequence Ьу the sine mixing sequence of 1,1, 1, 1. This sine mixing
sequence is out of phase with the cosine mixing sequence Ьу 900, as
shown in Figure 1O 5(b),
If we multiply the 1, 1, 1,1 cosine mixing sequence and an input
sequence х(n), we'll find that the i(n) product has а DFI' magnitude that's
related to the input's DFI' magnitude Х(n) Ьу
I I(т) 11 , 1 , 1 . 1 I Х(т) I
..J2
(10-8)
То see why, let's explore the cosine mixing sequence 1, 1, 1,1 in the fre
quency domain, We know that the DFI' of the cosine mixing sequence,
represented Ьу Fl. lA,1 (т), is expressed Ьу
390 Digital Sigпal Processiпg Tricks
-f2 - - - - - -
1 ." .
/ \
(а) о
3 Time
/
1 .,
v2 - - - - - -
12 - - - -
1 . .
(Ь) 2 3
О о i !
\ I i Time
\ i i/
1 . .
-f2 - - - - - - .
Figure 10-5 Quadroture mixing sequences for downconversion Ьу ',/4:
(о) cosine mixing sequence using 1, 1, 1'1....: (Ь) sine mlxing
sequence uslng 1, 1, 1, 1....'
N l
Fl, 1, 1.1(m)= L(l, l, l,l, ",)e j21tпт/N
п=О
(1O 9)
Бесаusе а 4-point DFТ is sufficient to evaluate Eq, (10-9), with N = 4,
F (m ) =e j21tOт/4 e j21tlт/4 e j21t2т/4 +е j21tзт/4. (10-10)
1, 1, 1,1
Notice, again, that the cosine mixing sequence is embedded in the signs
of the terms of Eq. (10 10). Evaluating Eq, (10-10) for т = 1, correspond-
ing to а frequency of l'fJN, or 1./4, we find
т = 1: F ( 1) = e jO e j1t/2 e j1t + е jЗ1t/2
1, 1, 1,1
=1+j1+1+j1=2+j2= L450.
(10-11)
Freqиeпcy Trans/at/oп W/thoиt Mи/tip/icatioп 391
50, in the frequeney domain, the eosine mixing sequenee has an fs/ 4 rnag
nitude оЕ 4/.fi at а phase angle оЕ 450. 5imilarly, evaluating Eq. (10-10)
for т = 3, eorresponding to а frequeney оЕ / 4, we find
m=3: F, (3)=е jО е jЗ1t/2 е jЗ1t+е j91t/2
l, l, l,l
= 1 j1 + 1 j1 = 2 j2 = Jz L 450.
(10-12)
The energetie reader should evaluate Eq. (10-10) for т = О and т = 2, to
eonfirm that the 1, 1, 1,! sequenee's DFТ eoefficients are zero for the fre-
queneies оЕ О and fs/2.
Beeause the 4 point DFT magnitude оЕ ап аН positive ones mixing
sequenee (1, 1, 1, 1) is 4 t , we see that the frequeney-dornain seale faetor for
the 1, 1, 1,1 eosine mixing sequenee is expressed as
I( ) 1 f eosine sequenee DFТ magnitude
т 1 l ll sea е aetor =
, , , аll ones sequenee DFТ magnitude
4/.fi 1
.fi '
(10-13)
whieh eonfirms the relationship in Eq. (10-8) and Figиre 1O 5(a), Likewise,
the DFТ scale factor for the quadrature-phase mixing sequence (1,l, 1, 1) is
1
Q(m) l l l l seale faetor = r;;; '
" , -..;2
thus
IХ(m)1
IQ(m)1 11 l l=
'" -..;2
(10 14)
50 what this аН means is that ап input signal's speetral magnitude,
after frequeney translation, will Ье redueed Ьу а faetor of .fi. There's
reaHy по harm done, however when the in phase and quadrature-phase
eomponents are eombined to find the magnitude оЕ а eomplex frequeney
t We сап show this Ьу letting К = N = 4 in Eq, (3-44) for а fOUNample аП ones sequence in
Chapter 3,
392 D/gita/ S/gпa/ Process/пg Tr/cks
sample Х(т), the fi scale factor is eliminated, and there's по overall
magnitude loss because
I scale factor I == (I(т) scale factor? + (Q(т) scale factor?
== (l/fi?+(l/fi? = (1/2)+(1/2) ==1.
(10-15)
We сап demonstrate this quadrature mixing process using the х(п)
sequence from Eq. (10-7) whose spectrum is shown in Figure 10-3(Ь). If
we multiply that 32-sample х(п) Ьу 32 samples of the quadrature mixing
sequences 1, 1, 1,1 and 1,l, 1, 1, whose DFT magnitudes are shown in
Figure 10-6(а) and (Ь), the new quadrature sequences wil1 have the
f MagnltudeoI1' 1' 1'1 6О ! Рhasвot1, 1, 1,1
1т . 40 "-......
I : , -45
() . . 20. 2'
а ' : O..+............+......... IIJI.IIJI...I[I.....,.....'t...,."...,'
0'.'...."'..+'.'.......'1[1.."..."+.......'.. --20 1 2 " 6 8 10 12 ,4 ,8 18 20 22 f 26 28 30
О 2 " 6 8 10 12 ,4 16 18 20 22 24 28 28 за --40 ....,:J1" .
-80 .....
1 Magn/fude 01 1, 1, 1, 1 t Phasв 01 1. -1, 1, 1
1/-12. 60
I . 40 i
(Ь) . 20 8 :
о ,.,., "IJI.IIJI I!I+I!IIIJI.,., ,.... ,.,., ,.,., ,.'+IIJI.,.IIJI ....,..
о ....I!I."'.....+..,I!I,..".I!II!II!I..I!II!I...+I!I.,."...I!I I 2 4 6 \ ,0 12 14 16 1620 22 24 26 28 за
О 2 4 6 8 10 12 14 16 18 20 22 24 26 28 за --40 .-
-80
1. r Megnitude ot Цт)
10 " . . . . i .. 161 . ' 150t Phase of Цт) in degrees .
(с) 5 : : ., 1, 1(:+ 2 4 1". 20 \ \ .
: ! i ! j'!' '!' : I о.,+н,.".II[I' ' ' ".'+1[I1,.' I[I'..'-++-+'
о ....+++.......,.+++..,.+++.........-н....... --50 1 .:: 6 8 10 12 .. 18 18 22 24 26 26 за
О 2 4 6 8 10 12 14 16 18 20 22 24 26 28 зо 100 .
-,50
1. ! Magn ude 01 Q(m) 150 } Phase 01 Q(m).n degree8
10 · .. 161 I!' 100 .
(d)." "" " i 50 " 4 1" " 18 20 30
. . . . I О .+...,................н.........+r+............ .,..
о .-.+++.......... ....+++...........щ.. 50 t 2 · 6 8 10 12 14 16. 22 24 26 28 .
О 2 4 6 8 10 12 14 16 16 20 22 24 26 28 за 1OO ..
-150
Flgure 10-6 Frequency translatlon Ьу ',/4: (а) normalized magnltude and phase of
coslne 1, 1, 1, 1 sequence; (Ь) normalized magnltude and phase of
sine 1, 1, 1,-1 sequence: (с) magnitude and phase of frequency-
translated, In phase /(т); (d) magnltude and phase of frequency
translated, quadrature phase Q(m),
Frequeпcy Traпs/Qtioп Without Mu/tip/icQtioп 393
frequency translated 1(т) and Q(m) spectra shown in Figure 10-6(с) and
(d). (Remember now, we don't асшаllу perform any multiplications; we
merely change the sign of appropriate х(п) samples to get the i(n) and
q(n) sequences.)
There' s а lot to learn from Figure 1O 6. First, the positive frequency сот-
ponents of Х(т) from Figure 1O 3(b) are indeed shifted downward Ьу f/ 4
in Figure 10 6(c). Because our total discrete frequency span ifs Hz) is
divided into 32 samples, f/4 is equal to eight, 50, for example, the Х(10)
component in Figure 10 3(b) corresponds to the 1(10---8) = 1(2) component
in Figure 10 6(c). Likewise, Х(11) corresponds to 1(11---8) = 1(3), and so оп,
Notice, however, that the positive and negative components of Х(т) have
each been repeated twice in the frequency span in Figure 10 6(c), This
effect is inherent in the process of mixing any discrete time-domain signal
with а sinusoid of frequency f/4. Verifying this gives us а good opportu-
nity to pull convolution out of our toolbox and use it to see why the 1(т)
spectral replication period is reduced Ьу а factor of 2 from that of Х(т),
Recall, from the convolution theorern, that the DFТ of the time-domain
product of х(п) and the 1, 1, l,l mixing sequence 1(т) is the convolution
of their individual DFТs, or 1(т) is equal to the convolution of Х(т) and the
1, 1, l,l mixing sequence's magnitude spectrum in Figure 10-6(а). Н, for
convenience, we denote the 1, 1, 1,1 cosine mixing sequence's magnitude
spectrum as 5 с (т), we сап say that 1(т) = Х(т)*5/т) where the "*" symbol
denotes convolution.
Let's look at that particular convolution to make sure we get the Цт)
spectrum in Figure 10 6(c). Redrawing Х(т) from Figure 10-3(Ь) to show
its positive and negative frequency replications gives us Figure 10-7(а),
We also redraw 5 с (т) from Figure 10-6(а) showing its positive and nega-
tive frequency components in Figure 10 7(b). Before we perform the con-
volution's shift and multiply, we realize that we don't have to flip 5 с (т)
about the zero frequency axis because, due to its syrnmetry, that would
have по effect, 50 now our convolution comprises the shifting of 5 с (т) in
Figure 10-7(Ь), relative to the stationary Х(т), and taking the product of
that shifted sequence and the Х(т) spectrum in Figure 10 7(a) to arrive at
1(т), No shift of 5 с (т) corresponds to the т = О sample of 1(т), The surns
of the products for this zero shift is zero, so 1(0) = о. If we shift 5 с (т) to
the right Ьу two sarnples, we'd have an overlap of 5/8) and Х(10), and
that product gives us 1(2), Onе more 5 с (т) shift to the right results in an
overlap of 5 с (8) and Х(l1), and that product gives us 1(3), and so оп. 50
shifting 5 с (т) to the right and summing the products of 5 с (т) and Х(т)
results in 1(1) to 1(14). If we return 5 с (т) to its unshifted position in Figure
10 7(b), and then shift it to the left two samples in the negative frequency
394 D/g/1Q/ S/gпQI Processlпg Trlcks
Magnitude 01 Х(т)
.
Negallve tr'quency
components
,.............,
.
Positive frequency
component&
.
.
(а)
j . I!I , I!I I!I i
;.! . \ \, ;:
..........I!I+++....,. .,.+44.,.",.., ""., .,.....I!II!I.-+++III' ,.,. III"+++""'
--28--26-24-22...2(H8...16 ...14-12 ...10-8 -84--2 О 2 4 6 8 10 12 14 '6 16 20 22 24 28 т
I I
'./2 './2
Sc(m) = Spвctral magnitude "\/
011, 1, 1, 1
.
.
------->
(Ь)
. .
. .
. .
. .
...I!II!I.I!II!I..,....I!I., '111""'" ..III..I!I.'I!I+I!II!II!I.I!I....
...16...14...12...10 -8 -8 4 --2 О 2 4 6 8 10 12 14 т
Magnltude 01 /(т)
{7
.
. .
.
(с)
. ..
. .. .
. '. .
. '. .
. . .
. '. .
. . .
j. .:!. .!
i 1. · : : : i. . i :
.,.Ш. .'. ..'....н.+.'....н.+.'. ..'...+i+..
...16...14...12...10-8 -84--2 О 2 4 6 8 10121416 т
Тheм r.sult from shlfting
Sc(т) to th. '-"
Тheae result from shlttlng
Sc(m) to the rlghl
F/gu.e 10-7 Frеquепсу-dоmQlп coпvolutloп resultlпg lп /(т): (а) mQgпltude of
Х(т); (Ь) spectral mQgnltude ofthe coslne's l,...l, l, 1 tlme-domalп
sequeпce, Sc(m); thls Is the sequence we'lI shlft to the left Qnd rlght to
perform the convolutloп; (с) convolutloп result: the mQgnltude of
frequeпcy-traпslafed, In-рhQsе /(т),
direction, we'd have an overlap of Sc( 8) and X( 10), and that product
gives us I( 2). One more Sc(m) shift to the left resu1ts in an overlap of
SC( ) and X( 11), and that product gives us Ц 3), and so оп. Continuing
to shift Sc(m) to the left determines the remaining negative frequency
components I( 4) to Ц 14). Figure 1O-7(c) shows which I(m) samples
resиlted from the left and right shifting of Sc(m). Ву using the convolu...
tion theorem, we сап see, now, that the magnitudes in Figure 1O-7(c) and
Figure 10-6(с) reaHy are the spectral magnitudes of the in-phase соmро-
nent I(m) with its reduced spectral replication period.
The upshot of аН of this is that we can change the signs of appropriate
х(п) samples to shift x(n)'s spectrum Ьу one quarter of the sample rate
Frequeпcy Traпs/Qtioп Without Mu/tiplicQtioп 395
without having to perform any explicit mu1tip1ications. Moreover, if we
change the signs of appropriate х(п) samples in accordance with the mix
ing sequences in Figure 1O 5, we can get the in phase i(n) and quadrature-
phase q(n) components of the original х(п). One important effect of this
digital mixing Ьу f/4 is that the spectral replication periods of [(т) and
Q(m) are ми the replication period of the origina1 X(m),t 50 we must Ье
aware of the potential frequency aliasing problems that mау occur with
this frequency translation method if the signal bandwidth is too wide rel-
ative to the samp1e rate, as discussed in Section 7,3,
Before we leave this particular frequency translation scheme, 1et's
review two more issues, magnitude and phase. Notice that the untrans
lated Х(10) magnitude is equal to 16 in Fi re 10 3(b), and that the trans
lated [(2) and Q(2) magnitudes are 16/..J2 = 11.314 in Figure 1O 6. This
validates Eq. (1O 8) and Eq. (10-14). If we use those quadrature соmро--
nents [(2) and Q(2) to determine the magnitude of the corresponding fre-
quency-trans1ated, соmр1ех spectra1 component from the square root of
the sum of the squares relationship, we'd find that the magnitude of the
peak spectral component is
peak component magnitude = (16/ .J2)2 +(16/ .J2)2 =.Ji56 = 16, (10 16)
verifying Eq. (10-15). 50 combining the quadrature components I(т) and
Q(m) does not resu1t in any 10ss in spectra1 amplitude due to the fre
quency translation. Finally, in performing the above convolution process,
the phase angle samples of Х(т) in Figure 10 3(b) and the phase samples
of the 1, 1, l,1 sequence in Figure 1О-6(а) add a1gebraically. 50 the resul-
tant [(т) phase ang1e samples in Figure 1О-6(с) result from either adding
or subtracting 450 from the phase samples of Х(т) in Figure 1O 3(b).
Another easily implemented mixing sequence used for f/4 frequency
translations to оЫаin [(т) is the 1, О, 1, О, etc" cosine sequence shown in
Figure 1O 8(a). This mixing sequence's quadrature companion О, 1, О, 1,
Figure 10-8(Ь), is used to produce Q(m). То determine the spectra of these
sequences, let's, again, usе а 4-point OFТ to state that
N l
Fl,O, l,O(m) = L (1,0, 1,0,...)e j21tпт/ N
п O
(10 17)
t Recall that we saw this reduction in spectra1 replication period in the quadrature sampling
resuIts shown in Figures 7-2(g) and 7-З(d),
396 DigitQ/ S/gпQ/ Process/пg Tricks
.
:L. /
(а) /
.
о 1 " /3 Time
1 1 /
...
't .. .
'"
,
, 3
(Ь) О " . I .
о 2' / Time
/
11 " I
'.,....
Flgure 10-8 Quodrature mixing sequences for downconversion Ьу ',/4: (о)
cosine mixing sequence using 1, О, 1, О, , , ,; (Ь) slne mlxlng
sequence uslng О, 1, О, 1, ' , ,
WhenN= 4,
F (т) = e j21COm/4 e j21C2m/4 ,
1,0, 1,0
(1O 18)
Again, the cosine mixing sequence is ernbedded in the signs of the terrns
of Eq. (10 18), and there are only two terrns for our 4--point DFТ, We eval
uate Eq. (10 18) for т = 1, corresponding to а frequency off./4, to find that
Fl,O, 1.0(1) = e jO e j1t = 1 + 1 = 2LO° ,
(1O 19)
Evaluating Eq, (1O 18) for т = 3, corresponding to а frequency of f./4,
shows that
F (3) jO jЗ1С 1 1 2LO °
1,o, 1,0 =е e = + = ,
(10-20)
Using the approach in Eq. (10 13), we сап show that the scaling factor for
the 1, О, 1, О cosine mixing sequence is given as
Freqиeпcy Troпs/otioп W/thoиt Mи/t/pllcQt/oп 397
2 1
1(т) 1 О 1 О scale factor = =
" , 4 2
So
IХ(т)1
I 1(т) Il,O, l,о= '
(1O 21 )
Likewise, if we went through the same exercise as above, we' d find that
the scaling factor for the 0,1, О, 1 sine mixing sequence is given Ьу
1
Q(m)O,l,o, l scale factor = '2
So
I Q( m) I = IX(m)1
O,l,O, l 2
(10-22)
So these mixing sequences induce а 10ss in the frequency translated sig
nal amplitude Ьу а factor of 2,
Ву way of ехатрlе, let's show this scale factor 10ss again Ьу frequency
translating the х(п) sequence from Eq, (1O 7), whose spectrum is shown in
Figure 10-3(Ь). If we multiply that 32-sample х(п) Ьу 32 samples of the
quadrature mixing sequences 1, О, 1, О and О, 1, О, 1, whose DFr mag
nitudes are shown in Figure 10-9(а) and (Ь), the resulting quadrature
sequences wil1 have the frequency translated 1(т) and Q(m) spectra
shown in Figure 10-9(с) and (d).
Notice that the untranslated Х(10) magnitude is equal to 16 in Figure
1O 3(b) and that the translated 1(2) and Q(2) magnitudes are 16/2 = 8 in
Figure 1O 6. This validates Eq. (1O 21) and Eq. (1O 22), If we use those
quadrature components 1(2) and Q(2) to determine the magnitude of the
corresponding frequency translated, сотрlех spectral component from
the square root of the sum of the squares relationsblp, we'd find that the
magnitude of the peak spectral component is
peak component magnitude = (1612)2 +(16/2)2 = (16)2 /2 = ' (1O 23)
When the in phase and quadrature-phase components are combined to
get the magnitude of а сотрlех value, а resultant ..fi scale factor, for the
1, О, 1, О and О, 1, О, 1 sequences, is not eliminated. An overa1l3 dB 10ss
remains because we eliminated some of the signal power when we multi
plied ha1f of the data samples Ьу zero,
398 Digital Slgпal Proeesslпg Trieks
t Magnilude 01 1, О, 1, О
112.
(а) '/4 '; i
' ., ".,.. '.' I!I." ,.,. ,.,.. ,..,.. ....1. ".,.. .'.
о 246 810121416 18
t Phaseol1,O, I,O
'00
о
50 I \
о ",." ,..,., .., ".,.. ,." ",.. ".,,,, IIJI."I!I"'"''
о 2 4 в 8 10 12 14 16 18 20 22 24 26 28 30
t Magnitude 01 О, 1, О, 1
'12 J PhasвoI0,1,O, 1
(Ь) '/4: I ': 9O....>
' .",." "1111.."..",.",.",...1..".."., о 111" '.'II!Н";'.' ,.".,.., ""."..' "'II!I-II!I'."
О 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 2 4 6 : 10 12 14 16 18 20 22 24 26 28 30
100 . ---90
t Magnitude 01 Цт)
8 7 loot Pha8e 01 I(m) in :egrees
(с)... . ;, . 50t 2 4 i 18 20 :.
2: ! ; , , . ,! О.'-I!I-Н-,..,..,.,.4+".".IJ++,."."."-i+,,
о ,..+Н--.,.".....+-f..+.,..+++.,..,....,++.+. ...БО f . 6 8 10 12 ,4 16 . 22 24 28 28 30
О 2 4 8 8 10 12 14 16 18 20 22 24 26 28 30. .
-100
t Magnitude 01 Q(m)
(d) 8 i 1 200t Phase 01 Q(m) in d reвs
: i : i I i 1, l:t 14 , , ,,,ll , ,,,.. ,..;;.
о ,.,+++".,II!I-II!I'.'+++'II!HI!I +H!I..".,..,+++, t 2 4 6 8 10 12 14 16 ,:' 22 24 26 .:
О 2 4 6 8 10 12 1. 18 18 20 22 2. 28 28 эо 100 . . ·
-200
Flgure 10-9 Frequency translotion Ьу ',/4: (а) normollzed mognltude ond phose of
eoslne 1, О, 1, О sequenee; (Ь) normolized mognitude ond phose of
sine О. 1, О. 1 sвquenee; Се) magпitude ond phose of frequeney-
transloted in-phose /(т); (d) mognitude ond phosв of frequency-
transloted quodrature-phose Q(m),
Тhe question is "Whywould the sequences 1, О, 1, О and 0,1, О, 1 ever
Ье used if they induce а signal amplitude 10ss in i(n) and q(n)?" The
answer is that the a1temating zero valued samples reduce the amount оЕ
follow on processing performed оп i(n) and q(n), Let's say, for ехатрlе,
tha t an application requires both i(n) and q(n) to Ье 10w pass fi1tered,
When a1temating samples оЕ i(n) and q(n) are zeros, the digital fi1ters have
only ha1f as тanу multiplications to perform because mu1tiplications Ьу
zero are unnecessary,
Another way to 100k at this situation is that i(n) and q(n), in а sense, have
been decimated Ьу а factor оЕ 2, and the necessary follow on processing
rates (operations/second) are also being reduced Ьу а factor оЕ 2, If i(n) and
q(n) are, indeed, applied to two separate FIR digital filters, we сan Ье clever
and embed the mixing sequences' plus and minus ones and zeros into the
Frequeпcy TrQпs/Qt/oп W/thout Mu/tip//cQt/oп 399
Demultiplexer
[ """
"' ;':
j,
Even
samplвs
i(n)
х(п)
cos=1, 1J1J 1,..
Odd
samples
q(n)
"'
sin = 1 , 1, 1, 1 , . .
Flgure 10-1 О Quadrature downconversion Ьу f,l4 uslng а demultlplexer
(demux) and the sequence 1. 1, 1. 1"",
fi1ters' coefficient values and avoid actually perforrning any mu1tip1ica
tions, Becaиse some coefficients are zero, they need not Ье иsed at all, and
the number of actual mu1tip1iers иsed can Ье reduced. In that way, we'll
have perforrned quadrature mixing and FIR filtering in the same process
with а simpler filter, This technique al80 forces us to Ье aware of the poten-
tial frequency a1iasing problems that mау occur if the input signal is not
sufficient1y bandwidth 1imited relative to the original sample rate.
Figиre 1O 10 illustrates an interesting hybrid technique using thefs/2
mixing sequence (1, 1, 1, 1) to perforrn quadrature mixing and down
conversion Ьу fs/ 4, Тhis scheme uses а demu1tiplexing process of routing
a1temate input samples to one of the two mixer paths[ ]. Although both
digital mixers use the same mixing sequence, this process is equivalent to
mu1tiplying the input Ьу the two quadrature mixing sequences shown in
Figиre 10-8(а) and 10-8(Ь) with their frequency domain magnitudes indi-
cated in Figure 10 9(a) and 10 9(b), Тhat's because a1temate samples are
routed to the two mixers, Although this scheme can Ье used for the quad-
rature sampling and demodulation described in Section 7,2, interpolation
fi1ters must Ье used to remove the inherent ha1f sample time delay
between i(n) and q(n) caused Ьу using the single mixing sequence of
1, 1,l, 1.
ТаЫе 10-1 summarizes the effect of mu1tiplying tim domain signal sam-
ples Ьу varioиs digital mixing sequences of ones, zeros, and minиs ones,
А "yes" in the last column in ТаЫе 10-1 indicates that a1temating sam
ples of i(n) and q(n) can Ье discarded with по adverse impact оп their
spectra, al10wing а fol1ow-on processing data rate that's ha1f the original
х(п) data rate,
400 Dig/to/ S/gпo/ Process/пg Tr/cks
ТаЫе 10-1 Dlgltol Mixlng Sequences
Iп phase Quadrature Frequency Scale Fiпal sigпal Decimatioп
sequence sequence traпslatioп ьу factor power loss сап occur
1, 1, 1, 1"" М2 1 OdB по
1, 1, 1, 1"" 1, 1,-1, 1"" М4 1/ ..fi OdB yes
1, О, 1, О, .. , 0,1, О, 1, ' , , М4 1/2 3dB yes
1, 1, 1, 1,.. , 1, 1, 1, 1, ' , , f/ 4 1/2 3dB по
(with demux) (with demux)
10.2 High-Speed Vector-Magnitude Approximation
The quadrature processing techniques employed in spectrum analysis,
computer graphics, and digital communications routinely require high-
speed dеtелninаtiоn оЕ the magnitude оЕ а complex vector V given its real
and imaginary parts; i.e., the in-phase part 1 and the quadrature-phase
part Q[4]. Тhis magnitude calculation requires а square root operation
because the magnitude оЕ V is
I V 1= 12 +Q2
(10-24)
Assuming that the sum 12 + Q2 is available, the problem is to efficiently
perform the square root operation.
Тhere are several ways to оЫаin square roots, but the optimum tech
niqиe depends оп the capabilities оЕ the available hardware and software.
For example, when performing а square root using а high-level software
language, we employ whatever software square root function is available.
Although accurate, software routines сап Ье very slow. In contrast, if а
system must accomplish а square root operation in 50 nanoseconds, high-
speed magnitude approximations are required[7,8]. Let's look at а neat
magnitude approximation scheme that's particularly efficient.
10.2.1 aMax+BMln Algorlthm
There is а technique called the аМах+ Мin (read as "alpha mах plus beta
min") algorithm for calculating the magnitude оЕ а complex vector. t It's а
t А "Мах+ Мin" algorithm had Ьееn in UБе, but in 1988 this author suggestecl expanding it
to the aМax+!3Мin form, where а could Ье а value other than unity[9],
Hlgh Speed Vector-МQgпltudе ApproxlтQtioп " 4т
«.! "'4'
'$1;}' ., ,
linear approximation to the vector magnitude problem that requi ,
determining which orthogonal vector, I or Q, has the greater absolu':;:;; :' '
value, If the maximum absolute value of I or Q is designated Ьу Мах and" ",
':,:"
the rninimum absolute value of either 1 or Q is Мin, аn approximation of
I V I , using the aМax+ Min algorithm, is expressed аБ
IVI =a.Мax+ Min.
(10-25)
There are several pairs for the а and constants that provide varying
degrees of vector magnitude approximation accuracy to within
O,ldB[7,10], The aМax+ Мin algorithms in reference [10] dеtепninе а vec-
tor magnitude at whatever speed it takes а system to реrfопn а magnitude
comparison, two multiplications, and one addition, But, as а minimum,
those algorithms require а 16-bit multiplier to achieve reasonably accurate
results. However, if hardware mu1tipliers are not avai1able, аН is not lost.
Ву restricting the а and constants to reciprocals of integral powers of 2,
Eq. (10 25) lends itself wel1 to implementation in binary integral arith-
metic. А prevai1ing application of the aМax+ Min algorithm uses а= 1.0
and = 0.5[11,12], The 0,5 multiplication operation is performed Ьу shift-
ing the minimum quadrature vector magnitude, Мin, to the right Ьу 1 bit.
We сап gauge the accuracy of anу vector magnitude estimation Ьу plotting
its error аБ а function of vector phase angle. Let's do that. The a.Мax+ Мin
estimate for а complex vector of unity magnitude, using
Мin
IVI=Max+
2 '
(10 26)
over the vector angular range of О to 900, is shown as the solid curve in
Figure 10 11. (The curves in Figure 10-11, of course, repeat every 900.)
An ideal estimation curve for а unity magnitude vector would have an
average value of one and an error standard deviation (о) of zero; that is,
having а е = О means that the ideal curve is flat because the curve's value
is one for all vector angles and its average error is zero, We'll иБе this ideal
estimation curve аБ а yardstick to measure the merit of various
aМax+ Min algorithms, Let's make sure we know what the solid curve
in Figure 10 11 is teHing ш. It indicates that а unity magnitude vector ori-
ented at an angle of approximately 260 will Ье estimated Ьу Eq, (10-26) to
have а magnitude of 1.118 instead of the correct magnitude of one. The
error then, at 260, is 11.8 percent, or 0,97 dB, Analyzing the entire solid
curve in Figure 10-11 results in а е = 0.032 and an average error, over the О
to 900 range, of 8.6 percent (0,71 dB),
402 DigitQI SlgпQI Processiпg Trlcks
Vector-magnitude estimate
1.15
1,1 О
-'
.....
1,()5
1.00
0,90
"-
,
.. ..,
Мах + Minl4 / \
,
, ,-
"
,-
/
/
/
/
/
0,95
0,85
О
10
20
ЗО 40 50 80
Vector phase angle (degrees)
70
80
90
Flgure 10-11 Normalized aMax+BMln estimates for а= 1, В = 1/2, and В = 1/4,
То reduce the average error introduced Ьу Eq. (1O 26), it is equally con
venient to use а value of 0.25, such as
Min
IVI",Max+ ,
4
(10--27)
Equation (10-27), whose multiplication is realized Ьу shifting the digital
valиe Min 2 bits to the right, results in the nonnalized magnitude approx
imation shown as the dashed curve in Figure lO l1, Although the largest
error of 11.6 percent at 450 is similar in magnitude to that realized from
Eq, (10-26), Eq, (1O 27) has reduced the average error to O,64 percent
( 0.06 dБ) and produced а slightly larger standard deviation of а е = 0.041,
Though not as convenient to implement as Eqs, (1O 26) and (10-27), а
valиe of 3/8 has been used to provide even more accurate vector mаgni
tude estimates[13]. Using
3.Мin
'V'",Max+
8
(1O 27')
provides the nonnalized magnitude approximation shown as the dotted
curve in Figure 10-11, Equation (10-27') resu1ts in magnitude estimates,
whose largest error is only 6,8 percent, and а reduced standard devia-
tion of а е = 0.026.
H/gh Speed Vector MQgп/tude Approx/тQt/oп 403
Vector-magnitude estimate
1,10
1,05
15(Мах + Мln/2)11б
,.- :...,... .... ...... ---....;../. ... ", ,.- ......... ...... .....
--
" ......, " ............
1.00
0,95
0,90
0,85
О
10
20
30 40 50 60
Vector phase angle (degrees)
70
80
90
Flgure 10-12 a.Max+BMin estlmates for а. = 7/8, В = 7/16 and а. = 15/16, В = 15/32,
Although the values for а and in Figure 10-11 yield rather accurate
vector magnitude estimates, there are other values for а and that
deserve our attention because they result in smaller error standard devia-
tions. Consider а = 7/8 and = 7/16 where
7 7, 7 ( Мin )
IVI", Max+ Mm= Max+
8 16 8 2
(10 28)
Equation (1O 28), whose norma1ized results are shown as the so1id curve in
Figure 1O 12, provides an average error of 5.01 percent and Cf e = 0,028. The
7/8ths factor app1ied to Eq. (1()",26) produces both а smaller Cf e and а reduced
average епоr it lowers and flattens out the епоr cшvе from Eq. (1O 26),
А further improvement сап Ье obtained with а = 15/16 and = 15/32
where
15 15, 15 ( Мin )
IVI", Max+ Mm= Max+ .
16 32 16 2
(10 29)
Equation (1O 29), whose norma1ized results are shown as the dashed curve
in Figure 10 12, provides an average error of 1.79 percent and Cf e = 0.030, At
the expense of а slightly larger Cf e ' Eq, (10 29) provides an average error
that is reduced below that provided Ьу Eq. (10 28).
Although Eq. (10 29) appears to require two multiplications and one
addition, its digital hardware implementation сan Ье straightforward, as
404 Digitol Sigпol Processiпg Tricks
11'
15(Мах + Minl2)
16
IQI
Мах + Мin12
16
10112
Flgure 10-13 Hordwore implementation of Eq, (10-29),
shown in Figure 1O 13. The diagonallines, \1 for example, denote а hard
wired shllt оЕ 1 bit to the right to implement а divide--by two operation Ьу
truncation. Likewise, the \4 8утЬоl indicates а right shift Ьу 4 bits to real
ize а divide by 16 operation, The 111> I Q I controlline is ТRUE when the
magnitude оЕ 1 is greater than the magnitude оЕ Q, 80 that Мах = I 1 I and
мin = I Q I , This condition enables the registers to apply the value8 111 and
I Q I /2 to the adder. When 111 > I Q I is FALSE, the registers apply the val-
ues I Q I and 111/2 to the adder. Notice that the output оЕ the adder, Мах
+ Мin/2, is the re8ult оЕ Eq. (10 26). Equation (1O 29) is implemented via the
subtraction оЕ (Мах + Мin/2)/16 from Мах + Мin/2.
In Figure 10 13, аН impIied multiplications from Eq. (10 29) are per
formed Ьу hardwired bit shifting, and the total execution time is limited
onIy Ьу the delay times as80ciated with the hardware components.
10.2.2 Overflow Errors
In Figures Ю-ll and 1(}..12, поЬсе that we have а potential overflow problem
with the results оЕ Eqs, (1(},,26), (1(},,27), and (1O 29) because the estimates сan
exceed the correct noпnalized vector magnitude values; i.e" Боте таgni
tude estimates are greater than опе. This теans that, although the correct
magnitude value тау Ье within the system's fuH scale word width, the
algorithm result тау exceed the word width of the system and cause over-
flow епоrs, With etМax+ Мin algorithms, the user must Ье certain that по
true vector magnitude exceed8 the value that will produce an estimated
magnitude greater than the maximum a110wable word width, For example,
High-Speed Vector-МQgп/tиdе ApproxiтQtioп 405
when using Eq. (lO 26), we mиst ensure that по Пuе vector magnitude
exceeds 89.4 percent (1/1.118) of the maximum allowable work width.
10.2.3 Truncatlon Errors
The penalty we рау for the convenience of having а and as powers of
two is the error induced Ьу the division by truncation process; and, thus
far, we haven't taken that error into account. The error curves in Figure
10 11 and Figиre 10 12 were obtained using а software simиlation with its
floating point accuracy and are useful in eva1uating different а and val
ues. However, the Пuе error introduced Ьу the aМax+ Min algorithm
will Ье somewhat different from that shown in Figиres 1O 11 and 1O 12
due to division errors when truncation is иsed with finНе word widths,t
For аМах+ Мin schemes, the truncation errors are а function of the
data' s word width, the algorithm иsed, the values of both 111 and I Q I ,
and the vector's phase angle. (These errors due to truncation compound
the errors already inherent in our aМax+ Мin a1gorithms.) Thus, а сот-
plete analysis of the truncation errors is beyond the scope of this book.
What we сап do, however, is illustrate а few truncation error examples.
Figure 1O 14 shows the percent truncation errors using Eq. (lO 29) for
vector magnitudes of 4 to 512, Two vector phase angles were chosen to
% Error
8
6
о
6c2=9 6_9 _O_O.O 9 O O O O O O O
4
2
% о
2
--4
--в
. 260
о 00
--в
10
4
85
165
245
325
405
485
v
Flgure 10-14 Equatlon (10-29) truncaHon error vs, vector тagnitude I VI,
t Errors associated with division-by-truncation are covered in more detail in Section 9.3,
.
406 D/g/ta/ S/gпQ/ Process/пg Tr/cks
ТаЫе 10-2 а MQx+SMin Algorlthm CompQrisons
Largest Largest Average Average Staпdard Мах
AIgorithm error error error error deviatioп IVi
IVi (%) (dB) (%) (dB) а, (% F.S.)
Мах + Мin/2 11,8% 0.97 dB 8,6% 0,71 dB 0,032 89,4%
Мах + Мin/4 11,6% 1Ш dB O,64% O,06 dB 0,041 97,0%
Мах + 3Мin/8 6,8% 0,57 dB 3.97% 0,34 dB 0,026 93,6%
7(Мах + Мin/2)/8 12,5% 1,16 dB 4,99% 0.45 dB 0,028 100%
15(Мах + Мin/2)/16 ,25% O,56 dB 1,79% 0,15 dB 0,030 95,4%
illustrate these truncation errors, The first is 260 because this is the
phase angle where the most positive algorithm error occurs, and the
second is 00 because this is the phase angle that introduces the greatest
negative algorithm error, Notice that, at small vector magnitudes, the
truncation errors are as great as 9 percent, but for ап eight bit system
(maximum vector magnitude = 255) the truncation error is less than 1
percent. As the system word width increases, the truncation errors
approach О percent, This means that truncation errors add very little to
the inherent aMax+ Min algorithm errors,
ТЬе relative реrfопnаnсе of the various algorithms is summarized in
ТаЫе 10 2, The last column in ТаЫе 10 2 illustrates the maximum allow
аЫе true vector magnitude as а function of the system's full-scale (Е5.)
word width to avoid overflow errors.
50, the aМax+ Мin algorithm enables high speed, vector magnitude
computation without the need for math coprocessor or hardware mu1ti
plier chips, Of course with the recent availability of high speed, floating
point multiplier integrated circuits with their ability to multiply or
divide Ьу nonintegral nurnbers in опе or two clock cycles and mау
not always need to Ье restricted to reciprocals of integral powers of two.
It's also worth mentioning that this algorithm сап Ье nicely implemented
in а single hardware integrated circuit (for example, а field programma
Ые gate array) affording high speed operation,
10.3 Data Windowing Tricks
There are two usefиl schemes associated with using window fиnctions оп
input data applied to а DFТ or ап FFТ. The first technique is an efficient
implementation of the Hanning (raised cosine) and Hamming windows
OQtQ Wiпdowlпg Trlcks 407
to reduce 1eakage in the FFf, Тhe second scheme is re1ated to minimizing
the amplitude 10ss associated with using windows,
10.3.1 Windowing in the Frequency Domain
Тhere's а c1ever technique for minimizing the calcu1ations necessary to
imp1ement FFf input data windowing to reduce spectra11eakage, Тhere
are times when we need the FFf о! unwindowed time-domain data, and
at the same time, we a1so want the FFТ о! that same time data with а win-
dow function applied, In this situation, we don' t have to perform two sep-
arate FFfs, We сап perform the FFf о! the unwindowed data, and then we
сап perform frequency domain windowing оп that FFf result to reduce
1eakage, Let's see how,
Recall from Section 3,9 that the expressions for the Hanning and the
Hamming windows were wиап(п) = O,5 O,5cos(21tn/N), and
wиаm(п) = 0.54 O,46cos(21tn/N), respective1y. They both have the gen-
era1 cosine function form о!
W(n) = а cos(21tn/N) ,
(10-30)
for п = О, 1, 2, ' , " N l, Looking at the frequency response о! the genera1
cosine window function, using the definition о! the DFf, the transform о!
Eq, (10-30) is expressed Ьу
N l
W(m) = )a ,В cos(21tn / N)]e j27tпт! N ,
п=О
(10-31)
ej27tп! N e j27tп! N
Because cos(21tn / N) = + , Eq, (10-31) сап Ье rewritten as
2 2
N l R N l R N l
W(m)= Lae j27tпт!N Lej27tп!Ne j27tпт!N Le-j27tп!Nе j27tпт!N
п=О п=О п=О
N l N-l R N-l
= Lae j27tпт!N Le-j27tп(т 1)!N Le j27tп(т+l)!N , (10-32)
п=О п=О п=О
Equation (10-32) 100kБ pretty complicated, but, using the derivation from
Section 3,13 for expressions like those summations, we find that Eq, (10-32)
mere1y results in the superposition о! three sin(x)/ х functions in the fre-
quency domain, Тheir amplitudes are shown in Figure 10-15,
408 DigitC11 Slgпol Processlпg rrlcks
FrSQ
т--1 т mt1
Flgure 10-15 General coslne window frequency-response amplltude,
Notice that the two translated sin(X)/X funсНоns have sidelobes with
phase opposite from that of the center sin(x) / х funсНоn. тhis mеаns that
а tirnes the mth Ыn output, minus /3/2 times the (m l)th Ьin output,
minus /3/2 times the (т+ l)th Ьin output will minimize the sidelobes of the
mth Ьin. Тhis frequency domain convolution process is equivalent to mul
tiplying the input time data sequence Ьу the N-valued window funсПоn
w(n) in Eq. (10 30)[14,l5].
For example, let's say the output of the mth FFf Ьin is Х(т) = а т + jb m ,
and the outputs of its two neighboring ЫNS are X(m 1) :::: a l + jb l алd
Х(т+l) = а+ 1 + jb+1' Тhen frequency-domain windowing for the mth Ьin of
the иnwindowed Х(т) is as follows:
Xwindowed(m):::: aX(т) x(т l) X(т+ 1)
= а(а т + jb m ) (a l + jb l) (a+l + jb+ 1 )
::::ал т (a l +a+l)+ ЛаЬ т (b l +Ь+ 1 )]
(10-33)
То get а windowed N point FFТ, then, we сал apply Eq, (1(},,33), requiring
4N additions алd 3N mшНрНсаНоns, to the unwindowed FFТ result and
avoid having to perform the N multiplications of time-domain windowing
and а second PFТ with its Mog 2 N additions and WogjJ multiplications.
.
DQtQ Wiпdow/пg Tricks 409
The neat situation here is the а. and values for the Hanning window,
They're both 0.5, and the products in Eq, (10-33) сап Ье obtained in hard
ware with binary shifts Ьу а single bit for а. and two shifts for /2, Тhus,
по multiplications are necessary to implement the Hanning frequency-
domain windowing scheme, Тhe issues that we need to consider are the
window function best for the application and the efficiency оЕ availabIe
hardware in реrfопning the frequency-domain multiplications,
Along with the Hanning and Hamming windows, reference [15]
describes а family оЕ windows known as Бlасkmаn and Бlасkmаn
Harris windows that are also very useful for performing frequency-
domain windowing. (Бе aware that reference [15] has two typographical
errors in the 4-Term ( 74 dБ) window coefficients column оп its page 65,
Reference [16] specifies that those coefficients shouId Ье 0.40217,
0.49703,0,09892, and 0,00188,) Let's finish our discussion оЕ frequency-
domain windowing Ьу saying that this scheme сап Ье efficient because
we don't have to window the entire set оЕ РРТ data, Frequency-domain
windowing need Ье реrfопnеd only оп those РРТ bins that are оЕ inter
est to us,
10.3.2 Minimlzing Wlndow-Processing Loss
In Section 3.9, we stated that nonrectangular window functions reduce
the overa11 signallevels applied to the PFТ, Recalling Figure 3-16(а), we
see that the peak response оЕ the Hanning window function, for ехатрlе,
is half that obtained with the rectangular window because the input sig-
nal is attenuated at the beginning and end edges оЕ the window sample
interval, as shown in Figure 10-16(а), In tепns оЕ signal power, this atten-
uation results in а 6 dБ 10ss, Going beyond the signal-power 10ss, window
edge effects сап Ье а probIem when we're trying to detect short duration
signals that тау occur right when the window function is at its edges,
Well, some early digital signal processing practitioners tried to get around
this probIem Ьу using dual window functions,
Тhe first step in the dua1 window process is windowing the input data
with а Hanning window function and taking the PFТ оЕ the windowed
data. Тhen the same input data sequence is windowed against the inverse
оЕ the Hanning, and another РРТ is performed, (The inverse оЕ the
Hanning window is depicted in Figure 10-16(Ь),) Тhe two FFТ results are
then averaged. Using the dual window functions shown in Figure 10-16
enabIes signal energy attenuated Ьу one window to Ье multiplied Ьу the
Ыl gain оЕ the other window, Тhis technique seemed like а reasonabIe
idea at the time, but, depending оп the original signal, there could Ье
.
41 О D/g/ta/ S/gпQI Process/пg Tr/cks
(а)
Hanning window:
\li.ian (п) = 0,5 О,5соs(2пn'N)
о
L...........-,-
Windowedge
L..........,---
Window edge
Time
(Ь)
Inverse 01 the
Hanning window:
w Han (п) = 0,5 + О,5соs(2пn'N)
о
Window edge
Windowedge
Time
Flgure 10-16 Dual windows used to reduce windowed-slgnalloss,
excessive leakage from the inverse window in Figure 10-16(Ь). Remember,
the purpose of windowing was to ensure that the first and last data
sequence samples, applied to an FFТ, had the same value. Тhe Hanning
window guaranteed this, but the inverse window could not, Although
this dual window technique made its way into the literature, it quickly fell
out of favor, Тhe most common technique used today to minimize signal
1055 due to window edge effects is known as overlapped windows,
The use of overlapped windows is depicted in Figure 1O 17, It's а
straightforward technique where а single good window function is
applied multiple times to an input data sequence, Figure 10-17 shows аn
N-point window function applied to the input time series data four times
resulting in four separate N point data sequences. Next, four separate N-
point FFТs are performed, and their outputs averaged, Notice that anу
input sample value that's fully attenuated Ьу оnе window will Ье multi-
plied Ьу the full gain of the following window, Тhus, all input samples
will contribute to the finаl averaged FFТ results, and the window function
keeps leakage to а minimum. (Of course, the user has to decide which
particular window function is best for the application.) Figure 10-17
shows а window overlap of 50 percent where еат input data sample con
tributes to the results of two FFТs, It's not uncommon to see an overlap of
FQst Mu/tip//cQt/oп of Сотр/ех Nuтbers 411
Inpu! 1(
!ime I
series
1s!FFТ
2,5 N samples
»
I
Time ..
I
N samples
Flgure 10-17 Windows overlapped Ьу 50 percent to reduce
windowed-signalloss,
75 percent being used where each input data sample would contribute to
the results о! the three individual FFТs. О! course the 50 percent and 75
percent overlap techniques increase the amount о! total signal processing
required, but, depending оп the application, the improved signal sensi-
tivity mау justify the extra numЪеr crunching.
10.4 Fast Multiplication о' Complex Numbers
Тhe multiplication о! two complex numbers is one о! the most common
functions performed in digital signal processing, It's mandatory in аН dis-
crete and fast Fourier transformation algorithms, necessary for graphics
transformations, and used in processing digital communications signals.
Не it in hardware or software, it's always to our benefit to streamline the
processing necessary to perform а complex multiplication whenever we
can. If the avai1able hardware сап perform three additions faster than а
single multiplication, there's а way to speed ир а complex multiplication
operation[17].
Тhe multiplication о! two complex numbers а + jb and с + jd, results in
the complex product
R + jI = (а + jb) . (с + jd) = (ас bd) + j(bc + ad) .
(10-34)
We can see that Eq, (10-34) required four multiplications and two addi-
tions. (From а computational standpoint, we'l1 assume that а subtraction
412 D/g/tQI S/gпQ/ Process/пg Tr/cks
is equivalent to an аddШоn,) Instead of using Eq, (1O 34), we сan ca1culate
the following intermediate values:
k 1 = а(с + d) ,
k 2 = d(a + Ь) ,
and
k3 = с(Ь а) .
(10 35)
Then we perform the fol1owing operations to get the final R and 1:
R=kl k2'
and
I=kl+kз,
(10 36)
The reader is invited to plug the k values from Eq. (10 35) into Eq.
(10 36) to verify that the expressions in Eq, (1(},36) are equivalent to Eq.
(10-34), Тhe intermediate values in Eq. (1(},35) required three аddШоns
and three multiplications, whereas the results in Eq. (10-36) required two
more аddШоns. 50 we traded one of the multiplications required in Eq.
(10-34) for three аddШоn operations needed Ьу Eq, (10 35) and Eq,
(1O 36), If ощ hardware uses fewer clock cycles to perform three аddШоns
than а single multiplication, we тау well gain overal1 processing speed
Ьу using Eq. (10 35) and Eq. (10-36) for complex multiplication, instead of
Eq. (1O 34),
10.5 Efficiently Performing the FFТ 01 Real Sequences
Upon recognizing its linearity property and understanding the odd and
even syrnmetries of the transform's output, the early investigators of the
fast Fourier transform (FFT) rea1ized that two separate, real N point
inpиt data sequences could Ье transformed using а single N point coт
plex FFТ, Тhey also developed а technique using а single N point сот-
plex FFТ to transform а 2N point real input sequence. Let's see how these
two techniques work.
10.5.1 Performing Two N Point Real FFТs
The standard FFТ algorithms were developed to accept complex inputs;
that is, the FFТ's normal input х(п) sequence is assumed to comprise real
and imaginary parts, such as
Efflcieпtly Perforтlпg the FFТ of Reol Seqиeпces 41 3
х(О) = хт(О) + jxi(O) ,
х(l) = Х т (1) + jx j (1) ,
х(2) = Х т (2) + jxP) ,
x(N l) = xr(N l) + jXi(N l) .
(10-37)
In typical signal processing schemes, РРТ input data sequences are
usuaHy real. Тhe most common example of this is the PFТ input samples
coming from an A/D converter that provides real integer values of some
continuous (analog) signal. In this case the FFTs imaginary xi(n)'s
inputs are аН zero, 50 initial PFТ computations performed оп the xi(n)
inputs represent wasted operations. Early FFТ pioneers recognized this
inefficiency, studied the problem, and developed а technique where two
independent N point, real input data sequences could Ье transformed Ьу
а single N-point complex РРТ. We саН this scheme the Two
N Point Real FFTs algorithm. The derivation of this technique is
straightforward and described in the Iiterature[18--20], If two N-point,
real input sequences are а(п) and Ь(п), they'H have discrete Fourier
transforms represented Ьу Ха(т) and Хь(т). If we treat the а(п) sequence
as the real part of an РРТ input and the Ь(п) sequence as the imaginary
part of the PFТ input, then
х(О) = а(О) + jb(O) ,
х(l) = а(l) + jb(l) ,
х(2) = а(2) + jb(2) ,
x(N l) =a(N l) + jb(N l) .
(10-38)
Applying the х(п) values from Eq, (10-38) to the standard DFТ,
N l
Х(т) = Lx(n)e j21tпm/N ,
п=О
(10-39)
we'H get an DFТ output Х(т) where т goes from О to N l, (We're assum-
ing, of course, that the DFТ is implemented Ьу way of an PFТ algorithm.)
Using the superscript * symbol to represent the complex conjugate, we
сап extract the two desired FFТ outputs Ха(т) and Хь(т) from Х(т) Ьу
using the following:
414 Digitol Sigпai Processiпg Trlcks
Ха(т) = Х' (N т) + Х(т) ,
2
(1O 40)
and
Хь(т) = ЛХ' (N т) Х(т)]
2
(10-41)
Let's break Eqs. (10-40) and (1O 41) into their real and imaginary parts
to get expressions for Ха(т) and Хь(т) that are easier to understand and
implement. Using the notation showing X(m)'s real and imaginary parts,
where Х(т) = Х,(т) + jXi(m), we can rewrite Eq. (10-40) as
Ха(т) = X,(N т) + Х,(т) +l[x j (m) Xj(N т)]
(1O 42)
where m = 1,2,3, ' . ., N l, What about the first Ха(т), when m = О? Well,
this is where we run into а bind if we actually try to implement Eq,
(10-40) direct1y. Letting m = О in Eq. (10-40), we quickly realize that the
first term in the nuтerator, X*(N O) = X*(N), isn't available because the
X(N) sample does not exist in the output of an N-point FFr! We resolve
this problem Ьу remembering that Х(т) is periodic with а period N, so
X(N) = X(O),t When m = О, Eq, (10-40) becomes
= + + = .
2
(10-43)
Next, simplifying Eq. (1O 41),
Хь(т) = j[X,(N т) jXj(N т) Х,(т) jXi(m)]
2
= Xi(N т) + Xj(m) + j[X,(N т) Х,(т)]
2
(10-44)
where, again, m = 1,2,3, ' , " N l. Ву the same argument used for Eq.
(10-43), when m = О, Хь(О) in Eq, (1O 44) becomes
t This fact is illustrated in Section 3.8 during the discussion of spectralleakage in DFТs,
Efflcieпtly Perforтiпg the FFТ о' Real Sequeпces 415
Хь(О) = Xj(O) + Xj(O) + ЛХт(О) Хт(О)] Xj(O) .
2
(1O 45)
This discussion brings ир а good point for beginners to keep in mind.
In the 1iterature Eqs. (10 40) and (1O 41) are often presented without anу
discussion of the т = О problem, 50, whenever you're grinding through
an algebraic derivation or have some equations tossed out at you, Ье а lit
t1e skeptical, Try the equations out оп an exampl ee if they're true.
(After аll, both authors and book typesetters are human and sometimes
make mistakes, We had an old saying in Ohio for this situation: "Trust
everybody, but cut the cards.") Following this advice, let's prove that this
Two N-Point Real FFTs algorithm really does work Ьу applying the
8 point data sequences from Chapter 3's DFТ Examples to Eqs. (10-42)
through (1O 45), Taking the 8 point input data sequence from 5ection 3.1 's
DFТ Example 1 and denoting it а(п),
а(О) = 0,3535,
а(2) = 0,6464,
а(4) = 0.3535,
а(6) = 1.3535,
а(1) = 0.3535,
а(3) = 1.0607,
а(5) = 1,0607,
а(7) = 0.3535 .
(10-46)
Taking the 8-point input data sequence from 5ection 3,6's DFТ Example 2
and calling it Ь(п),
Ь(О) = 1.0607,
Ь(2) = 1,0607,
Ь(4) = 0.3535,
Ь(6) = 0.3535,
Ь(1) = 0.3535,
Ь(3) = 1.3535,
Ь(5) = 0.3535,
Ь(7) = 0.6464 .
(1O 47)
Combining the sequences in Eqs, (1O 46) and (1O 47) into а single complex
sequence х(п),
а(n)
.j.
х(п) = 0.3535
+ 0.3535
+ 0.6464
+ 1,0607
+ 0.3535
1,0607
1.3535
0.3535
Ь(n)
.j.
+ j 1,0607
+ j 0,3535
j 1,0607
j 1.3535
j 0.3535
+ j 0.3535
+ j 0.3535
+ j 0.6464 ,
(1O 48)
. 416 DigitQI SlgпQI Processiпg Tricks
Now, taking the 8 point PFТ of the complex sequence in Eq, (1O 48) we get
Х,(т)
J.
Х(т) = 0.0000
2.8283
+ 2.8282
+ 0.0000
+ 0.0000
+ 0.0000
+ 0.0000
+ 2,8283
50 from Eq. (10 43),
Х;(т)
J.
+ j 0.0000
j 1.1717
+ j 2.8282
+ j 0.0000
+ j 0.0000
+ j 0.0000
+ j 0.0000
+ j 6.8282
Ха(О) = Х/О) = о ,
....m=Ot.rm
....m=lt.rm
.....m=2term
.....т.зtеrm
.....m-4term
.....m=5term
.....т=6term
.... т = 7 term . (1O 49)
То get the rest of Ха(т), we have to plug the PFТ output's Х(т) and
X(N m) values into Eq. (1O 42).t Doing so,
Ха(1) = X r (7) +X r (1) +l[Xi(l) X i (7)1 2.8283 2.8283 + 1,1717 б.8282J
0 j7.9999 0 j4.0=4L 900,
2
Х ( 2 ) = Хr(б) + X r (2) + j[X i (2) Хi(б)J = 0.0+ 2.8282 + j[2.8282 0.0]
а 2 2
2.8282 + j2,8282 = 1.414 + j1.414 = 2L45 o ,
2
Х ( 3 ) = Xr(5) + Хr(З) + j[Xi(3) Xi(5)] = 0,0+ 0.0 + ло.о о.о] OLOO,
а 2 2
Х ( 4 ) = Xr(4)+Xr(4)+ j[Xi(4) Xi(4)1 0.0+0.0+ j[O.O O,O] OLO O ,
а 2 2
t Remember, when the FFТ's input is complex, the FFТ outputs тау not Ье conjugate sym-
metrie; tha! is, we сan'! assume that F(m) is equa1 !о F*(N m) when the FFТ input sequence's
real and imaginary рartБ are both nonzero,
Efflcleпt/y Perforтiпg the FFТ of ReQI Sequeпces 417
Х ( 5 ) = Х т (3) + Х т (5) + j[X j (5) X j (3)] 0,0 + 0.0 + ДО.О 0.0] = OL00 ,
а 2 2
Х ( 6 ) = Х т (2) + Х,(6) + j[X j (6) X j (2)] = 2,8282 + 0,0 + ло.о 2.8282]
а 2 2
2.8282 j2.8282 1,414 j1,414 = 2L 450 , and
2
Х ( 7 ) = Х,(l) + Х,(7) + j[X j (7) Xj(l)] = 2.8282 + 2,8282 + Д6.8282 + 1,1717]
а 2 2
0.0 + j7.9999 = 0+ j4.0 = 4L900 .
2
50 Eq. (10 42) real1y does extract Х.(т) from the Х(т) sequence in Eq.
(1O 49). We сап see that we need not solve Eq. (10-42) when т is greater
than 4 (or N /2) because Ха(т) wil1 always Ье conjugate symmetric,
Because Х.(7) = Х.(l), Х а (6) = Х а (2), etc., only the first N /2 elements in
Ха(т) are independent and need Ье calculated,
ок, let's keep going and use Eqs, (10-44) and (10 45) to extract Хь(т)
from the PFТ output. From Eq. (10 45)
Хь(О) = Х;(О) = о .
Plugging the FFТ's output values into Eq. (10-44) to get the next four
Xb(m)s, we have
Х ( 1 ) = X j (7) + X j (l) + ЛХ,(7) X,(l)] 6,8282 1,1717 + Д2,8283 + 2.8283]
ь 2 2
5.656 + j5,656 2,828 + j2,828 = 4L450 ,
2
Х ( 2 ) = X j (6) + X j (2) + ДХ,(6) Х,(2)] 0,0 + 2.8282 + до.о 2,8282]
ь 2 2
2.8282 j2,8282 = 1,414 j1,414 = 2L 450, and
2
-418 Dig/to/ SigпQ/ Process/пg Tr/cks
Х ь (3) = Хj(5)+Хj(З)+ j[Хr(5) Хr(З)]
2
0,0 + 0,0 + j[O.O 0.0] = OL00 and
2 - '
Х ь (4) = X j (4)+X j (4)+ j[Xr(4) Xr(4)]
2
0.0+0.0+ j[O,O O,O] OLO O .
2
ТЬе question arises "With the additiona1 processing required Ьу Eqs,
(10 42) and (10 44) after the initial FFТ, how much computational saving
(or 10ss) is to Ье had Ьу this Two N-Point Real FFТs algorithm?" We Сan
estimate the efficiency оЕ this algorithm Ьу considering the number оЕ
arithmetic operations required relative to two separate N-point radix 2
FFТs. First, we estimate the number of arithmetic operations in two sepa
rate N point complex FFТs.
From 5ection 4.2, we know that а standard radix 2 N point complex
FFТ comprises (N /2) .10g2N butterfly operations. If we use the optimized
butterfly structure, each butterfly requires one complex mu1tiplication
and two соmр1ех additions. Now, one соmр1ех mu1tiplication requires
two real additions and four real multip1ications, and one complex addi
tion requires two real additions. t 50 а sing1e FFТ butterfly operation com
prises four rea1 mu1tip1ications and six rea1 additions. Тhis means that а
single N point соmр1ех FFТ requires (4N /2) .10g 2 N rea1 multiplications,
and (6N /2) 'log 2 N real additions, Finally, we сап say that two separate
N-point complex radix-2 FFТs require
two N point complex FFrs .....
4N .10g2N rea1 multip1ications, and
(10 50)
6N .10g2N real additions.
(10 50')
Next, we need to determine the computational workload of the Two
N-Point Rea1 FFТs algorithm. If we add ир the number of rea1 mu1tiplica
tions and rea1 additions required Ьу the algorithm's N-point complex FFТ,
plus those required Ьу Eq, (10 42) to get Ха(т), and those required Ьу Eq.
(10-44) to get Хь(т), the Two N Point Rea1 FFТs algorithm requires
two N Point Real FFrs algorithm..... 2N .10g 2 N + N rea1 multiplications, and (10 51)
3N. 10g2N + 2N real additions.
(10-51')
t Пlе complex addition (a+jb) + (c+jd) = (а+с) + j(b+d) requires two real additions, А complex
multiplication (a+jb) . (c+jd) = ac bd + j(ad+bc) requires two real additions and four real
multi plications,
Efflcieпtly Perforтiпg the FFТ о' ReQI Sequeпces 419
Equations (1O 51) and (10 51') assume that we're calculating only the first
N /2 independent elements оЕ Ха(m) and Хь(m). Тhe single N term in Eq,
(10-51) accounts for the N /2 divide Ьу 2 operations in Eq, (10-42) and the
N /2 divide Ьу 2 operations in Eq. (1O 44).
ОК, now we сап find out how efficient the Two N-Point Real FFIs algo
rithm is compared to two separate complex N point radix 2 FFfs, Тhis
comparison, however, depends оп the hardware used for the calculations.
If our arithmetic hardware takes mаnу more clock cycles to perform а
multip1ication than an addition, then the difference between multip1ica-
tions in Eqs, (1O 50) and (1O 51) is the most important comparison, In this
case, the percentage gain in computational saving оЕ the Two N-Point Real
FFTs algorithm relative to two separate N point complex FFIs is the dif-
ference in their necessary multiplications over the nиmber оЕ multiplica
tions needed for two separate N point complex FFIs, or
4N .log2 N (2N .10g2 N + N) .100%:::: 2 .log2 N 1 .100%
4N .log2 N 4 .log2 N
(1 О-52)
The computational (multiplications only) saving from Eq, (10-52) is plot
ted as the top curve оЕ Figure 10 18. In terrns оЕ multiplications, for N?::32,
% Computational saving 01 the Two N-point Aeal FFТs algorithm
50
48
48
44
%
42 .
40
38
36 ..
з4
100 1 102 Э 4 5 N
10 10 10 10
Flgure \0- \8 Computotlono\ soving of the Two N-Point Reol FFТs olgorithm
over thot of МО seporote N-point complex FFТs, Тhe top сшуе
indicates the soving when only multipllcotions ore consldered,
Тhe bottom сшуе is the soving when both odditions ond
multlplicotlons ore used in the compQrison,
420 DlgitQI SigпQI Processlпg Tricks
the Two N-Point Real FFfs algorithm saves us over 45 percent in compu
tational workload compared to two separate N point complex FFfs,
For hardware using high-speed multiplier integrated circuits, multipli-
cation and addition can take rougbly equivalent clock cycles. Тhis makes
addition operations just as important and Нmе consuming as multiplica
tions, Thus the difference between those combined arithmetic operations
in Eqs. (1O 50) plus (10 50') and Eqs, (10-51) plus (10 51') is the appropri
ate comparison. In this case, the percentage gain in computational saving
of our algorithm over two FFfs is their total arithmetic operational differ-
ence over the total arithmetic operations in two separate N-point complex
FFfs, or
(4N .log 2 N +6N 'log 2 N) (2N .log 2 N + N + 3N 'log 2 N +2N) .100%
4N ,1og 2 N + 6N ,1og 2 N
= 5 ,1og 2 N 3 .100% ,
10.10g 2 N
(10 53)
The full computational (multiplications and additions) saving from Eq.
(10-53) is plotted as the bottom curve of Figure 10-18, ОК, that con-
cludes our discussion and illustration of how а single N-point complex
FFT сап Ье used to transform two separate N-point real input data
sequences.
10.5.2 Performing а 2N-Point Real FFТ
Similar to the scheme above where two separate N point real data
sequences are transformed using а single N-point FFT, а technique
exists where а 2N point real sequence can Ье transformed with а single
complex N point FFT. This 2N-Point Real FFT algorithm, whose deriva-
tion is also described in the literature, requires that the 2N-sample real
input sequence Ье separated into two parts[20,21]. Not broken in two,
but l.lnzipped separating the even and odd sequence samples, The N
even-indexed input samples are loaded into the real part of а complex
N-point input sequence х(п), Likewise, the input's N odd-indexed sam-
ples are loaded into x(п)'s imaginary parts, То Шustrаtе this process,
let's say we have а 2N-sample real input data sequence а(п) where
О:::; 11 :::; 2N 1. We want a(п)'s 2N-point transform Ха(т). Loading a(п)'s
odd / even sequence values appropriately into an N point complex
FFT's input sequence, х(п),
Efflcleпtly Perforтlпg the FFТ of Rвol Sвquвnces 421
х(О) = а(О) + ja(l) ,
х(l) = а(2) + jа(З) ,
х(2) = а(4) + ja(5) ,
x(N l) = a(т 2) + ja(2N l) ,
(10-54)
Applying the N сотрlех values in Eq. (10-54) to an N-point сотрlех РП,
we'll get an FFf output Х(т) = Xr(m) + jXi(m), where т goes from О to N 1.
То extract the desired 2N Point Real FFf algorithm output Х.(т) =X.,real(т)
+jX.,imas<m) from Х(т), let's define the following relationships
Х:(т) = Xr(m) + r(N т) ,
(1O 55)
X;(m)= Xr(m) r(N m) ,
(1O 56)
Х + ( ) Xi(m) + Xi(N т) d
i т ,an
2
(10 57)
X;(m)= Xi(m) i(N m) ,
(1 О-58)
Тhe values resu1ting from Eqs. (10-55) through (1O 58) are, then, used as
factors in the fol1owing expressions to obtain the real and imaginary parts
of our finаl Х.(т):
X.,rea1 (т) = Х:(т) + cos( ТC;:; ), xt(m) Sin( ТC;:; ), х; (т),
(10-59)
and
X.,imag(m) = Х;(т) sin( ТC;:; ), xt(m) cos( тс;:; ). Х;(т) ,
(10-60)
Remember now, the original а(п) input index п goes from О to 2N 1, and
our N point РП output index т goes from О to N l. We аррlу 2N real
input time-domain samples to this algorithm and get back N сотрlех
frequency-domain samples representing the first ha1f of the equivalent
2N point сотрlех FFf, Х.(О) through X.(N l). Because this algorithm's
422 DigitQI SigпQI Processiпg Tricks
Unzip Ihe
а(п) 2N-poinl (еа' а(п) х(п)
sequence and
e.tabIi.h the
N-poinl complex
х(п) sequance,
Calculate the
N poinl
complex FFТ о!
х(п) 10 get Х(m).
Ca/culale the !ои,
Х(m) N po/nl .equence.
Х;(m), Х;(m),
X (m), and Х; (m).
Calculale the !ina!
N-poinl complex Х.(m)
.equence Х.(m) =
Х.,,.., (m)
+ jX .,,,,.g(m),
Flgure 10-19 ComputatlonQI flow of the 2N-Point Real FFТ Qlgorlthm,
а(п) input is constrained to Ье real, Xa(N) through Xa(2N 1) are merely
the complex conjugates of their Ха(О) through Xa(N l) counterparts and
need not Ье calculated. То help us keep аН of this straight, Figure 1O 19
depicts the computational steps of the 2N-Point Real FFТ algorithm,
То demonstrate this process Ьу way of example, let's apply the 8 point
data sequence from Eq. (1O 46) to the 2N-Point Real FFT algorithm.
Partitioning those Eq. (10-46) samples as dictated Ьу Eq, (1O 54), we have
our new FFТ input sequence:
х(О) = 0.3535 + j 0.3535,
х(l) = 0.6464 + j 1,0607,
х(2) = 0,3535 j 1,0607,
х(3) = 1.3535 j 0.3535,
(1O 61)
With N = 4 in this example, taking the 4 point FFT of the complex
sequence in Eq. (1O 61) we get
Х,(т) Xj(m)
J, J,
Х(т) = 0,0000 + j 0,0000 ..... т = О lenn
+ 1.4142 j 0.5857 ..... т = 1 lenn
+ 1.4141 j 1.4141 t-----т =2 term
1.4142 + j 3.4141 .....т=зlеnn, (10-62)
Using these values, we now get the intermediate factors from Eqs.
(1O 55) through (1O 58), Calculating our first Х;(О) value, again we're
reminded that Х(т) is periodic with а period N, so Х(4) = Х(О), and
Х;(О) = [Х, (О) + Х, (0)]/2 = О, Continuing to use Eqs. (10 55) through (1(}58),
х:(о)=о, х;(о)=о, Х((О) =0, Xi(O)=O,
Х:(1) =0, Х;(l) = 1.4142, Xt(l) = 1.4142, Xi(1)= 1.9999,
Х:(2) = 1.4141, Х;(2)=0, X((2)= 1.4144, Xi(2) =0,
Х: (3) = О, Х;(3) = 1.4142, Х((3) = 1.4142, Xi(3) = 1.9999. (1O 63)
E{fIc/eпtfy Perform/пg the FFТ of Rвa/ Sвqueпces 423
Using the intermediate values from Eq, (10 63) in Eqs. (1O 59) and (1O 60),
( п.о ) ( п.о )
Xa.reai (О) '" (О) +сов "'"'4 . (О) Вin "'"'4 . (О)
( п.о ) ( n'О )
Xa,irnag(O) = (О) Вin "'"'4 . (О) сов "'"'4 . (О)
( n'1 ) ( n'1 )
X a ,reai(l) = (О) + сов """'4 .(1,4142) Bin"'"4 '(1,4142)
( n'1 ) ( n'1 )
Xa,imag(1)=( l,9999) sin"'"4 '(1,4142) COB""4 .(1,4142)
( n'2 ) ( п'2 )
X a ,reai(2) = (1,4141) + сов "'"'4 .( 1.4144) Bin"'"'4 .(0)
, ( n'2 ) ( n'2 )
Xa,imag(2)=(0) sm "'"'4 .( 1.4144) COB "'"'4 .(0)
( n.з ) ; (1t.з )
X a ,reai(3) = (О)+сов "'"'4 '(1.4142) B \"'"'4 .( 1,4142)
( n.з ) ( n.з )
Xa,imag(3)=(l,9999) sin "'"'4 .(1,4142) COB '4 .( 1,4142)
(10-64)
Evaluating the Вinе and СОВinе terms in Eq, (10 64),
Xa,reai(O) = (О) + (1). (О) (ОНО) '" О,
xa,irnag(O) = (О) (о). (О) (1НО) = О,
X a ,reai(l) = (О) + (0,7071). (1,4142) (0,7071)' (1.4142) = О,
X a ,imag(l) '" Н.9999) (0.7071). (1.4142) (0,7071)' (1,4142) = 3.9999,
X a ,reai(2) = (1,4141) + (о). Н.4144) (1). (О) = 1,4141,
X a ,imag(2) = (О) (1). Н,4144) (о). (О) = 1.4144,
X a ,reai(3) = (О) + ( O,7071). (1,4142) (0,7071)' Н,4142) = О, and
X a ,irnag(3) = (1.9999) (0,7071)' (1,4142) ( 0.7071). Н,4142) = О ,(1O 65)
Combining the results of the terms in Eq, (1O 65), we have our final cor-
rect answer of
Ха(О) = Xa,reai(O) +jXa,imag(O) = О + jO = О LO°,
Х а (l) = X a ,reai(l) +jX a ,imag(l) = О j3,999 = 4 L 90°,
Ха(2) = X a ,reai(2) +jX a ,imagC2) = 1,4141 + j1.4144 = 2 L45°, and
Х а (3) = X a ,reai(3) +jX a ,imag(3) = О + jO = О LO°, (10-66)
424 Dlgltal Slgnal Processlng Trlcks
After going through аН the steps required Ьу Eqs. (1O 55) through
(1O 60), the reader might question the efficiency of this 2N Point Real
FFТ algorithm. Using the same process as the аЬоуе Тwo N-Point Real
FFTs algorithm analysis, let's show that the 2N Point Real FFТ algo
rithm does provide some modest computational saving. First, we know
that а single 2N Point radix-2 FFT has (2N /2) .log22N = N, (log2N+1)
butterf1ies and requires
2N-point complex FFТ ....
4N. (1og:fl+1) real mu1tiplications
(1O 67)
and
6N. (1og2N+1) real additions.
(1(),,67')
If we add ир the number of real multiplications and real additions
required Ьу the algorithm's N point complex FFТ, plus those required Ьу
Eqs_ (1O 55) through (10-58) and those required Ьу Eqs. (10 59) and
(10 60), the complete 2N-Point Real FFf algorithm requires
2N-Point Real FFТ algOrithm.... 2N .log2N + 8N real multiplications
(10-68)
and
3N.log 2 N + 8N real additions.
(1(),,68')
ОК, using the same hardware considerations (mu1tiplications only) we
used to arrive at Eq. (10-52), the percentage gain in mu1tiplication saving
of the 2N-Point Real FFТ algorithm relative to а 2N point complex FFТ is
4N. (10g2 N + 1) (2N .log2 N + 8N) .100%
4N . (log2 N +1)
= 2N .log2 N + 2N N .log2 N 4N .1000;'
2N .log2 N +2N
= log2 N 2 .100%.
2.1og 2 N+2
(1(),,69)
The computational (mu1tiplications only) saving from Eq. (10-69) is plot
ted as the bottom curve of Figure 1 (),,20. In terms of mu1tiplications, the
2N-Point Real FFТ algorithm provides а saving of >30% when 128 or
whenever we transform input data sequences whose lengths are 56.
CQ/cu/Qtiпg the /пverse FFТ Us/пg the ForwQrd FFТ 425
% 20
to
100
% ComputatJonal savJng 01 the 2N-point Ава' FFТ algorithm
50
105 N
40
за
10
О. .
10'
103
10'
102
Figure 10-20 ComputationQ/ saving of the 2N-Polnt Real FFТ algorithm over
that of а slпgle 2N-point сотр/ех FFТ. The top curve is the
savlng when both addltions and multiplications are used 'п the
comparison, The bottom curve Indlcates the savlng when only
mu/tlpllcations are considered,
Again, for hardware using high-speed mu1tipliers, we consider both
multip1ication and addition operations. The difference between those сот-
bined arithmetic operations in Eqs. (10 7) plus (10-67') and Eqs, (10-68)
plus (10-68') is the appropriate comparison. In this case, the percentage
gain in computational saving of our algorithm is
4N . (log 2 N + 1)+ 6N . (log 2 N + 1) (2N .log 2 N + 8N + 3N .log 2 N +8N) .100%
4N ,(log 2 N +1)+6N .(log 2 N +1)
= 10. (log 2 N + 1) 5 .log 2 N 16 .100%
10. (log 2 N + 1)
= 5 .log 2 N 6 .100%.
10. (log 2 N + 1)
(10-70)
The full computational (multiplications and additions) saving from Eq,
(10-70) is plotted as а function of N in the top curve of Figure 10-20,
10.6 Calculating the Inverse FFT Using the
Forward FFТ
There are many signal processing applications where the capability to
perform the inverse FFТ is necessary, тhis can Ье а problem if available
426 DIg'tQ/ SlgпQ/ Processiпg Trlcks
hardware or software routines have the capability to perform only the for
ward FFТ. Fortunately, there are two s1ick ways to perform the inverse FFТ
using the forward FFТ algorithm,
10.6.1 Flrst Inverse FFT Method
Тhe first inverse FFТ calculation scheme is implemented following the
processes shown in Figure 1O 21, То see how this works, consider the
expressions for the forward and inverse DFТs:
Forward DFf ....
N l
Х(т) = Lx(n)e j21tnт/ N ,
n=О
(1O 71 )
and
lnverse DFf ....
1 N l
х(п) = N LX(m)ej27tmn/N ,
т=О
(1O 72)
То reiterate our goal, we want to use the process in Eq, (1O 71) to imple
ment Eq, (1O 72),
Тhe first step оЕ our approach is to use complex conjugation, Remember,
conjugation (represented Ьу the superscript .. symbol) is the reversal оЕ the
sign оЕ а complex number's imaginary exponent if х = e jf2J , then, х" = r jf2J ,
50, as а first step, we take the complex conjugate оЕ both sides оЕ Eq,
(10-72) to give us
[ ..
N l
х" (п) = x(m)ej21tтn/N]
(10 73)
Forward
X (
+ N Х lmag(п)
1
:::: I
1
Flgure 10-21 Processlng dlagram of first Inverse FFТ cQlculatlon method,
Ca/cu/at/пg the /пverse FFТ Us/пg the Forward FFТ 427
Onе of the properties of complex numbers, discussed in Appendix А, is
that the conjugate of а product is equal to the product of the conjugates;
that is, if с = аЬ, then с* = (аЬ)* = а*Ь*, Using this fact, we сап show that the
conjugate of the right side of Eq. (1O 73) is given Ьу
N l
х* (п) = LX(m)* (ej21tmn/N)
т=О
N l
= LX(m)* e j21tmn/N
т=О
(1O 74)
Hold оп, we're almost there, Notice the similarity of Eq. (1O 74) to our orig-
inal forward ОРТ expression Eq, (1O 71), If we perform а forward ОРТ оп
the conjugate of the Х(т) in Eq, (1O 74) and divide the results Ьу N, we get
the conjugate of our desired time samples х(п). Taking the conjugate of
both sides of Eq. (1(),,74), we get а more straightforward expression for х(п):
[ N l ] *
х(п) = x(m)*e j21tmn/N
(1O 75)
50, to get the inverse FFf of а sequence Х(т) using the first inverse FFТ
algorithm,
Step 1: Conjugate the Х(т) input sequence.
Step 2: Calculate the forward FFf of the conjugated sequence.
Step 3: Conjugate the forward FFf's resиlts,
Step 4: Oivide each term of the conjugated results Ьу N to get х(п),
10.6.2 Second Inverse FFT Method
The second inverse FPТ calcиlation technique is implemented following
the interesting data flow shown in Figure 1O 22, In this clever inverse FFf
scheme, we don't bother with conjugation. Instead, we merely swap the
real and imaginary parts of sequences of complex data[22] , То see why
this process works, let's look at the inverse OFТ equation again while sep
arating the input Х(т) term into its real and imaginary parts and remem
bering that &0 = cos(e) + jsin(e):
.
428 Digital Sigпal Processlпg Tricks
Inverse DFr
1 N l
х(п) = N LX(т)ej2I1mпIN
т=О
1 N l ,
= N L [X real (т) + jXimag(т)]
т=О
. [cos(21tтn / N) + j sin(21tтn / N)] .
(1O 76)
Multiplying the complex terms in Eq. (10-76) gives us
1 N l
х(п) = N L[Xreal(т)cos(21tтn/ N) X imag (т)sin(21tтn/ N)]
т=О
+ j[X real (т)sin(2xтn/ N) + Ximag(т) cos(21tтn / N)]. (10-77)
Equation (1D--77) is the general expression for the inverse DFf, and we'll
now quickly show that the process in Figure 10-22 implements this equa-
tion. With Х(т) = Xreal(т) + jXimag(т) and, swapping these tenns,
Xswap(т) = Ximag(т) + jXrea1(т) .
(1O 78)
Тhe forward DFf of our Xswaiт) is
N l
Forward DFf L[Ximag(т) + jXreal(т)]
п=О
. [cos(21tтn / N) j sin(21tтn / N)]
(10-79)
Forward
FFТ
Х reaI (n)
Ximag(т)
.", ximag(n)
Flgure 10-22 Processlng diagram of second Inverse FFТ calculatlon method,
Fast FFТ Averagiпg 429
Multiplying the complex terms in Eq, (10-79) gives us
N l
Forward DFf L [Ximag(m) cos(21tmn/ N) + X real (m)sin(21tmn / N)]
п=О
+ j[X real (m)cos(21tmn / N) X imag (m)sin(21tmn/ N)] . (10 80)
Swapping the rea1 and imaginary parts оЕ the results оЕ this Eorward ОРТ
gives us what we're after:
N l
Forward DFf.w.. )Xreal(m)cos(21tmn/ N) X imag (m)sin(21tmn/ N)]
п=О
+ j[Ximag(m)cos(21tmn/ N)+ X real (m)sin(21tmn/ N)] (10 81)
If we divide Eq, (10-81) Ьу N it would Ье equal to the inverse DPТ expres-
sion in Eq. (1O 77), and that's what we set out to show, То reiterate, we cal
culate the inverse FFТ оЕ а sequence Х(т) using this second inverse FPТ
algorithm in Figure 10-22:
Step 1: Swap the real and imaginary parts of the Х(т) input sequence,
Step 2: Calculate the forward РП оЕ the swapped sequence.
Step з: Swap the real and imaginary parts оЕ the forward FFf's results.
Step 4: Divide each term оЕ the swapped sequence Ьу N to get х(п).
10.7 Fast FFТ Averaglng
Section 8.3 discussed the integration gain possible when averaging
multiple FFT outputs to enhance signal detection sensitivity. Well,
there's а smart way to do this iE we recall the linearity property оЕ the
ОРТ (which of course applies to the FPТ) introduced in Section 3.3. If an
input sequence x1(n) has an FFf оЕ X1(m) and another input sequence
х 2 (п) has an FFf оЕ Х 2 (т), then the FFТ оЕ the sum оЕ these sequences
xsum(n) = х 1 (п) + х 2 (п) is the sum оЕ the individual FFfs, or
Xsum(m) = Х 1 (т) + Х 2 (т) .
(1O 82)
430 D/gita/ S/gпal Process/пg Tr/cks
50, if we want to average multiple FFТ outputs, we сап save considerable
processing effort Ьу averaging the individual PFТ input sample sequences
(frames), first, and then take а single FFT, 5ау, for ехаmрlе, that we
wanted to average 20 FFТs to improve our PFТ output signaHo-nоisе
ratio, Instead of taking 20 FFTs of 20 frames of input signal data, we
should average the 20 frames of input data, first, and then take а single
РП of that average. Тhis avoids the number crunching necessary for 19
FFrs. Ву the way, for this technique to improve an FFТ's signal-detection
sen.sitivity, the original signal sampling must meet the criterion of coher-
ent integration as described in 5ection 3.12,
That's the good news. Тhe bad news is that this technique only works
for periodic signals whose initial samples, xk(O), are collected synchro-
nously, Тhat iБ, the beginning of each new block of time-domain data is
col1ected at а constant phase relative to the periodic signal.
10.8 Simplified FIR Filter Structure
If we need to implement an FIR digital filter using the standard structure
in Figure 10-23(а), there's а way to simplify the necessary calculations
when the filter has an odd number of taps. Let's 100k at the top of Figure
10-23(а) example where the 5-tap filter coefficients are h(O) through h(4)
and the у(п) output is given Ьу
у(п) = h(4)x(n 4) + h(3)x(n 3) + h(2)x(n 2) + h(l)x(n 1) + h(O)x(n) , (1O 83)
If the FIR filter's coefficients are symmetrical, we сап reduce the number
of necessary multipliers; that is, if h(4) = h(O), and h(3) = h(l), we сап
implement Eq, (1O 83) Ьу
у(п) = h(4)[x(n 4)+x(n)] + h(з)[х(п 3)+х(п 1)] + h(2)' x(п 2) , (1O 84)
where only three multiplications are necessary, as shown at the bottom of
Figure 1O 23(а), In our 5 tap filter case, we've eliminated two multipliers
at the expense of implementing two additional adders,
In the general case of symmetrical coefficient FIR filters with 5 taps, we
can trade (5 1)/2 mu1tipliers for (5 1)/2 adders when 5 is an odd num
ber. 50, in the case of an odd number of taps, we need perform only
(5 1)/2 + 1 multiplications for each filter output sample, For an even
number of symmetrical taps as shown in Figure 10-23(Ь), the saving
afforded Ьу this technique reduces the necessary number of multiplica-
tions to 5/2, For the half-band filters discussed in 5ection 5,7, with their
Simplified FIR Filter Structure 431
11(0) = 11(4)
у(п)
11(0)
11(0)
.
I
(а)
Х(п)
У(п)
11(0) = 11(5)
у(п)
...
Х(п)
11(0)
11(2) I1(З)
11(0) ) i i 4) 11(5)
.' I!,
(Ь)
х(п)
Flgure 10-23 Slmplifled FIR fllter Implementotlons: (о) conventlonol ond
slmpllfled, structures of оп FIR fllter with оп odd number of tops;
(Ь) conventlono\ ond s\mplified structures of оп FIR filter wlth оп
even number of tops,
432 Dlgftal Slgпal Processlпg Trlcks
alternating zero valued coefficients, the simplified FIR structure in Figure
1O 23(b) allows us to get away with only (5+1)/4 + 1 multiplications for
each filter output sample when 5 is odd and the first filter coefficient h(O)
is not zero.
We always benefit whenever we can exchange multipliers for adders.
Because multiplication often takes а longer time to perform than addition,
this symmetrical FIR filter simplification scheme тау speed filter calcula-
tions performed in software, For а hardware FIR filter, this scheme сап
either reduce the number of necessary multiplier circuits or increase the
effective number of taps for а given number of available hardware multi
pliers. Of course, whenever we increase the effective number of filter taps,
we improve our filter performance for а given input signal sample rate.
10.9 Accurate A/D Converter Testing Technique
The manufacturers of A/D converters have recently begun to take advan-
tage of digital signal processing techniques to facilitate the testing of their
products. А traditional test method involves applying а sinusoidal analog
voltage to an A/D converter and using the FFТ to obtain spectrum of the
digitized samples. Converter dynamic range, missing bits, harmonic dis
tortion, and other nonlinearities сan Ье characterized Ьу analyzing the
spectral content of the converter output. These nonlinearities are easy to
recognize because they show ир as spurious spectral components and
increased background noise levels in the FFТ spectra,
То enhance the accuracy of the spectral measurements, window func-
tions were originally used оп the time-domain converter output samples
to reduce the spectralleakage inherent in the FFТ. This was fine until the
advent of 12- and 14 bit A/D converters, These converters have dynamic
ranges so large that their small nonlinearities, evident in their spectra,
were being swamped Ьу the sidelobe levels of even the best window func-
tions. (From Figure 9 4 we know that а 14-bit A/D converter сan have an
SNR ratio of well over 80 dB.) The clever technique that circumvents this
problem is to use an analog sinusoidal input voltage whose frequency is
an integral fraction of the A/D converter's sample frequency as shown in
Figиre 10-24(а), That frequency is mfJN where т is an integer, fs is the
sample frequency, and N is the FFТ size. Figure 1O 24(a) shows the х(п)
time domain output of an ideal A/D converter under the condition that
its analog input is а sinewave having ехасНу eight cycles over 128 output
samples. In this case, the input frequency normalized to the sample rate fs
is 81./128 Hz. Recall, from Chapter 3, that the expression mfslN defined
the analysis frequencies, or bin centers, of the DFТ; and а DFТ input
whose frequency is at а bin center results in по leakage even without the
Accurate А/О Coпverter Testiпg Techпique 433
(а)
Б Х(П)
1 ", 7'\ " " (, . " " ,
.. .. .. .. .. . .. ..
O,5. Jl Jl Jl Jl Jl Jl Jl
° I l \ f f \ f f \ f \ t \ f \
...(),5 ii ii '; '; ii ii ii \7
" " " , " " ., .,
у у у у у у
(Ь)
1 F(т) in dB
о .
20 I
: I
100 ,'IJ'."' 'IJ"I!'''I!!' I!I' '' '' ''''' 'I!''IJ''I!''I!'''I!!''I!'I!''I!"I!',IJ'
О 3 6 9 12 15 18 21 24 27 30 зз 36 39 42 45 48 51 54 57 60 63 т
Figure 10-24 Ideal A/D converter output whose input is оп analog 8f,l128 Hz
slnusold: (а) time-domain samples; (Ь) frequency-domain
spectrum In dB.
use оЕ а window fиncHon, Another way to look at this situation is to real-
ize that the analog mf/N frequency sinusoid wШ have exactly т complete
cycles over the N PFТ input samples, ав indicated Ьу Figure 3-7(Ь) in
Chapter 3,
The first ha1f оЕ а 128-point FFТ оЕ х(п) is shown in the logarithmic plot
in Figure 10-24(Ь) where the input tone liев exact1y at the т = 8 Ьin center,
and DFТ leakage has Ьееп avoided altogether. Specifically, if the sample
rate were 1 MHz, then the A/D's input analog tone would have to Ье
exact1y 8. 106/128 = 62,5 kHz. То implement this scheme, we need to
ensure that the analog test generator Ье synchronized, exact1y, with the
A/D converter's clock frequency of!. Hz, Achieving this synchronization
is why this A/D converter testing procedure is referred to ав coherent sam-
pling[ 25]. The analog signal generator and the A/D clock generator
providingf s mиst not drift in frequency relative to еаш other they must
remain coherent. (We must take care here from а semantic viewpoint
because the quadrature sampling schemes described in Sections 7,1 and
7.2 are also вотеНтев called coherent sampling, but they are unrelated to
this A/D converter testing procedure,)
Ав it turns out, воте values оЕ т are more advantageous than oth-
ers. Notice in Figure 10-24(а), when т = 8, only пinе different атрН-
tude values are put out Ьу the A/D converter. Those values are
434 Digital Sigпal Processiпg Trlcks
1 + n) { {., j.1 . (, i;
0,5 . . , . . \ . , . ' . i' .. i
· \ . \ . '. I i I \ ,.
о V · 1 ; I \ · \ I · i i · \
i . \ .... f i 1 \ .. / i n
.5 11 lJ i'T,J 1i ,;
, iJ 1 " 1 ] "
1 ,. ,:
Flgure 10-25 Seveп-cycle slnusoldol А/О converter output,
(а)
l Х(П)
1f ("l "1 \"1\ ("l "I ,
.. .. .. .. .. .. .. .,
0,5 " f 1 i' f 1 l' f 1 f 1 J i
о .................................... . ....................... ...........................J........
i \ I \ ' I \ I \ I \ I \ I \ n
.............. ..
--0,5 l J l J \ J l J l J l J l J l J
, , \ I '\ I , I \ I , , \ I \ I
1 у у у
Dropped blts
(Ь)
} F(т) in dB
О .
10
20
O · :.. .. I.:i IШii i :11: П.. Щ 11: 11 i7 i il! i I i l11i1
50
о 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 т
Figure 10-26 Nonldeol А/О converter output showing several dropped blts:
(о) tlme--domoin somples; (Ь) frequency-domoin spectrum In dB,
repeated over and over, As shown in Figure 1O 25, when т = 7, we exer
cise тапу more than nine different A/D output values. Because it's
best to test aS тапу A/D output binary words as possible, in practice,
users of this A/D testing scheme have found that making т an odd
prime number (3, 5, 7, 11, etc.) minimizes the number of redundant
A/D output word values,
Fost F/R F//ter/пg Usiпg the FFТ 435
Figure 10 26(a) illиstrates an extreme example of nonlinear A/D con
verter operation with several discrete output samples having dropped
bits in the time domain х(п) with m = 8. The FFТ of this distorted х(п) is
shown in Figиre 10 26(b) where we сап see the greatly increased back
ground noise level due to the A/D converter's nonlinearities compared to
Figиre 10 24(b).
То fиlly characterize the dynamic performance of ап A/D converter,
we' d need to perform this testing technique at тanу different input fre-
quencies and amplitudes.+ In addition, applying two analog tones to the
A/D converter's input is often done to quantify the intermodulation dis-
tortion performance of а converter, which, in tиrn, characterizes the соп-
verter's dynamic range. In doing so, both input tones must comply with
the mfj N restriction. Тhe key issue here is that, when апу input frequency
is mfjN, we сап take fиll advantage of the FFТ's processing sensitivity
whi1e completely avoiding spectralleakage.
10.1 О Fast FIR Filtering Using the FFT
Whi1e contemplating the convolution relationships in Eq, (5 31) and
Figиre 5 41, digital signal processing practitioners realized that convolu
tion could sometimes Ье performed more efficient1y using FFТ algorithms
than it could Ье using the direct convolution method[26,27], This FFT
based convolution scheme, called fast convolution, is diagrammed in
Figиre 10-27. The standard convolution equation, for ап M tap FIR filter,
given in Eq. (5 6) is repeated here for reference as
M l
у(п) = L h(k)x(n k) = h(k) * х(п)
k=O
(10-85)
where h(k) is the impulse response sequence (coefficients) of the FIR filter
and the "*" symbol indicates convolution. It has Ьееп shown that, when the
final у(п) output sequence has а length greater than 30, the process in Figиre
10 27 requires fewer multiplications than implementing the convolution
expression in Eq. (1O 85) directly. Consequently, this fast convolution tech-
nique is а very powerfu1 signal processing tool, particи1arly, when used for
t The ana!og sinewave applied to аn А/О converter must, of course, Ье as рите as possible,
Anу distortion inherent in the analog signa! will show ир in the finа! FFТ output and cou!d
Ье mistaken for А/О nonlinearity,
436 D/g/tol Sigпal Process/пg Tr/cks
FFТ
r? Щт). т) I ;""" h(k).x(n)
FFТ Н(т) FFТ
Flgure 10-27 Processlng dlagram of fast convolutlon,
digital filtering. Very efficient FIR filters can Ье designed using this tech
nique because, if their impиlse response h(k) is constant, then we don't have
to bother recalcиlating Н(т) each time а new х(п) sequence is filtered. In
this case, the Н(т) sequence сап Ье precalcиlated and stored in memory,
Тhe necessary forward and inverse FFТ sizes must, of course, Ье equal
and are dependent оп the length of the original h(k) and х(п) sequences.
Recall from Eq. (5 29) that, if h(k) is of length Р and х(п) is of length Q, the
length of the final у(п) sequence will Ье (P+Q---1), For valid resиlts from this
fast convolution technique, the forward and inverse РРТ sizes must Ье
equal and greater than (P+Q---1). This means that h(k) and х(п) mиst both Ье
padded (or stиffed) with zero valued samples at the end of their respective
sequences, to make their lengths identical and greater than (P+Q---1). This
zero padding will not invalidate the fast convolution results. 50, to use fast
convolution, we must choose an N point FFТ size such that N ;:: (P+Q---1)
and zero pad h(k) and х(п), so that they have new lengths equal to N.
An interesting aspect of fast convolution, from а hardware standpoint,
is that the РРТ indexing bit reversal problem discussed in 5ections 4.5 and
4,6 is not an issue here. If the identical РРТ structиres used in Figure 10-27
result in Х(т) and Н(т) having bit reversed indices, the multiplication
сап still Ье performed directly оп the scrambled Н(т) and Х(т)
seqиences. Then, an appropriate inverse РРТ structиre сап Ье used that
expects bit-reversed input data. That inverse РРТ then provides an output
у(п) whose data index is in the сопесt order!
10. 11 Calculation of Sines and Cosines
of Consecutive Angles
There are times in digital signal processing when we need our software to
cakulate lots of sine and cosine values, particularly in implementing cer
tain FFT algorithms[28,29], Because trigonometric calculations are time
consuming to perform, а clever idea has been used to calculate the sines
Calcu/atloп of Slпes aпd Coslпes of Coпsecutlve Aпgles 43i
and cosines of consecutive angles without having to actually call ироп
standard trigonometric functions. t Illustrating this scheme Ьу way оЕ
example, let's say we want to ca1culate the sines and cosines of all angles
from 0° to 900 in 1 degree increments, Instead of performing those 91 sine
and 91 cosine trigonometric operations, we сап use the following identi-
ties found in our trusty math reference book,
sin(A+B) = sin(A). cos(B) + cos(A). sin(B)
(10-86)
and
cos(A+B) = cos(A). cos(B) sin(A). sin(B) ,
(10-87)
to reduce our computational burden. То see how, lets make А = а and
В = па, where а = 1° and n is an integra1 index O n 89. Equations (10-86)
and (10-87) now Ьесоте
sin(a +nа) = sin(a [1+n]) = sin(a). cos(na) + cos(a). sin(na), (10-88)
and
cos(a +nа) = cos(a [1+n]) = cos(a). cos(na) sin(a. sin(na), (1 89)
ОК, here's how we ca1culate the sines and cosines. First, we know the sine
and cosine for the first angle of 0°; i,e., sin(OO) = О, and cosine(OO) = 1. Next
we need to use а standard trigonometric function call to ca1culate and
store, for later use, the sine and cosine оЕ 1 о; that is, sin(l О) = 0.017452 and
cos(l О) =0.999848, Now we're ready to ca1cu1ate the sine and cosine оЕ 2°
Ьу substituting n = 1 and а = 1 О into Eqs, (10-88) and (10-89) giving us
sin(l °[1 + 1]} = sin(l О) . cos(l . 1°) + cos(l О) . sin(l. 1°),
or
sin(2°) = (0.017452). (0.999848) + (0.999848). (0.017452) = 0.034899, (10-90)
and
cos(l °[1+ 1]) = cos(l О) . cos(l. 1 О) sin(l O),sin(l. 1 О),
or
cos(2°) = (0.999848). (0.999848) + (0.017452)' (0.017452) = 0.999391, (10-91)
t We're assuming here that there's insufficient memory space ауаilаЫе to store аН the
required sine and cosine values for later recaH,
438 Digital Sigпa/ Processiпg Tr/cks
Веса use we've already calculated sin(l О) and cos(l О), Eqs, (1O 90) and
(10-91) each required on1y two multiplies and an add, No trigonometric
function needed to Ье called Ьу software.
Next we're аЫе to calculate the sine and cosine of 3° Ьу substituting
п = 2 and а = 1° in Eqs, (1O 88) and (10-89); that is,
sin(l °[1 +2]) = sin(l О) . cos(2 .1 О) + cos(l О) . sin(2.1 О),
or
sin(3°) = (0,017452)' (0.999391) + (0.999848)' (0,034899) = O,052336,(1O 92)
and
cos(l °[1+2]) = cos(l О) . cos(2.1 О) sin(l о). sin(2.1 О),
or
cos(3°) = (0.999848). (0.999391) + (0.017452). (0,034899) = 0.998629, (1O 93)
Again, because we previously calculated sin(2°) and cos(2°), Eqs, (10-92)
and (1O 93) on1y require us to perform four multiplications and two addi
tions. Тhe pattem of our calculations is clear now. For successive angles,
we rnerely use the sine and cosine of 1° and the sine and cosine values
obtained during the previous angle calcu1ation. In our example, the angle
increment was а= 1°, but it's good to know that Eqs. (10-88) and (10-89)
apply for anу fixed angle increment,
10. 12 Generating Normally Distributed
Random Data
Section 0.4 in Appendix О discusses the normal distribution curve as it
relates to random data. А problem we mау encounter is how to actually
generate random data samples whose distribution follows that normal
curve, There's а slick way to solve this problem using any software
package that сап generate uniformly distributed random data, as most
of them do[30], Figure 10-28 shows our situation pictoriaHy where we
require random data that's distributed normally with а mean (average)
of J.l' and а standard deviation of а', as in Figure 10-28(а), and аН we
have available is а software routine that generates random data that's
uniformly distributed between zero and one as in Figure 10-28(Ь). As it
turns out, there's а principle in advanced probability theory, known as
the Ceпtral Liтit Theorem, that says, when random data from an arbi
trary distribution is summed over М samples, the probability distribu
Geпeratiпg Norma//y Distribиted Raпdom Data 439
Normal probability distribution
Uniform probability distribution
шшш. 1
3cr' 2a' --о' 1-1' +а' +2а' +Зcr'
о
(а)
(Ь)
Flgure 10-28 ProbabIllty dlstributioп fuпctioпs: (а) пormal distributloп with
теап = ', and standard deviation о'; (Ь) uпlform dlstribution
ЬеМееп zero aпd опе,
Probability dis!ribu!ion 01 se! у вит
о
1-I=М12
м
.
Тhis distance is
assumod \о ье \о 60.
Flgure 10-29 ProbabIlity dlstributioп of the summed set of raпdom data
derlved from uпiformly distrlbuted data,
tion о! the sиm begins to approach а normal distribиtion, as М
increases[31,32], In other words, if we generate а set о! N random sam-
ples that are иniformly distribиted between zero and one, we сап begin
adding other sets о! N samples to the first set, As we continиe sиmming
additional sets, the distribиtion о! the N element set о! sиms becomes
more and more пorтal. We сап soиnd impressive and state that "the
sиm becomes asymptotically normal," Experience has shown that, for
practical pиrposes, if we sиm М ;:: 30 times, the sиmmed data distribu-
tion is essential1y normal. With this rиle in mind, we're halfway to solv-
ing oиr problem.
After sиmming М sets of uniformly distribиted samples, the sиmmed
set У ошn will have а distribution like that shown in Figure 10-29, Becaиse
we've sиmmed М sets, the mеan о! У ошn is 11 = Mj2. То determine Ysum's
standard deviation о, we assиme that the six sigma point is eqиal to M I1;
that is
60=M I1'
(1O 94)
440 Digitol Sigпal Processlпg Tricks
That assumption is valid because we know that the probability of an ele-
ment in У оит being greater than М is zero, and the probability of having
а normal data sample at six sigma is one change in б billion, or essen-
НаНу zero. Because Jl = Mj2, then from Eq, (1O 94), Ysum's standard devi
ation is set to
(1= M f1 = M Mj2 =Mj12
б б
(1 O 95)
То convert the У оит data set to our desired data set having а mean of Jl'
and а standard deviation оЕ а',
· subtract Mj2 from each element of У оит to shift its mean to zero,
· next, ensure that ба' is equal to Mj2 Ьу multiplying еаш element in
the shifted data set Ьу 12а ' j М, and
· finaHy, center the new data set about the desired Jl' Ьу adding Jl' to
еаш element of the new data.
ТЪе steps in our algorithm are shown in Figure 10 30. If we саН our
desired normally distributed random data set Ydesired' then the nth element
of that set is described m.athematically as
12а' [( J М ] ,
Ydesired(n)=M' t(k(n) 2 +Jl
(10 9б)
our discussion thus far has had а decidedly software algorithm.ic fla
vor, but hardware designers also occasionally need to generate normally
distributed (Gaussian) random data at high speeds in their designs. For
your hardware designers, reference [33] presents an efficient hardware
design technique to generate normally distributed random data using
fixed point arithmetic integrated circuits.
Gener&le М .еtз
01 r&ndom d&I&,
wilh e&ch заl
contalning N
elements.
Sum the М .el.,
elemenl to'
elemenl, 10 gel
the .ummed заl
у aum containing
N elements,
Subtract м/2
Irom e&ch
elemenl 01
У,Wn'
Muiliply e&ch
elemenl In the
.hifled .еl ьу
12о'/М,
Add 1" 10 each
elemenl 01 Ihe
.hifted d&l&
заl,
Flgure 10-30 Processlng steps requlred to generate normally dlstrlbuted random
data from unlformly dlstrlbuted data,
Refereпces 441
References
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Processing," IЕЕЕ Coттuпicatioпs Magaziпe, January 1980,
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1983,
[3] Harris Semiconductor Corp. "А Digital, 16-Bit, 52 Msps Halfband Filter,"
Microwave Jourпal, September 1993,
[4] Hack, Т, "IQ Sampling Yields Flexible Demodulators," RF Desigп Magaziпe,
April1991,
[5] Pellon, L, Е, "А Double Nyquist Digital Product detector for Quadrature
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[6] Waters, W, М" and Jarrett, В, R. "Bandpass Signal Sampling and Coherent
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November 1982,
(7) Palacherls, А. "DSP-mP Routine Computes Magnitude," EDN, October 26,1989,
[8] Mikami, N" Kobayashi, М" and Yokoyama, У, "А New DSP Oriented
Algorithm for Ca1culation of the Square Root Using а Nonlinear Digital
Filter," IЕЕЕ Traпs, оп Sigпal Processiпg, Vol, 40, No, 7, July 1992,
[9] Lyons, R. G, "Turbocharge Your Graphics Algorithm," ESD: The Electroпic
Systeт Desigп Мagaziпe, October 1988,
[10] Adarns, W. т., and Brady, J, "Magnitude Approximations for Мicroprocessor
Implementation," IЕЕЕ Micro, Vol, 3, No, 5, October 1983,
[11] Eldon, J, "Digita! Correlator Defends Signa! Integrity with Multibit
Precision," Electroпic Desigп, Мау 17, 1984.
[12] Smith, W, W. "DSP Adds Performance to Pulse Compression Radar," DSP
Applicatioпs, October 1993,
[13] Нзrris Semiconductor Corp, HSP50110 Digital Qиadrature Tuner Data Sheet,
File Number 3651, February 1994,
[14] Bingham, с., Godfrey, М" and Tukey, J, "Modern Techniques for Power
Spectrum Estimation," IЕЕЕ Traпs, оп Audio aпd Electroacoust" Vol, AU-15, No,
2, June 1967,
[15] Harris, F. J, "On the Use of Windows for Harmonic Analysis with the Discrete
Fourier Transform," Proceediпgs о/ the IЕЕЕ, Vol, 66, No, 1, January 1978,
[16] Nuttall, А. Н, "Some Windows with Very Good Sidelobe Behavior," IЕЕЕ
Traпs, оп Acoust, Speech, aпd Sigпal Proc" Vol, ASSP 29, No. 1, February 1981,
[17] Сох, R, "Complex-Multiply Code Saves Clocks Cycles," EDN, June 25, 1987,
[18] Rabiner, L, R" and Gold, В, Theory aпd Applicatioп о/ Digital Sigпal Processiпg,
Prentice-Hall, Englewood Cliffs, New Jersey, 1975, рр, 356,
442 Olgltol SigпQI Processlпg Trlcks
[19] Sorenson, Н, V" Jones, О, L" Heideman, М, т., and Burrus, С. 5, "Real-Valued
Fast Fourier Transform Algorithms," IЕЕЕ Trans, оп Acoust, Speech, aпd Sigпa/
Proc" Vol, ASSP 35, No, 6, June 1987,
[20] Сооlеу, J, W., Lewis, Р, А" and We1ch, Р, О, "The Fast Fourier Transform
Algorithm: Programming Considerations in the Ca1culation оЕ Sine, Cosine
and Laplace Transforms," Jourпa/ Souпd Vib" Vol, 12, July 1970,
[21] Бrighаm, Е, О, Тhe Fast Fourier Тransforт and Its Applicatioпs, Prentice-НаН,
Englewood Cliffs, New Jersey, 1974, рр, 167,
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НаН, Englewood Cliffs, New Jersey, 1994, рр, 53,
[23] Coleman, В" Meehan, Р" Reidy, J" and Weeks, Р, "Coherent Sampling Helps
When Specifying OSP А/О Converters," EDN, October 1987,
[24] Ushani, R. "Classical Tests Are Inadequate for Modem Нigh-Speed
Converters," EDN Magazine, Мау 9,1991,
[25] Meehan, Р" and Reidy, J, "FFТ Techniques Give Birth to Digital Spectrum
Analyzer," E/ectronic Design, August 11, 1988, рр. 120,
[26] Stockham, Т. G, "Нigh-speed Convolution and Сопеlаtiоn," in Digita/ Signa/
Processing, Ed, Ьу L, Rabiner and С. Rader, IEEE Press, New Jersey, 1972, рр, 330,
[27] Stockham, Т, G, "Нigh-Speed Convolution and Сопеlаtiоn with Applications
to Digital Fi1tering," Chapter 7 in Digita/ Processing о! Signa/s, Ьу В, Gold et al"
McGraw-НiН, New York, 1969, рр. 203,
[28] DоЬЬе, J, G, G, "Faster FFТs," Dr, Dobb's Jourпa/, February 1995, рр, 125, (Ве
carefиl here, Тhe last equation in Ехаmрlе 5 is incorrect оп page 133 оЕ this
reference, so Ье sure and use our Eq, (10-86) above.)
[291 Crenshaw, J, W. "АН About Foиrier Analysis," Eтbedded Systeтs Prograттing,
October 1994, рр, 70.
[30] Beadle, Е, "Algorithm Converts Random Variables to Normal," EDN
Magazine, Мау 11, 1995,
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McGraw-Нill Book Со" New York, 1961, рр, 142.
[32] Davenport, W. В" Jr" and Root, W, L. Randoт Signa/s and Noise, McGraw-Нill
Book Со" New York, 1958, рр, 81,
[33] Salibrici, В, "Fixed point DSP СЫр Сап Generate Real-time Random Noise,"
EDN Magazine, April29, 1993,
IAPPENDIX А/
The Arithmetic of
Complex Numbers
То understand digital signal processing, we have to get cornfortable using
cornplex numbers, Тhe first step toward this goal is learning to rnаnфи-
late cornplex numbers arithrnetically. Fortunately, we сап take advantage
of our knowledge of real numbers to rnake this job easier. Although the
physical significance оЕ cornplex numbers is discussed in Appendix С, the
following discussion provides the arithrnetic rules governing cornplex
numbers.
А.l Graphical Representation о' Real and
Complex Numbers
То get started, real numbers are those positive or negative numbers we're
used to thinking about in our daily lives, Exarnples of real numbers are
0.3, 2.2, 5.1, etc. Keeping this in rnind, we see how а real nurnber сап Ье
represented Ьу а point оп а one--dirnensional axis, called the real axis, as
shown in Figure A 1.
We сап, in fact, consider that аll real numbers correspond to аН of the
points оп the real axis line оп а one to one basis,
Thls polnt represents the
/ real number 2,2
I 8, I I r I I I ......
7 6 5 4 з 2 1 О 1 2 3 4 5 6 7 Real axis
Flgure д-! The representotlon of о reol number os о polnt оп the one-
dimensionol reol oxis,
443
444 Appeпd/x А: Тhe Ar/fhтef/c о' Сотр/ех Nuтbers
........ " This point represents the
complex number С = R + j/
о
R
Real axis
Flgure А-2 The phasor representatlon of the complex number С = R + }/ оп
the complex plane,
А complex number, unlike а real number, has two parts: а real part and
an imaginary part. Just as а real number сап Ье considered to Ье а point
оп the one-dimensional real axis, а complex number can Ье treated as а
point оп а complex plane as shown in Figure А-2. We'll use this geomet
rical concept to help us understand the arithmetic of complex numbers. t
А.2 Arithmetic Representation 01 Complex Numbers
А complex number С is represented in а number of different ways in the
literature, such as
Rectangular [ОrПI: .... С = R + jI , (A-l)
Trigonometric [ОrПI: .... С = M[COS(0) + jsin(0)] , (A-l')
Exponential [ОrПI: .... C=Mei e , (A-l")
Magnitude and angle [ОrПI: .... C=ML0, (A l"')
Equations (A l") and (А-1''') remind us that the complex number С сап
also Ье considered the Нр of а phasor оп the complex plane, with magni-
tude М, in the direction of 0 degrees relative to the positive real axis as
shown in Figиre A 2. (We'll avoid calling phasor М а vector because the
term vector means different things in different contexts, In linear algebra,
vector is the term used to signify а one-dimensional matrix, on the other
hand, in mechanical engineering and field theory, vectors are used to sig-
nify rnagnitudes and directions, but there are vector operations (scalar or
t The complex plane repre5entation of а complex number i5 sometime5 called an Argand dia
gram named after the French mathematician Jean Robert Argand (1768---1825),
Ar/ff1тetic RepreseпtQt/oп о( Сomр/вх Nuтbers 44f
dot product, and vector or cross product) that don't apply to our definition
оЕ а phasor,) Тhe relationships between the variables in this figure fol1ow
the standard trigonometry оЕ right triangles. Кеер in mind that С is а
complex number, and the variables R, 1, М, and 0 are аН real nиmbers, Тhe
magnitиde оЕ С, sometimes cal1ed the modulus оЕ С, is
M=ICI= R2+12 ,
(А-2)
and, Ьу definition, the phase angle, or argumeпt, оЕ С is the arctangent оЕ
I/ R, or
1 ( 1 )
0=tan R '
(А-3)
Тhe variable 0 in Eq. (A 3) is а general angle term. It сап have dimensions
оЕ degrees or radians. ОЕ course, we сап convert back and forth between
degrees and radians using 1t radians = 180., 50, if 0, is in radians and 0d is
in degrees, then we can convert 0. to degrees Ьу the expression
1800 .
0d=
1t
(A 4)
Likewise, we сап convert 0d to radians Ьу the expression
o = П0 d
r 180
(A 5)
Тhe exponential form оЕ а complex number has an interesting charac
teristic that we need to keep in mind, Whereas only а single expression in
rectangular Еолn сап describe а single complex nиmber, ап infinite nиm
ber оЕ exponential expressions сап describe а single complex nиmber; that
is, while, in the exponential Еолn, а complex number С сап Ье represented
Ьу С = м!,д, it can also Ье represented Ьу
с = Me i " = м!(" + 27tn) ,
(А-б)
where n = ::t1, х2, ::t3, ' . , and 0 is in radians. When 0 is in degrees, Eq.
(А б) is in the form
с = Me i " = Md(0+ пЗ60') ,
(А-7)
446 Appeпdix А: Тhe Arithтetic о' Coтpiex Nuтbers
Equations (А-б) and (А-7) are almost self explanatory, They indicate that
the point оп the complex plane represented Ьу the tip of the phasor С
remains unchanged if we rotate the phasor some integral mu1tigl: of 2х
radians or an integral multiple of 3БО . 50, for example, if С == Ме 20 >, then
с == Md(200) == мd(ЗВОО) == Md(7400) .
(А-В)
The variable о, the angle of the phasor in Figure A 2, need not Ье constant.
We'll often encounter expressions containing а complex sinusoid that
takes the form
с == Md rot ,
(A 9)
Equa tion (А-9) represents а phasor of magnitude М whose angle in Figure
A 2 is increasing linearly with time at а rate of (j) radians each second. If
(j) = 2х, the phasor described Ьу Eq, (A 9) is rotating counterclockwise at
а rate of 2х radians per second ne revolution per second and that's
why (j) is called the radian frequency. In terms of frequency, Eq, (A 9)'s
phasor is rotating counterclockwise at (j) == 2х! radians per second, where
! is the cyclic frequency in cycles per second (Нz), If the cyclic frequency
is! == 10 Hz, the phasor is rotating 20х radians per second. Likewise, the
expression
с = Me jrot
(A 9')
represents а phasor of magnitude М that rotates in а clockwise direction
about the origin of the complex plane at а negative radian frequency of--ro
rаdiалs per second.
А.З Arithmetic Operations о' Complex Numbers
А.З.1 Addition and Subtraction of Complex Numbers
Which of the above forms for С in Eq. (А-l) is the best to use? It depends
оп the arithmetic operation we want to perform. For example, if we're
adding two complex numbers, the rectangu1ar form in Eq. (А -1) is the eas
iest to use. The addition of two complex numbers, С 1 == R 1 + jl 1 and
С 2 == R 2 + jI 2 , is merely the sum of the real parts plus j times the sum of the
imagmary parts as
+ == + + + == + +M+ ,
(А-10)
Arithmet/c Operat/oпs о( Complex Numbers 447
Imaginary
axis И)
Imaginary
axis И)
Real
axis
Real
axis
(а)
(Ь)
Flgure А-З Geometrical representation of the sum of two complex numbers,
Figure А З is а graphical depiction оЕ the Sum оЕ МО complex numbers
using the concept of phasors. Here the sum phasor С 1 + С 2 in Figure А-3(а)
is the new phasor from the beginning of phasor С 1 to the end of phasor С 2
in Figure A 3(b). Remember, the Rs aпd the 1s сап Ье either positive or neg-
ative numbers. Subtracting one complex number from the other is straight
forward as long as we find the differences between the МО real parts and
the мо imaginary parts separately. Thus
С 1 С 2 = (R 1 + Р1) (R 2 + Р2) = R 1 R 2 + j(1 1 12) ,
(А-ll)
Аn example of complex number addition is discussed in Section 8,3
where we covered the topic of averaging fast Fourier transform outputs,
А.3.2 Multiplication of Complex Numbers
We сап use the rectangular form to multiply МО complex numbers as
С 1 С 2 = (R 1 + j1 1 ) (R 2 + Р2) = (R 1 R 2 1112) + j(Ri2 + R 2 11) .
(А-12)
However, if we represent the МО complex numbers in exponential form,
their product takes the simpler form
с с м e illJ , М d llJ2 М М e i (llJ,+llJ 2 )
1 2 1 2 1 2
(A 13)
because multiplication results in the addition of the exponents.
448 Appeпdix А: The Arifhтef/c of Сотр/ех Nuтbers
As а special case of mu1tiplication of two complex numbers, scaling is
multiplying а complex number Ьу another complex number whose imag-
inary part is zero. We сап use the rectangu1ar or exponential forms with
equal ease as fol1ows:
kC = k(R + jI) = kR + jkI ,
(А-14)
or in exponential form,
kC = k(Md") = kМd" .
(А-15)
А.3.3 Conjugation of а Complex Number
Тhe complex conjugate of а complex number is obtained Ьу merely chang-
ing the sign of the number' s imaginary part. 50, if we denote с' as the сот-
plex conjugate of the number С = R + jI = Md", then, с' is expressed as
с' = R jI = Me j0 .
(А-16)
There are two characteristics of conjugates that occasional1y соте in
handy. First, the conjugate of а product is equal to the product of the con-
jugates. Тhat iБ, if С = С 1 С 2 ' then from Eq, (А-13),
С* = (С 1 С 2 )* = (M1M2ej(01+02)* = M1M2e j(01+02)
М j01 M j02 С * с *
1 е 2 е 1 2 .
(А-l7)
Second, the product of а complex number and its conjugate is the complex
number's magnitude squared. It's easy to show this in exponential form as
сс' = Mt!0. Me j0 = M 2 tf = м 2 .
(А-18)
(Тhis property is often used in digital signal processing to determine the
relative power of а complex sinusoidal phasor represented Ьу Mt!rot,)
А.3.4 Division of Complex Numbers
Тhe division of two complex numbers is also convenient using the ехро-
nential and magnitude and angle forms, such as
Arlthmetic Operatloпs of Complex Nиmbers 449
.s;. = M 1 e i01 = М 1 ei(01 02} ,
С 2 М 2 е/ 02 М 2
(А-19)
and
С[ M 1
= L( I 2)'
С 2 М 2
(А-19')
Although not nearly so handy, we can perform complex division in rec-
tangиlar notation Ьу mu1tiplying the numerator and the denominator Ьу
the complex conjugate of the denominator as
C 1 = RI + j1 1
С 2 R 2 + j1 2
= RI + j1[ . R 2 Р2
+ j1 2 R 2 Р2
Щ + 1112) + j(R 2 1[ R(1 2 )
R; + 1;
(A 20)
А.3.5 Inverse of а Complex Number
А special form of division is the inverse, or reciprocal, of а complex num-
ber, If С = Mt!0, its inverse is given Ьу
..!. = = J..e j0
с Me J0 М
(А-21)
In rectangular form, the inverse of С = R + j1 is given Ьу
1 1 R j[
C R+j[ R 2 +[2
(А-22)
We get Eq. (А-22) Ьу substituting R 1 = 1, [1 = О, R 2 = R, and [2 = [in Eq, (А-20),
450 Appeпdix А: The Ar/fhтefic о( Сотр/ех Nuтbers
А.3.6 Complex Numbers Raised to а Power
Raising а complex number to воте power is еавНу done in the exponen-
оа! fопn. If С = Me i0 , then
С' = м" (e i0 / = м" e ik0 ,
(A 23)
"1250
For example, if С = зeI , then С cubed is
(qЗ = ЗЗ(еiН250) = 27еiЗ75° = 27ei15° ,
(А-24)
We conclude this appendix with four finаl complex arithmetic opera-
tions that are not very common in digital signal processing but уои тау
need them sometime,
А.З.7 Roots 01 а Complex Number
Тhe kth root of а complex number С is the number that, multiplied Ьу
itsel1 k times, resu1ts in С. Тhe exponentia! fопn of С is the best way to
explore this process, When а сотр!ех number is represented Ьу С = Me i0 ,
remember that it can also Ье represented Ьу
с = Mei(0 + nЗ60 0 ) ,
(А-25)
In this case, the variable 13 in Eq. (А-25) is in degrees, Тhere are k dis-
tinct roots when we're finding the kth root of С. Ву distinct, we mean
roots whose exponents are !ess than 3600. We find those roots Ьу using
the fol1owing:
vc = v Ме i (0+nЗ600) = VМе i (0+nЗ600)/k
(A 26)
Next, we assign the values О, 1, 2, 3, ' . ., k l to п in Eq. (A 26) to get the k
roots of С. ОК, we need an example here! Let's вау we're looking for the
.(75°)
сиЬе (3rd) root of С = 125е l , We proceed ав follows:
vc = 125ej(750) = 125e i (75° +nЗШ) = VU5ei(75° +n3600) / з ,
(A 27)
Next we assign the values п = О, п = 1, and п = 2 to Eq, (А-27) to get the
three roots of С. So the three distinct roots are
Ar/thтetic Operatioпs of Сотр/ех Nuтbers 451
1s1 rool .... = 5e j (75° +0'360°) 13 = 5e j (250) ;
2nd rool.... = 5e j (75° +1'360°) 13 = 5e j ( 435°) 13 = 5ej(l450) ;
and
3rd rool.... = 5e j (75°+2'3600)/3 = 5e j (795°)/3 = 5e j (265°)
А.З.8 Natural Logarithms о, а Complex Number
Taking the natura110garithm of а complex number С = Me j " is straightfor-
ward using exponential notation; that is
lnС = ln(Me j ,,) = 1nМ + ln(e j ")= 1nМ + j0 ,
(А-28)
where О $ 0 < 2х , Ву way of example, if С = 12e j1t/4 , the naturallogarithm
of С is
lnС = ln(12e]1t/4) = ln(12) + j1t/4 = 2.485 + jO.785 .
(А-29)
Тhis means that i 2 ,485 + jfJ,785) = ;.485 . O,785 = 12e j1t/4 .
А.З.9 Logarlthm to the Base 1 О о, а Complex Number
We сап calculate the base 10 logarithm of the complex number С = Me j "
using
log10C = loglO(Me i ") = logl + loglO(e j ") = logl + j0 . loglO(e) .t (А-30)
Of course е is the irrational number, approximately equal to 2.71828,
whose log to the base 10 is approximately 0.43429. Keeping this in mind,
we сan simplify Eq, (А-30) as
log10C '" logl + j(O,43429 . 0) ,
(А-31)
Repeating the above example with С = 1 1t/4 and using the Eq. (А-31)
approximation, the base 10 logarithm of С is
t For те second term оЕ те result in Eq, (А ЗО) we used loga(x ") = n ./ogax according to те
law оЕ logarithms,
452 Appeпdix А: The Arlthтetlc of Coтplex Nuтbers
log10 С::: lOg10(12e i 1!/4) ::: log10(12) + j(0.43429 . те/4)
::: 1,079 + j(0.43429 .0,785) ::: 1,079 + jO.341 ,
(А-32)
The resиlt from Eq, (А-32) means that
10(1.079 + iO.341) ::: 101.079 , 10 jO ,341 ::: 12 . (e 2 ,302)i O ,341
::: 12ei(2.302 . 0,341) ::: 12eiO.785 ::: 12e i 1!; 4 ,
(A 33)
А.З. 1 О Log to the Base 1 О of а Complex Number Using
NaturaJ Logarithms
Unfortиnately, some software mathematics packages have по base 10 log
arithmic function and сап calcиlate only naturaI Iogarithms. In this sitиa
Ноп, we jиst иse
ln(x)
loglO(X)::: l
п(10)
(A 34)
to caIcиlate the base 10 logarithm of х, Using this chaпge о/ base formula,
we сап find the base 10 logarithm of а complex nиmber С ::: Me i "'; that is,
InC
Iog 10 С ::: In = (lOg10 e)(In С).
10
(А-35)
Веса иse Iog 10 (e) is approximately eqиal to 0.43429, we иse Eq, (А-35) to
state that
log10C'" 0.43429 . (InC) ::: 0.43429 . (InМ + jfJ) ,
(А-36)
Repeating, again, the example above of С::: 12e i 1!/4, the Eq, (А-36) approx
imation aIlows иs to take the base 10 logarithm of С иsing natиraI logs as
loglO С::: 0,43429 ' (In(12) + jтe/4)
::: 0.43429 . (2.485 + jO,785) ::: 1,079 + jO,341 ,
(A 37)
giving иs the same resиlt as Eq, (А-32),
Some Proctico//mp//cot/oпs of Us/пg Сотр/вх Numbers 453
А.4 Some Practicallmplications of Using
Complex Numbers
At the beginning of Section А.3, we said that the choice of using the rec
tangular versus the polar form of representing complex numbers
depends оп the type of arithmetic operations we intend to perform. It's
interesting to note that the rectangular form has а practical advantage
over the polar form when we consider how numbers are represented in
а computer. For example, let's say we must represent our complex nиm-
bers using а four bit sign magnitude binary nиmber format. This means
that we сan have integral nиmbers ranging from 7 to +7, and our range
of complex nиmbers covers а square оп the complex plane as shown in
Figure A 4(a) when we use the rectangular form. Оп the other hand, if
we used 4 bit numbers to represent the magnitude of а complex nиmber
in polar form, those numbers must reside оп or within а circle whose
radius is 7 as shown in Figure A 4(b), Notice how the four shaded cor
ners in Figure A 4(b) represent locations of valid complex values using
the rectangular form, but are out о! bouпds if we use the polar form, Put
another way, а complex number calculation, yielding аn acceptabIe result
in rectangular form, could result in an overf1ow error if we used polar
notation in our computer. We could accommodate the complex value
7 + j7 in rectangular form but not its polar equivalent because the mag-
nitude of that polar nиmber is greater than 7,
Imaginary
axis и)
Imaginary
axls (J)
. . .. ." .. . .".,. .
,- ............,.".... ... ..... .......... ........
. .. . I!I ...... . ." Ii I!I ." , I!I
........,....". ....... ... ............ -.. ..........
. . . . ." ... ...
............ ......."...... ....... ...t,:....... ........
'........... ".,...
.....,.
....,,,...,,,.... ... .... .
.f. .
." ........ .......... ...
:..1: :..1...:...::..:..
...... ..
......i ReaJaxis
...,....................,....
...:..+...:.:.Z.:.:..;.::Z
........
...I!I...;...... .......+...
........
Аваl axis
(8)
(Ь)
Flgure А-4 Complex Integral numbers represented as polnts оп the complex plane
uslng о four-bIt slgn-magnltude data format: (а) using rectangular
notatlon; (Ь) uslng polar notatlon,
454 Appeпdix А: The Arithтetic о' Сотр/ех Nuтbers
Although we avoid аnу further discussion here of the practical impli-
cations of performing complex arithmetic using standard digital data for-
mats, it is аn intricate and interesting subject, То explore this topic fиrther,
the inquisitive reader is encouraged to start with the references,
References
[1] Plaиger, р, J, uComplex Math Fиnctions,u Eтbedded Systeтs Prograттing,
Aиgust 1994,
[2] Kahan, W, uBranch Cиts for Соmрlех Elementary Functions or Mиch Ado
Aboиt Nothing's Sign Bit,U Proceedings о/ the Joint lМA/SIAМ Con/erence оп the
State о/ the Art in Nuтerica/ Ana/ysis, Clarendon Press, 1987.
[3] Plaиger, Р. J, uComplex Made Simple,u Eтbedded Systeтs Prograттing, July
1994,
!APPENDIX вl
Closed Form of а
Geometric Series
In the literature of digital signal processing, we often encounter geomet-
ric series expressions like
N l Р N
L n r r
r
l r '
n=р
(В-1)
or
N l 1 i2mп
i21tnm/N e
-,-,е 1 e i21tm/N .
n=о
(В-2)
Unfortunately, тапу authors make а statement like "and we know that,"
and drop Eqs, (В-1) or (B 2) оп the unsuspecting reader who's expected to
accept these expressions оп faith, Assuming that уои don't have а Ph,D,
in mathematics, уои тау wonder exactly what arithmetic sleight of hand
allows us to amve at Eqs. (Б--1) or (B 2), То answer this question, let's con
sider а general expression for а geometric series such аБ
N l
5= Larn =ar P +ar P + 1 +ar P + 2 +...+arN l ,
n=р
(В-З)
where п, N, and р are integers and а and r are апу constants, Multiplying
Eq. (В-З) Ьу r, gives иБ
N l
Sr = L ar n + 1 = ar P + 1 + ar P + 2 +'" + arN l + ar N
n=р
(В-4)
455
456 Appeпdix В: C/osed Forт of Q Geoтetr/c Ser/es
5ubtracting Eq. (В-4) from Eq. (B 3) gives the expression
5 5т = 5(1 т) = атР ar N ,
or
тР r N
5=a'
1 T
(В-Б)
50 here's what we're after, Тhe closed forт of the series is
Clos..,d {оrш of а general
geometric series:
N l" rP rN
Lar =a'
1 T
"=р
(В б)
(Ву closed form, we mean taking an infinite series and converting it to а
simpler mathematical form without the surnrnation.) When а = 1, Eq. (В-б)
validates Eq, (B l), We сап quickly verify Eq. (В б) with an example,
Letting N = 5, Р = О, а = 2, and r = 3, for example, we can create the fol
lowing list:
п ат"=2.3"
О 2 . з0 = 2
1 2. 31 =6
2 2 . 32 = 18
3 2'33=54
4 2 . з4 = 162
The 5um of thi5 column 15
4
L2'3" =242
"=0
Plugging our example N, р, а, and r values into Eq, (В б),
N l т р rN з0 з5 1 243
ат" =a. =2. =2. =242,
k.,; 1 r 1 3 2
"=р
( 7)
Appeпdix В: C/osed Forт of Q Geoтetric Series 457
which equals the sum of the rightmost соlurnn in the list above.
As а final step, the terms of our earlier Eq. (B 2) are in the form of Eq,
(В б) as р = О, а = 1, and r = e j2"m/N,t 50 plugging those terms from Eq,
(B 2) into Eq. (В-б) gives us
N l j2"mOIN j21<mNIN
L j2"пmIN 1 е e
е .
1 e j2"mIN
п=О
1 e j27tm
1 e j21tmIN '
(В-8 )
confirming Eq, (B 2),
t From the math identity аХУ = (аХ)У, we сап say e 2""тIN = (e j2.тIN)п, so r = e J2.тIN.
IAPPENDIX с!
Complex Signals and
Negative Frequency
Complex nurnbers are used in just about every field оЕ science and engi-
neering. t For us, complex signal notation is necessary to describe the
effects оЕ quadrature and Fourier processing. It mау Ье fair to say that по
other topic in digital signal processing causes more initial discomfort for
the beginner than coтplex numbers with their real and iтagiпary parts.
Why, еуеn the terminology is bizarre. 50 here's where we review the fun
damentals оЕ complex nurnbers, get comfortable with how they're used to
represent complex signals, and ехаminе the notion оЕ negative frequency
as it relates to complex signal notation.
То start this discussion, it's important for us to Ье comfortable with the
concept оЕ negative frequency because it's essential in understanding the
spectral replication effects оЕ periodic sampling, discrete Fourier trans
forms, and the various quadrature signal processing techniques discussed
in Chapter 7. Тhe convention оЕ negative frequency serves as а consistent
and powerfи1 mathematical tool in our analysis оЕ signals. In fact, the use
оЕ negative frequency is mandatory when we represent real sigпals, such
as а sine or cosine wave, in complex notation.
ТЬе difficulty in grasping the idea оЕ negative frequency mау Ье, for
some, similar to the consternation fe1t in the parlors оЕ mathematicians
in the Middle Ages when they first encountered negative numbers,
Until the thirteenth century, negative numbers were consideredfictitious
because numbers were normal1y used for counting and measuring. 50
ир to that time, negative numbers just didn't make sense [1,2], In those
days, it was valid to ask, "How сап уои ho1d something in your hand
that is less than nothing?" Тhe idea оЕ subtracting six from four must
have seemed meaningless. Math historians suggest that negative nиm
t 1'ha t's because complex sinusoids are solutions to the second-order linear differential equa-
tions used to describe so much of nature,
458
сотр/ех SigпQ/s Qпd NegQtive Frequeпcy 459
bers were first analyzed in Italy. As the story goes, around the year 1200,
the НаНаn mathematician Leonardo da Pisa (known as Fibonacci) was
working оп а financial problem whose only va1id solution involved а
negative number. Undaunted, Leo wrote, "This problem, 1 have shown
to Ье insoluble unless it is conceded that the first таn had а debt," Thus
negative numbers arrived оп the mathematics scene, never again to Ье
disregarded.
Modern теn and women сап now appreciate that negative numbers
have а direction associated with them, ТЬе direction is backward from
zero in the context that positive numbers point forward from zero. For
example, negative numbers сап represent temperatures measured in
degrees below zero, minutes before the present if the present is consid-
ered as zero time, or тоnеу we owe the tax collector when our income
is considered positive dollars. 50 the notion of negative quantities is
perfect1y valid if we just define it properly, As comfortable as we now
are with negative numbers, negative frequency remains а troublesome
and controversial concept for таnу engineers [1,2], This author оnсе
(а) о
2rr.ft
(OJ
1
О 2 3
Time
(Ь) О
2rr.ft
(OJ
1
О 2 3
Flgure с-] Posltlve Qnd negQtive frequency sinusolds: (а) sine Qnd coslne
woves at Q positive radlan frequency (2rr.ft); (Ь) slne Qnd coslne
WQves whose Qrguments Qre Q negQtive frequency ( 2rr.ft),
460 Appeпdlx с: coтplex Slgпals aпd Negatlve Frequeпcy
encountered а paper in а technical joumal which stated: "since negative
frequencies cannot exist ," Well, like negative numbers, negative fre
quency is а perfectly valid concept as long as we define it properly rel
ative to what we're used to thinking of as positive frequency, In radians,
а negative frequency sinusoid is merely опе whose angu1ar argument is
negative, as shown in Figure C-l(b).t
С.1 Development 01 Imaginary Numbers
Before we go further in our discussion of negative frequency Ьу way of
complex notation, we must first take а deep breath and enter the Twilight
Zone of the 'т operator and imaginary numbers, You've seen the defini
tion j = н before. Stated in words, we say that j represents а number
which, when multiplied Ьу itself, results in а negative one, Well, this def-
inition causes initial difficulty for the beginner because we know that anу
number mu1tiplied Ьу itself always resu1ts in а positive number.
(Unfortunately digital signal processing textbooks typically define j and,
then, with justified haste, swiftly carry оп with аll the ways that the j
operator сan Ье used to analyze sinusoidal signals. Readers soon forget
about the question: What does j = н actually mean? Well, Н had
Ьееп оп the mathematical scene for some time but wasn't taken seriously
until it had to Ье used in the sixteenth century to solve сиЫс equations
[3, 4]. Mathematicians reluctantly began to accept the abstract concept of
Д without having to visualize it, as long as its mathematical properties
were consistent with the arithmetic of пorтal real numbers, It was Euler's
introduction of the complex plane, used to solve second order differential
equations, that further legitimized the notion of Н to Europe's mathe
maticians in the eighteenth century,t Euler showed how complex num
bers using the Н operator had а clean, consistent relationship to the
well-known trigonometric functions of sines and cosines [3].
We сап get more comfortable with the complex plane representation
of irn inary numbers Ьу examining the mathematical properties of the
j = ..J 1 operator as shown in Figure C 2(a). Multiplying апу number оп
t Shortly, we'lI represent negative frequency sinusoids graphically using phasor notation
and see that the negative frequency phasors are merely rotating in the direction opposite
from an arbitrary positive reference direction,
t Leonhard Euler, Ьот in Switzerland in 1707, is considered Ьу many historians as the
world's greatest mathematician, Ву the way; in case уои don't speak German, the пате Eиler
is pronoиnced as "oiler,"
Deve/opmeпt of /mQgiпQry Numbers 461
Imaginary
)8 axis
Imaglnary
axis и)
о
1/
j8
(а)
с= 8 + jb
... Ьш шш
}
the +8 Ilne
;k"
the +jb line
о
8 Rв.аl
aXls
(Ь)
Flgure С-2 Numbers оп the complex plone: (о) the 900 rototlons when о number Is
multlplied Ьу J; (Ь) о complex number, с = Q+Jb, hoving both reol ond
lmoginory parts оп the complex p/one,
the real axis Ьу j resu1ts in an imaginary product that 1ies оп the imagi-
nary axis. Тhe example in Figure С-2(а) shows that if +8 is represented
Ьу the dot lying оп the positive real axis, multiplying +8 Ьу j resu1ts in
an imaginary number, +j8, whose position has been rotated 900 counter-
clockwise from +8, putting it оп the positive imaginary axis. Similarly,
mu1tiplying +j8 Ьу j resu1ts in another 900 rotation yielding the 81ying
оп the negative real axis because j2 = 1, Mu1tiplying Ьу j resu1ts in а
further 900 rotation giving the j8 lying оп the negative imaginary axis,
Whenever any number represented Ьу а dot is multip1ied Ьу j, the result
is а counterclockwise rotation of 900 оп the complex plane. (Conversely,
multip1ication Ьу j resu1ts in а clockwise rotation of 900 оп the сот-
plex plane.) Тhe complex plane enables us to represent complex num-
bers having both real and imaginary parts. For example in Figure
С-2(Ь), the complex number с = a+jb is а point lying оп the complex
plane оп neither the real nor the imaginary axis. We locate point с Ьу
going +а units along the real axis and ир +Ь units along the imaginary
axis. Oust to refresh the reader's memory, Appendix А provides а dis-
cussion of the arithmetic of complex numbers.) We сап, if we care to,
think of аН numbers as being complex. Thus, real numbers are complex
numbers with по imaginary part, and imaginary numbers are complex
numbers with по real part.
Let's pause for а moment here to catch our breath, Don't worry if the
ideas of imaginary numbers and the complex plane seem а 1itt1e myste-
rious, It's that way for everyone at first you'll get comfortable with
462 Appeпdix с: Сотр/ех s/gпQ/s Qпd NegQt/ve Frequeпcy
them the more уои use them. (Remember, the Н operator puzzled
Europe's heavyweight mathematicians for hundreds of years,) Not only
is the mathematics of complex numbers а bit strange at first, but the ter-
minology is unfortunate, Although the term iтagiпary is an unfortunate
one to use, the term coтplex is downright weird, When first encountered,
the phrase coтplex пuтbers makes one think "complicated numbers."
This is regrettable because the concept of complex numbers is not really
аН that complicated,t
С.2 Representing Real Signals Using
Complex Phasors
ОК, now we сап tum our attention back to the original discussion of
understanding negative frequency when using the mathematics of сот-
plex numbers. We start Ьу representing а real sinusoid in the context of
the complex plane. То see how this is done, consider the waveform in
Figure с-з. тhink of а cosine wave defined Ьу 2cos((Ot) oscillating back
and forth оп the real axis as time passes, (Here the (О term is frequency in
radians/s, and it corresponds to а frequency of 2тс! cycles/s where j is а
single cycle/s or 1 Hz.) Now, consider two complex phasors rotating in
opposite directions about the time axis, one rotating counterclockwise
with а positive angular change (Ot and the other phasor rotating with а
negative angle, t. The angles, of course, change as а function of time
making the ej{j)t and e j{j)t phasors rotate, Figure С-4 attempts to show, as
time passes, one complete rotation of the ej{j)t phasor, for example. That
phasor's tip follows а corkscrew path spiraling along and centered about
the time axis.
Let's show how the cosine wave is the phasor sum of the two complex
phasors. We сan visualize the phasors' (Ot and (Ot angles better if we ori-
ent the three-dimensional view in Figure с-з so that we're looking right
straight down the time axis in the negative time direction as shown in
Figure С-5, The time arrow is coming straight out of the page toward the
reader and the two complex phasors, frozen in time, are represented Ьу
the bold arrows, Figure C 5 looks reasonably simple, but it contains а
wealth of information, Let's first consider the ej{j)t phasor. Using the real
and imaginary axis, we сап see why the e jrot phasor is defined Ьу
t The brilliant American engineer Charles Р, Steinmetz, who pioneered the use of real and
imaginary numbers in electrical circuit analysis in the early twentieth century, refrained
from using the term сотр/ех пuтber5 he cal1ed them "general numbers,"
Represeпtiпg ReQI S/gпQls Usiпg Coтplвx PhQsors 463
Euler's equation:
ejO)t = cos(rot) + jsin(rot) ,
(C l)
Named in his honor, Eq. (C-l) is now called Euler's equatioп, This expression
defines the position of the tip of the ejO)t arrow in the complex plane, The
cos(rot) term in Eq, (C l) is the phasor's component a10ng the rea1 axis, and
the sin(rot) term is the phasor's component along the imaginary, or j, axis,
It's very important to realize that the phasor's component projected оп the
imaginary ахiБ is the real number sin(rot), and пot jsin(rot). The +j in the sec
ond term of Eq, (C l) merely means that the phasor has а component along
Imaginary
axis (j)
Figure с-з Coslne represented Ьу two rotating complex phasors,
! Imaginary axis ( j)
I eal axis
900
Time
Flgule С-4 The rotQtion of thв е Jшt phasor Qbout the t/me аХ/5.
464 Appendix с: сотр/ех SJgпa/s aпd Negat/ve Freqиeпcy
, Imaginary
I axis (j)
t COS(ro 7
Тipofthe Sin (rot) ejrot
time arrow
____(J).t_______ ш _____
rot Аеаl axis
sin(ro е
+ cos(ro
Flgure С-б Complex phasors representing а coslne wave In the complex
plane,
the positive j axis whose magnitude is sin{rot). Likewise, the e jro/ phasor has
а component along the negative j axis whose magnitude is sin{rot), We
describe the e jro/ phasor with the expression
Another version of Euler's equation: --+
e--jro/ = cos(rot) jsin(rot) t
(C 2)
Тhe positive angle rot is (arbitrarily) defined as the counterclockwise
angle between the real axis and the phasor, As time t increases, angle ro t
increases and the e jro / phasor rotates counterclockwise, Тhe e jro/ phasor
rotates in the clockwise direction at the same angular frequency as the e jro /
phasor; that is, when rot is zero radians, both phasors are at 3:00 o'clock,
along the real axis. As times passes, when the e jro / phasor is at twelve
о' clock, the e jro/ phasor is at six о' clock. As the phasors continue rotating,
they pass each other, again, exactly at ninе о' clock.
Let's look further and see how the sum of the e jro / and e jro/ phasors rep
resents а real cosine wave, In Figure С..б(а), the phasor angle is rot = тс/4 (45
degrees), The bold arrow representing the real cosine wave, 2cos(rot), is the
phasor sшn of the e jro / and e jrot phasors. Notice how the cosine wave' s arrow
t Ot:her versions of Eu\er's equation, which we сan derive from Eqs, (C-l) and (С-2), асе
sinto/) = (ei"" e- i "")/2j, and cos((J)/) = (e i "" + e- i "")/2,
!I
Represeпt/пg Rea/ Sigпa/s Usiпg Сотр/ех Phasors 465
+ Imaglnary + Imaginary Imaglnary +
ах!. и) axl. и) axl. и)
t, /' .
ejrot ejrot
",t=1tI4 j"'t= ,,/2
7
...,t Аеаl Real /...,t :1
axi8 axis
в --fwt е --fw t в --fwt I
'" i
(а) (Ь) (с)
Flgure С-6 Complex phosors representing о cosine wove: (о) when oot= тс/4;
(Ь) when oot = тс/2; (с) when oot = 3тс/4,
tip is along the real зxis. At а later tiше, when ro t = п/2 radians, the ei(J)t pha-
sor has rotated counterclockwise, the e i(J)t phasor has rotated clockwise, and
the 2cos(rot) wave's arrow tip has moved to the left and sits exactly at the
zero point оп the real axis as shown in Figure C-б(Ь), We know this is correct
because 2cos(1t/2) = о. As tiше passes and rot = З1t/4 radians, the ei(J)t and
e i(J)t phasors and their resulting 2cos(rot) phasor are shown in Figure С.б(с),
which is real and negative, As the complex phasors continue rotating, the
2cos(rot) phasor tip merely slides back and forth always remaining оп the
real зxis. Figure С-З attempts to illustrate this Ьу showing the 2cos(rot) wave
always lying in the plane of the real axis in our three-dimensional depiction.
Тhinking about these phasors, it's clear, now, why the cosine wave сап
Ье equated to the sum of two complex phasors Ьу
ei(J)t + e i(J)t
cos(rot) = ,
2
(С-З)
which is ha1f the sum of Eqs. (С-l) and (С-2), (Тhe example in Figure С-З
used а cosine wave with а peak amplitude of 2 merely to avoid cluttering
Figures С-З through С-б with а denominator of 2 below each of the ei"'t
and e j(J)t terms.) If Eq, (С-З) describes а real cosine wave in the complex
plane, how do we represent а real sinewave? Well, we сап combine Eqs.
(С-l) and (С-2) and solve for sin(rot) to get the following standard expres-
sion for а sinewave in complex notation:
ej(J)t e i(J)t
sin(rot) = 2j
(С-4)
466 Appeпdix с: Сотр/ех Sigпo/s ond Negotive Frequeпcy
То investigate the meaning of this expression, we сan multiply both
sides of Eq. (С-4) Ьу 2 to get the following altemate expression for а
sinewave in terms of complex exponentials:
ejo>t e jo>t . ,
2sin«Ot)= , = j(eJo>t e JO)t)
]
(С-5)
Equation (С-5) looks а little strange, doesn't it? It states that а real
sinewave is equal to j times the sum of two complex numbers! Well, we
сan decipher Eq. (С-5) Ьу graphically illustrating the complex ехроnеn-
tials as phasors just as we did in Figure С-5. А depiction of just the
(ejo>t e jO)t) factor from Eq, (С-5) is the phasors in Figure C 7(a). То
account for the j term in Eq. (C 5), we rotate the phasors in Figure C 7(a)
Ьу Л/2, or 900, in the clockwise direction. 50 the full j(ejo>t e jo>t) expres-
sion оп the right side of Eq. (C 5) is shown in Figure C 7(b). Because the
joot phasor rotates clockwise and the ejo>t phasor rotates counterclock
wise, their phasor sum representing 2sin(O)t) will always lie оп the real
axis. We know this is true because the 2sin(O)t) siriewave in Eq. (C 5) is
real like а cosine wave, it has по imaginary part,
То belabor this point а bit further, consider Figure C 8. When the angle
о) t == О, both the je jo>t and jejo>t phasors lie оп the imaginary axis, and their
phasor sum sits exactly at the zero point of both axes, as shown in Figure
C (a). We know this is сопесt because 2sin(O) = О. At а later time when
O)t == п/4 radians, the je jo>t phasor has rotated clockwise, the jejo>t phasor
Imaginary
axis и)
\maginary
axis и)
/
/
j ,---e jo>t= je jo>t
Ава\
axis
je
Аваl
axis
jo>t
B jo>t
""
(а)
(Ь)
Flgure С-7 Coтplex phasors: (о) phosor represeпtotion of (e}o>t e-}o>t); (Ь) phosor
representotion of j(e}o>t e- pt ). or 2sin(w t) = je}o>t + je-}o>t,
Represeпtiпg Rea/ SigпQ/s USiпg Negatlve Frequeпeies 46j
Imaginary
axis и)
t
je ",t= ,соо(О) + sin(O)
irot :j1 . axis
jejrol = jcos(O) + sin(O)
. = j1
(8)
+ Imaginary
. axis и)
sin(rol)7- sin(roI) ,
je--jrot
+ Imaginary
axis (j)
Аааl
axis
Aaal
axis
(Ь)
(с)
Figure С-8 Complex phQsors representing а 2slп(юt) sinewove: (а) when юt = О; (Ь)
when юt = 1t/4; (е) when юt = 31t/4,
has rotated counterclockwise, and the 2sin(фt) wave's arrow tip has
moved to the right to а positive point оп the real axis, as shown in Figure
С-8(Ь), As time passes and фt = 31[/4 radians, the jej(J)1 phasor is now in
the northeast quadrant, and je j(J)1 phasor is directed southeast. They and
their 2sin(фt) phasor sum are shown in Figure C 8(c). As the complex
phasors continue rotating, the 2sin(фt) phasor oscillates back and forth,
always remaining оп the real axis,
То keep the reader's mind from spinning like our complex phasors,
please realize that the sole purpose of Figures C 6 through C 8 is to vali
date the complex expressions of the cosine and sinewave given in Eqs.
(С-3) and (C 4),
С.3 Representing Real Signals Using
Negative Frequencies
ОК, let's keep in mind one of the goals of this discussion why the concept
of negative frequency is valid when rea1 signa1s are represented in complex
notation. We сап do this Ьу comparing the relationship between а соs(фi)
wave and а sin(Фi) wave, using Eqs. (C 3) and (С-4) and combining them
graphically оп а three dimensional complex frequency axis as shown in
Figше C 9. If we say that the angular frequency is ФО radians/second = 21[10
Нz, the time--varying composite angie is Фi radians = 2rifot cycles at Шnе t,
Look at Figure C 9 very carefully it's а snapshot of the complex phasors
that comprise а rea1 cosine wave (solid arrows) from Eq. (C 3) and а real
sinewave (shaded arrows) from Eq. (C 4) when Шnе equals zero, or Фi = о.
468 Appeпdix с: Сотр/ех SigпQ/s Qпd Negative Frequeпcy
Jmaginary ! R I
' ( ' ) I ;а
axJS J V XiS
je O 2
t= О, ао root= О
Jro 0 t/ 2
J
Freq
Figure С-9 Relatlonsh/p of the сотplех representatlons of the
cos(root) = (ej())ot + e j())ot)/2 (solid arrows) and the
sin(roj) = (eJ())ot e-jo>ot)/j2 = He)olot + je-)oIоt)/2 (shaded
arrows), when roj = О,
тhi.5 Нgше shows that, when we use complex notation, e j ())/ and e j())t are the
fundamental constituents of а sinusoid, not sin(O)t) or cos(O)t) ЬесаuБe both
sin«(i)t) and cos(O)t) are made ир of e j ())/ and e j())t components, on the fre-.
quencyaxis, the notion of negative frequency is seen as those phasors located
at Шо radians оп the frequency axis, (Н we were to take the discrete Fourier
transform of а cosine wave and plot the complex results, we'd get exactly
those bold phasors in Figure C 9, Likewise, the discrete Fourier transform
of а sinewave results in the shaded phasors in Figure С-9,) Figure С-9 also
reiterates the fact that а cosine wave is merely а sinewave sblfted in time, If
we let the phasors in Figure С-9 rotate 900 (1t/2 radians, or а quarter of а
cycle), they're oriented as shown in Figure C 10, 50 the sinewave's shaded
phasors, in Figure С -10, are oriented exactly as the cosine wave' s phasors in
Figure С-9, The relationsblp between Figures C 9 and C 10 illustrates the
мо following trigonometric identities:
Sin(0) = COS(0 1t/2) ,
(С-б)
and
COS(0) = ---sin(0 1t/2) .
(С-7)
Represeпt/пg ReQ/ S/gпQ/s Us/пg Negative Frequeпcies 46
Imaginary t al
axI8(/) V i8
t = п/2ОО 0 = 1/4f o , or 000 t = п/2
еfщ:,t/ 2
е fщ:,t/2
Freq
Flgure C-IO Relationship of the complex representations of cos(fiJot) (solid arrows)
and sln(fiJot) (shaded arrows) when fiJot = 1t/2,
Putting Eqs. (C 6) and (C 7) into words relative to the two figures above,
we can say that the sine phasors in Figure С-I0 are equal to the cosine
phasors in Figure С-9, and the cosine phasors in Figure C 10 are equal to
the negative of the sine phasors in Figure C 9. This tel1s us that а sinu-
soidal wave сап Ье described as а sinewave or а cosine wave depending
оп the time we start looking at it; that is, а delay in the time domain man-
ifests itself as а phase shift in the frequency domain. This effect is very
important in digital signal processing, so remember it,+
If we use the same axis as before, we can illustrate а sinusoidal wave
at an arbitrary phase angle 0, Acos((Oi + 0), as shown Ьу the bold pha-
sors in Figure C-l1(a). The magnitude of the two bold phasors is, of
course, А/2, so they'll sum to А when they're aligned along the real axis.
Figure C ll(a) shows that the phasors are the phasor sum of their real and
imaginary parts. Figure C ll(b) shows the real and imaginary сотро-
nents a1igned with the real and imaginary axis. Why bother showing the
real and imaginary parts as in Figure C-ll(b)? Because this signal repre
sentation leads us to the graphical form used so often in the digital signal
processing literature of quadrature signal processing.
тhink for а moment of а real signal comprising seven sinusoids over а
bandwidth В centered about а carrier frequency of (00' We could represent
that signal' s spectrum Ьу the bold phasors in the complex frequency
domain of Figure С-12(а). If we had another signal with an unlimited
number of sinusoids over the bandwidth В, we could forego drawing
+ For additional examples of this shifting property, check ои! Section 3.6,
470 Appeпd/x с: Сотр/ех S/gпQ/s Qпd NegQt/ve Frequeпcy
Imaginary
iS(j)
Аеа'
axis
F ,maginar y
axis (j)
Аеа'
axis
I
юо
Freq
Freq
(а)
(Ь)
Figure с-!! Two depictions of the complex representatlon of Acos(wot + 0)
indlcated Ьу the bold phQsors, The thin solid Qrrows ше the real PQrts,
and the thin shQded arrows are the imQginary PQrts,
individual phasors and merely show the spectrum Ьу joining the tips of
the phasors with the bold lines in Figure C 12(b), Тhis signal has а con
tinuous spectral envelope over the bandwidth В, and that's the typical
signal spectrum representation used in the literature of quadrature pro
cessing. Figure C 12(b) shows the projection of the signal's constant
amplitude rectangular spectrum оп both the real and the imaginary axis.
In quadrature processing, Ьу convention, the real part of the spec
trum is called the iп-phase component, and the imaginary part of the
spectrum is called the quadrature-phase component. Тhe signal in Figure
C 12(b) is real, and, in the time domain, it сап Ье represented digitally
Ьу а series of amplitude values that have nonzero real parts and zero-
valued imaginary parts, We're not forced to use complex notation to rep
reseI1t it in the time domain the signal is real. (If it was а continuous
physical signal, it could Ье transmitted over а single conductor.) Real
signals always have positive and negative frequency spectral compo
nents, For any real signal, the positive and negative frequency compo
nents of its in phase (real) spectrum always have even symmetry about
the zero frequency point; that is, the in-phase part's positive and nega
tive frequency components are mirror images of each other, Conversely,
the positive and negative frequency components of its quadrature
(imaginary) spectrum are always complex conjugates of each other [5].
This means that the phase angle of any given positive quadrature fre
queIlcy component is the negative of the phase angle of the correspond-
Сотр/ех SigпQ/s Qпd QUQdroture Mixiпg 471
Imaginary
t/ axiS(j)
Aeal
аюв
о
(а)
(Ь)
Original signal Freq
In-phase (real) рап
Quadra!ura phase (imaginary) par!
imaginary
l/ iS(j)
Аеаl
axis
о
Freq
о
(с)
Freq
Flgure С-12 Quadrature representation of bandpass signals: (а) real signal
containlng seven sinusoids over bandwldth В; (Ь) real signal
contalnlng an Infinite number of sinusoids over bandwldth В;
(с) complex slgnal of bandwidth В,
ing quadrature negative frequency component, as shown Ьу the dashed
arrows in Figure С-12(Ь). Тhis is the invariant nature of real signals
when their spectra are represented using complex notation,
С.4 Complex Signals and Quadrature Mixing
Complex signals, оп the other hand, are а different story. Figure C 12(c)
shows the spectrum of an arbitrary complex signal of bandwidth В. In
472 Appeпdix с: Сотр/ех S/gпa/s aпd Negatlve Freqиeпcy
the tllnе domain, it сап only Ье represented Ьу amplitude values that
have nonzero real and nonzero imaginary parts, We're forced to use
complex notation to represent it in the Нте domain the signal is coт
plex. (As а continuous physical signal, а complex signal could only Ье
transmitted иsing two condиctors.) Complex signals aren't forced to
have the symmetric real and the conjugate symmetric imaginary spec
tral components characteristic of real signals omplex signals сап have
just positive (or just negative) frequency components, as shown in
Figure С-12(с).
We can summarize аН that we've covered in this chapter Ьу consider-
ing an interesting topic the spectrиm translation effects of quadrature
mixing [7]. Figure С-13(а) shows the triangular spectrиm of an arbitrary
real signal centered about 000 radians/s, In quadrature mixing, we trans
late the original signal's spectrum down to the zero frequency point
before digitization, This is typicaHy done Ьу multiplying the original sig-
nal Ьу а cosine wave оЕ 000 radians/s, in the time domain, to get the in-
phase spectral component of the input signal as shown in Figure С-13(Ь).
It's irnportant to see that Figure С-13(Ь) is the product of Figure С-13(а)
and the solid cosine phasors in Figure C 9, Тhe e jro"t /2 complex cosine
phasor from Figure С-9 translates both Figure C-13(a)'s original input sig
nal spectral bandwidths toward the negative frequency direction, as
shown in Figure C 13(b), Likewise, the ej(fJ,t /2 complex cosine phasor
from Figure C 9 translates both original inpиt signal spectral bandwidths
toward the positive frequency direction, also shown in Figure С-13(Ь),
Although not obvious in the figure, the magnitude in Figure С-13(Ь) is
half the magnitude of that in Figure С-13(а) because of the 1/2 scale fac-
tors in the cosine expressions in Eq. (С З) and Figure C 9,
То get the quadrature-phase spectral components of our inpиt signal,
we multiply the input signal Ьу а sinewave of 000 radians/s to get the
spectrиm shown in Figure С-13(с), Again, Figure С-13(с) is the product of
Figure С-13(а) and the dashed sine phasors in Figure C 9. Тhe je j(fJ,t /2
dashed complex sine phasor from Figure С-9 translates both Figure C
13(a)'s original input signal spectral bandwidths toward the negative fre-
quency direction, but aligned with the imaginary +j axis as shown in
Figure С-13(с), Тhe jeiro"t /2 dashed complex sine phasor from Figure С..-9
translates both original input signal spectral bandwidths toward the pos-
itive frequency direction aligned with the imaginary j axis, also shown in
Figure С-13(с), Figure C-13(d) shows both the in-phase and quadrature
spectral bands combined оп the complex frequency plane axis. (Examples
of this quadrature mixing scheme are discussed in Sections 7,1 and 7,2,)
Сотр/ех SigпQ/s Qпd QUQdrQture Mixiпg 473
Jmaginary
iS(j)
Ава\
axis
(а)
Imaginary
iS(j)
Aeal
axis
, +«\)
в---
Freq
(Ь)
(с)
о
Freq
(d)
+2<00
"'=,
Freq
Freq
Flgure С-13 Quadrature mixlng of а real bandpass slgnal with even symmetrie
speetral eomponents: (а) original real input signal of bandwidth В;
(Ь) in-phase speetral eomponents of the input signal; (е) quadrature-
phase speetral eomponents of the input signal; (d) eombined in-
phase and quadrature phase eomponents,
Providing another example to demonstrate the frequency translation
that takes place in complex signal mixing, Figure C 14 shows the spectral
results of quadrature mixing when the input signal is real and has conju-
gate positive and negative frequency components, Section 10,1, with its dis
cussion of digital mixing sequences, provides several practical examples of
474 Сотр/ех S/gпa/s aпd Quadrature M/x/пg
Imaginary
iS(j)
Аваl
axis
(а)
lmaginary
'S(j)
Авal
axis
2<%
Freq
(Ь)
(с)
(d)
Freq
2щ,
Freq
Flgure С-14 Quadrature mlxlng of а real bandpass signal with eomplex eonjugate
symmetrie speetral eomponents: (а) original real input signal of
bandwidth 8; (Ь) In-phasв speetra/ eomponents of the input signa/;
(е) quadrature-phase speetral eomponents of the Input signal;
(d) eombined In-phase and quadrature-phase eomponents,
the spectral translation effects of quadrature mixing. Тhat concludes our
introduction to complex signals and negative frequency. For further mater-
ial оп these topics, the reader is encouraged to review reference [7], а tепifiс
series of papers detailing how complex signals and negative frequencies
relate to modem-day digital communications signals.
Refereпces 475
References
[1] Lewis, L, J" et al, ипеат Systeтs Aпalysis, McGraw Hill Inc., New York, 1969,
рр, 193,
[2] Schwartz, М. Iп/orтatioп, Traпsтissioп, Modulatioп, aпd Noise, McGraw-НШ
Inc., New York, 1970, рр, 35,
[3] Struik, D, А Coпcise History о/ Matheтatics, Dover Publications Inc" New
York, 1967, рр. 92,
[4] Вergarnini, D, Мatheтatics, Ше Science Ubrary, Tune Inc" New York, 1963, рр,
152.
[5] Нsи, Н. Fourier Aпalysis, Simon and Schuster, New York, 1970, рр, 75.
[6] Chester, D, В., and Phi1lips, G. "Single СЫр Digital Down Converter
Simplifies RF DSP Applications," RF Design, November 1992,
[7] Воиtin, N" "Соmрlех Signals," RF Design, December 1989,
I APPENDIX D
Mean, Variance, and
Standard Deviation
In our studies, we're often forced to consider noise functions. Тhese are
descriptions of noise signals that we cannot explicitly describe with а
time domain equation. Noise functions сan Ье quantified, however, in а
worthwhile way using the statistical measures of mean, variance, and
standard deviation. Although we only touch, here, оп the very broad and
important field of statistics, we will describe why, how, and when to use
these statistical indicators, so that we сап add them to our collection of
signal analysis tools, First we'll determine how to calculate these statisti
cal values for а series of discrete data samples, cover an example using а
continuous analytical function, and conclude this appendix with а dis
cussion of the probability density functions of several random variableE
that are common in the field of digital signal processing, 50 let's proceec.
Ьу sticking our toes in the chilly waters of the mathematics of statistics to
obtain а few definitions.
D.l Statistical Measures
Consider а continuous sinusoid having а frequency of /0 нz with а peak
amplitude of Ар expressed Ьу the equation
x(t) = A p sin(21tfot) ,
(D l)
Equation (D l) completely specifies x(t that is, we сan determine x(t)'s
exact value at anу given instant, For example, when time t = 1/4/0' we know
that x(t)'s amplitude will Ье Ар and, at the later time t = 1/210, x(t)'s ampli
tude will Ье zero, on the other hand, we have по deftnite way to express
476
Stat/stica/ Measures 477
the successive values of а random fиnction or of random noise. t Тhere's по
equation like Eq. (D l) available to predict future noise amplitude values
for example. (Тhat's why they сall it random noise.) Statisticians have, how-
ever, developed powerful mathematical tools to characterize several prop
erties of random fиnctions, Тhe most important of these properties have
been given the names теап, variaпce, and staпdard deviatioп,
Mathematically, the mеап, or average, of N separate values of а sequence
х, denoted x ave ' is defined as [1]
N
xave =..! 2: х (п) = х(1)+х(2)+х(З)+,..+х(N)
N п=l N
(D-2)
Equation (D 2), already familiar to most people, merely states that the
average of а sequence of N numbers is the sum of those numbers divided
Ьу N. Graphically, the average сап Ье depicted as that value about which
а series of sample values cluster, or congregate, as shown in Figure D-1. If
the eight values depicted Ьу the dots in Figure D-1 represent some теа-
sured quantity and we applied those values to Eq. (D-2), the average of the
series is 5.17, as shown Ьу the dotted linе.
6.0 t х(п)ш
5,5t .
50 !
4.5
4,0
3,5
ХаУе= 5,17
. \
.
.
.
.
.
.
.
I I
2 3
I
4
I
5
I
6
I
7
1.
8 п
Flgure D-I Average of а sequence of eight values,
t We deftne raпdoт пoise to Ье unwanted, unpredictable, disturbances contaminating а sig-
nal or а data sequence of interest,
478 Appeпd/x о: Меоп Vor/oпce, oпd Stoпdord Dev/ot/oп
Now that we've defined average, another key deftnition is the variance of а
sequence, о' 2, defined as
1 N
0"2 = N ' )x(п) XaYe]2
п=l
[x(1) XaYe]2 +[x(2) XaYe]2 +[Х(З) Хаvе]2 +",+[x(N) Xave]2 (D З)
N
Sometimes, in the 1iterature, we'll see 0"2 defined with а l/(N l) factor
before the summation instead оЕ the l/N factor in Eq, (D З), Тhere are
subtle statistical reasons why the l/(N l) factor sometimes gives more
accUIate resu1ts [2], However, when N is greater than, say 20, as it wШ Ье
for our purposes, the difference between the twofactors wi1l have по
practical significance.
Variance is а very important concept because it's the yardstick with
which we measure, for example, the effect of quantization епоrs and the
usefulness of signal-averaging algorithms, It gives us an idea how the
aggregate values in а sequence fluctuate about the sequence's average
and provides us with а well defined quantitative measure of those fluc-
tuations, (Бесаusе the positive square root of the variance, the standard
deviation, is typically denoted as о' in the 1iterature, we'll use the conven
tional notation оЕ 0"2 for the variance.) Equation (D-З) looks а bit perplex-
ing if you haven't seen it before. Its meaning becomes clear if we examine
it carefully, Тhe x(l) Хауе value in the bracket, for example, is the differ
ence between the х(l) value and the sequence average Хауе' For any
sequence value х(п), the х(п) Хауе difference, which we denote as д(п),
сап Ье either positive or negative, as shown in Figure D-2. Specifically, the
differences д(l), д(2), Д(З), and д(8) are negative because their corre
sponding sequence values are below the sequence average shown Ьу the
dotted line, If we replace the х(п) Хауе difference terms in Eq, (D-З) with
д(п) terms, the variance сап Ье expressed as
1 N
0"2 = N )д(п)]2, where д(п) = х(п) Хауе
п=l
(D 4)
The reader might wonder why the squares of the differences are
summed, instead of just the differences themselves. If we just add the dif-
ferences, some оЕ the negative д(п)s wШ сanсеl some of the positive д(п)s
Statistica/ Measures 479
6,0 t х(п)
5,5 t
5,01
4,5
A(8)
2
3
Х. уе = 5,17
5 6 7 8 п
4,0
3,5
4
Figure О-2 Dlfference values (п) of the sequence in Figure D-l,
resu1ting in а sum that тау Ье too small. For example, if we add the /\(n)s
in Figure D-2, the positive /\(6) and А(7) values and the negative /\(3)
value will just about cancel each other out and we don't want that,
Because we need an unsigned measure of each difference, we use the dif
ference-squared terms as indicated Ьу Eq. (D-4). In that way, individual
/\(п) difference terms will contribute to the overall variance regardless of
whether the difference is positive or negative. Plugging the А(п) values
from the example sequence in Figure D-2 into Eq. (D-4), we get а variance
value of 0,34. Another useful measure of а signal sequence is the square
root of the variance known as the standard deviation. Taking the square
root of Eq. (D-3) to get the standard deviation о,
N
а = Ы = )x(п) XaYe]2
п l
(D 5)
So far, we have three measurements to use in evaluating а sequence
of values: the average ХаУе' the variance 02, and the standard deviation
о. Where Хауе indicates about what constant level the individual
sequence values vary, 02 is а measure of the magnitude of the noise fluc-
tuations about the average Хауе. If the sequence represents а series of
random signal samples, we сап say that Хауе specifies the average, or
constant, value of the signal. The variance о 2 is the magnitude squared,
or power, of the fluctuating component of the signal. The standard devi
ation, then, is аn indication of the magnitude of the fluctuating coтpo
nent of the signal.
480 Appendix о: Меаn, Variaпce, aпd Staпdard Deviatfoп
D.2 Standard Deviation, or RMS, 01 а Continuous
Sinewave
For sinewaves, electrical engineers have taken the square root of Eq.
(D 3), with Хауе = О, and defined а useful parameter, called the rтs value,
that's equal to the standard deviation, For discrete samples, that parame
ter, x rms ' is defined as
Х nn. =
1 N
Lx(п)2
N п=1
(D 6)
Тhe х пns in Eq. (D 6), obtained Ьу setting Хауе = О in Eq. (D 5), is the square
root of the тean (average) of the squares of the sequence х(п). For а con-
tinuous sinusoid x(t) = A p sin(21tft) = Apsin(Q)t) whose average value is
zero, x(t)rms is x(t)rms-sinewave defined as
2.
x(t)rms sinewave = 2 f [Ар sin(Q)t)]2 d(rot)
2 2.
А f 1
....L [1 cos( ш)] d( Q)t)
2х 2
А; . [ rot .!(Sin(Q)t)) ] 27t = А; . [ 2Х ]
2х 2 2 2х 2
о
Ар
.J2
(D-7)
Тhis .x(t)rms-sinewave = A/.J2 expression is а lot easier to use for ca1culating
average power dissipation in circuit elements than taking the integral of
more complicated expressions for instantaneous power dissipation. Тhe
variance of а sinewave is, of course, the square of Eq, (D-7), or Ai/2,
We've provided the equations for the mean (average) and variance of а
sequence of discrete values, introduced an expression for the standard
Тhe Меап aпd Var/aпce of Raпdom Fuпctioпs 481
deviation or rms of а sequence, and given an expression for the rms оЕ а
continuous sinewave. The next question is "How сan we characterize ran-
dom functions for which there are по equations to predict their values and
we have по discrete sample values with which to work?" The answer is
that we must use probability density functions,
D.З The Mean and Variance о' Random Functions
То determine the теan or variance of а random function, we use what's
called the probability deпsity fuпctioп. The probabi1ity density function
(PDF) is а measure of the likelihood of а particular value occurring in
some function. We Сan explain this concept with simple examples of Шр
ping а СОin or throwing dice as i11ustrated in Figure D 3(a) and (Ь). The
result of f1ipping а СОin сan only Ье one оЕ two possibilities: heads or tai1s.
Figure D 3(a) indicates this PDF and shows that the probabi1ity (like1i-
hood) is equal to one half for both heads and tails. That is, we have an
equal chance that the СОin side facing ир wi1l Ье heads or tai1s. The sum
of those two probabi1ity values is one, meaning that there's а 100% prob-
abi1ity that either а head or а tai1 wi1l occur.
Figure D 3(b) shows the probabi1ity of а particular sum оЕ the upper
faces when we throw а pair of Фсе. This probabi1ity function is not uni
form because, for example, we're six times more likely to have the die
faces add to seven than add to two (snake eyes). We сan say that after toss-
ing the dice а large number of times, we should expect that 6/36 = 16,7
percent of those tosses would result in sevens, and 1/36 = 2.8 percent of
the time we'll get snake eyes. The sum of those eleven probabi1ity values
CoJn lace PDF
Sum 01 dice PDF
Order 01 birth PDF
6IЭ6
"" '/З
2/з8
Gender
order
Coin
Hoada !асе
(а) (Ь)
(с)
Flgure D.3 Slmple probobIllty density functions: (о) the probobIllty of flipping о
slngle coin; (Ь) the probobIllty of о portlculor sum of the upper
foces of two die; (с) the probobility of the order of bIrth of the glrl
ond her slbIlng,
482 Appeпd/x о: Меаn Var/aпce, aпd staпdard Dev/atloп
in Figure D-3(b) is also one, telling us that this PDF accounts for all (100%)
of the possible outcomes of throwing the dice.
ТЬе fact that PDFs must account for аН possible results is emphasized
in ал interesting way in Figure D 3(c). If а woman says, "Of ту two chil
dren, one is а girl, What's the probability that she has а sister?" Ве careful
now uriously enough, the answer to this controversial question is not а
50 50 chance. There are more possibilities to consider than just the girl
havi.ng а brother or а sister, We сап think of all the possible combinations
of birth order of two children such that one child is а girl, Because we
don't know the gender of the first born child, there are three gender order
possibilities: girl, then Ьоу; Ьоу, then girl; and girl, then girl as shown in
Figure D 3(c). 50 the possibility of the daughter having а sister is 1/3
instead of 1/2! (Believe Н,) Again, the sum of those three 1/3rd probabil
ity values is one.
Two important features of PDFs are illustrated Ьу the examples in
Figure D 3: PDFs are always positive, and the areas under their curves
must Ье equal to unity. The very concept of PDFs make them а positive
like/ihood that а particular resu1t will оссщ and the fact that some result
must occur is equivalent to saying that there's а probabi1ity of one (100%
chance) that we'll have а result, For continuous probability density func
tions, РИ, we indicate these two characteristics Ьу
PDF va1ue are never negative: .....
p(f) ? О ,
(D 8)
and
Тhe "ит of аН the PDF values is one: .....
J p(j)dJ= 1 ,
(D 8')
In 5ection D,l we illustrated how to calculate the average (mean) and
variance of discrete samples, We сап also determine these statistical теа-
sures for а random function if we know the PDF of the function, Using Jlf
to denote the average of а random function ofJ, then, Jlfis defined as
Jlf = J!.p(f)dJ,
(D 9)
and the variance of J is defined as [3]
Тhe Меап and Var/ance о' Randoт Fиncf/ons 483
J Continous P(f)
1 1
. ...............
Ь ( a) Ь+а
I ....
a О Ь f
Flgure D.4 Continuous, unlforт probabillty density function,
(11= f(f f)2.p(f)df= ff2.p(f)df 7'
(D 10)
In digital signal processing, we'll encounter continuous probability
density functions that are uniform in value similar to the examples in
Figure D-З. In these cases, it's easy to use Eqs, (D 9) and (D-10) to deter
minе their average and variance. Figure D 4 illustrates а uniform, contin
uous PDF indicating а random function whose values have an equal
probability of being anywhere in the range from a to Ь. From Eq. (D-8),
we know that the area under the curve must Ье unity (i.e., the probability
is 100% that the value will Ье somewhere under the сшvе), 50 the аmрli-
tude ofp(f) mustbe the area divided Ьу the width, orp(f) = l/(Ь + а). From
Eq. (D 9), the average of this p(f) is given Ьу
! = f f. p(f)df= f f' df
Ь+а
ь [ 2 ] Ь
= f fdf= , L
Ь+а Ь+а 2
a a
ь 2 а 2
2(Ь + а)
(b+a)(b a)
2(Ь +а)
b a
2
(D 11)
which happens to Ье the midpoint in the range from a to Ь. Тhe variance
of the PDF in Figure D 4 is given Ьу
484 Appendix о: Меоn, Vorionce, ond Stondord Deviotion
2 f 2 2 b f 2 1 (Ь а)2
а f = f . p(f)df 11 f = f' df
_ Ь+а 4
= . [ i ] b (b a)2 = '(b3 +a3) (b a)2
Ь+а 3 a 4 3(Ь+а) 4
(Ь+а)(ь 2 ab+a2)
=
3(Ь+а)
ь 2 2ab+a2
4
= ь 2 +2аЬ+а 2 (b a)2
12 12
(D 12)
We use the results оЕ Eqs. (D l1) and (D 12) in Chapter 9 to analyze the
errors induced Ьу quantization from analog to-digital converters and the
effects оЕ finНе word lengths оЕ hardware registers.
D.4 The Normal Probability Density Function
А рrоЬаЬШtу density ЕипсНоп that's so оНеп encountered in nature
deserves our attention. This function is so common that it's actually called
the пorтa! probability density function,t Тhis function, whose shape is
shown in Figure D-S, is important because random data having this dis-
tribution is very useful in testing both software algorithms and hardware
processors. The normal PDF is defined mathematically Ьу
р(х) = e (X X"")' /20' .
а .f2it
(D-13)
The area under the curve equals опе, and the percentages at the bottom оЕ
Figure D S tell us that, for random functions having а поrшаl distribution,
there's а 68.27 percent chance that anу particular value оЕ х wШ differ
from the mеan Ьу $а. Likewise, 99.73 percent of all the х data values wШ
Ье within 3а оЕ the mеan Хауе'
t The пorтal probabili/y deпsity fипctioп is sometimes called the Gaussian function, А scheme
for generating data 10 fit this function is discussed in Section 10,12,
Refвrences 485
р{х)
х
-<
Ха"" о' Хауо + о'
68.27%
95.45%
99,73%
Ха"" + 3cr
Ха"" 3cr
>
Flgure О-5 А normal PDF wlth mean = Xove and standard deviation = 0',
References
[1] Papoulis, А. ProЬability Random Variables, aпd Stochastic Processes, McGraw-Нill,
New York, 1965, рр. 266---68.
[2] МШеr, 1" and Freund, J, Probability aпd Statistics for Eпgiпeers, 2nd Ed"
Prentice-Ha11, Englewood Cliffs, New Jersey, 1977, рр, 118,
[3] Bendat, J" and Piersol, А. Measuremeпt aпd Ana/ysis о/ Raпdom Data, John
Wileyand Sons, New York, 1972, рр. 61,
APPENDIX Е
Decibels
(dB and dBm)
This appendix introduces the 10garithmic function used to improve the
magnitude reso1ution of frequency-domain p10ts when we eva1uate signa1
spectrums, digita1 filter magnitude responses, and window function mag-
nitude responses, When we use а 10garithmic fиnction to p10t signa11ev-
e1s in the frequency domain, the vertica1 axis unit of measure is decibels.
Е.l Using Logarithms to Determine Relative
Signal Power
In discussing decibe1s, it's interesting to see how this unit of measure
ev01ved, When comparing continuous (ana1og) signa11eve1s, ear1y spe-
cia1ists in e1ectronic communications found it usefи1 to define а measure
of the difference in powers of two signa1s, If that difference was treated as
the 10garithm of а ratio of powers, it cou1d Ье used as а simp1e additive
measure to determine the overall gain or 10ss of cascaded e1ectronic cir
cuits, The positive 10garithms associated with system components having
gain cou1d Ье added to the negative 10garithms of those components hav
ing 10ss to quick1y determine the overall gain or 10ss of the system. With
this in mind, the difference between two signa1 power 1eve1s (Р} and Pz),
measured in be1s, was defined as the base 10 10garithm of the ratio of
those powers, or
Power difference = 10g10( ;: J be1s,+
(E-l)
t Тhe dimensionless unit of measure bel was named in honor of Alexander Graham ВеН,
486
Using LogQrithms to Determine Re/Qtive SignQ/ Power 487
Тhe use of Eq, (E l) led to another evolutionary step because the иnа of
bel was soon found to Ье inconveniently large, For example, it was dis
covered that the human ear could detect audio power level differences of
one tenth of а bel, Measured power differences srnaller than оnе bel were
so comrnon that it led to the use of the decibel (Ьеи10), effectively mak-
ing the unit of bel obsolete. Тhe decibel (dB), then, is а unit of rneasure of
the relative power difference of two signals defined as
Power difference = 10 .10glO( J dB ,
(E 2)
The logarithrnic function 10 'loglO' plotted in Figure Е-1, doesn't seem too
beneficial at first glance, but its application turns out to Ье very useful,
Notice the large change in the function's value, when the power ratio
(Pl/P2) is srnall, and the gradual change when the ratio is large. Тhe effect
of this nonlinearity is to provide greater resolution when the ratio P 1 /P 2
is srnall, giving us а good way to recognize very srnall differences in the
power levels of signal spectrurns, digital filter responses, and window
function frequency responses.
Let's dernonstrate the utility of the logarithrnic function's variable res-
olution, First, rernernber that the power of аnу frequency domain
sequence representing signal rnagnitude I Х(т) I is proportional to
10'log(P1 IP 2 ), in dB
4
2
О
2
4
--6
--6
10
12
14
16
18
1,3
1,5
1,7 1,9 2,0 Р 1 /Р 2
Figure Е-l Logarithmic decibei function of Eq, (Е-2),
488 Append/x Е: Dec/be/s (dB ond dBm)
'Х(т) I squared. For convenience, the proportionality constant is
assumed to Ье one, so we say the power оЕ / Х{т) , is
discrete power spectrum оЕ Х(т) = / Х(т) /2.
(Е З)
Although equation (E 3) mау not actually represent power (in watts) in
the classical sense, it's the squaring operation that's important here,
because it's analogous to the traditional magnitude squaring operation
used to determine the power оЕ continuous signaIs. (ОЕ course, if Х(т) is
complex, we сап caIculate the power spectrum sequence using
I Х(т) 12 = X reaI (m)2 + X ima s<m)2,) Taking 10 times the log оЕ Eq. (E 3)
allows us to express а power spectrum sequence XdB(m) in dB as
XdB(m) = 10' log10 ( 1 Х{т) 12) dB.
(E 4)
Because log(x 2 ) = log(x) + log(x) = 2Iog(x), we сап eliminate the squaring
operation in Eq, (Е-4) Ьу doubIing the factor оЕ 10 and represent the
power spectrum sequence Ьу the expression
XdB(m) = 20' log10 (1 Х(т) 1) dB ,
(Е-Б)
Without the need for the squaring operation, Eq. (Е Б) is а more conve
nient way, than Eq. (E 4), to caIculate the XdB(m) power spectrum
sequence from the Х(т) sequence.
Equations (Е-4) and (Е Б), then, are the expressions used to convert а
linear magnitude аxiБ to а logarithmic magnitude--squared, or power, axis
measured in dB, What we most often see in the literature are normalized
log magnitude spectral plots where еаф value оЕ 1 Х(т) 12 is divided Ьу
the first I Х(О) 12 power value (for т = О) as
, ( 1 Х(т) 12 ) ( ' Х(т) ' )
normallzedX dB (m)=10.log lO 2 =20.log lO dB.
I Х(О) I 1 Х(О) 1
(E 6)
Тhe division Ьу the 1 Х(О) 12 or 1 Х(О) I уaIие aIways forces the first уaIие in
the normalized log magnitude sequence XdВ(т) equaI to О dB.t This makes
it easy for us to compare multiple log m.agnitude spectraI plots. То illus
trate, let's look at the frequency domain representations оЕ the Hanning
t Тhat's because ]ogI0( J Х(О) J / I Х(О) 1) = logloi1) = О,
Us/ng Logar/thтs to Deterтine Re/at/ve Signa/ Power 489
and triangular window functions, The magnitudes of those frequency
domain functions are plotted оп а linear scale in Figure Е-2(а) where we've
arbitrarily assigned their peak values to Ье two. Comparing the two linear
scale magnitude sequences, Wиanrring(т) and W triangиJar(т), we сan see some
minor differences between their magnitude values, If we're interested in the
power associated with the two window functions, we square the two mag-
nitude functions and plot them оп а linear scale as in Figure E 2(Ь). The dif
ference between the two window function's power sequences is impossible
to see above the frequency of, say, т = 8 in Figure E 2(b), Here's where the
dB scale helps иs out, If we plot the normalized log magnitude versions of
the two magnitude squared sequences оп а logarithmic dB scale using Eq.
(E 6), the difference between the two functions will Ьесоте obvious,
Normalization, in the case of the Hanning window, amounts to calcu
lating the log magnitude sequence normalized over 1 WиanniniО) I as
( 1 WИanning(т) 12 J ( 1 WИanning(т) ' )
WH(т) = 10.10g 10 2 =20.10g lO dB. (Е-7)
I Wиаnning(О) I 1 Wиаnning(О) 1
The normalized log magnitude sequences are plotted in Figure Е-2(с), We
сап now clearly see the difference in the magnitude squared window
functions in Figure E 2(c) as compared to the linear plots in Figure E 2(b).
Notice how normalization forced the peak values for both log magnitude
functions in Figure E 2(c) to Ье zero dB. (The dots in Figure E 2 are соп-
nected Ьу lines to emphasize the sidelobe features of the two log таgni-
tude sequences.)
Although we've shown the utility of dB plots using window function
frequency responses as examples, the dB scale is equally useful when
we're plotting signal power spectrums or digital fi1ter frequency
responses, We сап further demonstrate the dB scale using а simple digital
fi1ter example, Let's say we're designing ап ll tap highpass FIR fi1ter
whose coefficients are shown in Figure E 3(a). If the center coefficient h(5)
is O.48, the fi1ter' s frequency magnitude response 1 H ,48(т) 1 сап Ье plot
ted as the white dots оп the linear scale in Figure E 3(b). Should we change
h(5) лот O,48 to O.5, the new frequency magnitude response I H .5(т) I
woиld Ье the black dots in Figure E 3(b), It's difficиlt to see much of а dif-
ference between 1 H O,48(т) I and I H O.5(т) 1 оп а linear scale, If we иsed
Eq. (E 6) to calcиlate two normalized log magnitude sequences, they coиld
Ье plotted as shown in Figure E 3(c) where the filter sidelobe effects of
changing h(5) from O,48 to O,5 are now easy to see.
490 Appвnd/x Е: Dec/be/s (dB ond dBm)
2,0
1,8
1,6
1,4
1,2
(а) 1,0
0,8
0,6
0,4
0,2
О
О
4,0
3,6
3,2
2,8
2.4
(Ь) 2,0
1,6
1,2
0,8
0,4
О
О
10
20
O
(с) --40
50
--60
70
Window function magn ude responses
on а linear scale
/1 WHann;ng (т)'
I WtrfangUla, (т)'
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Frequency
(т)
Window functlon magnltude squared
responses on а linear scale
2
IWtrfangula, (т)'
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Frequency
(т)
W;ndow functlon тagnltudв squarвd responses
О on а normalized logarlthmlc scale (dB)
о 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Frequency
(т)
Flgure Е-2 Honnlng (whlte squares) ond trlongular (bIack squares) wlndow functlons
In the frequency domain: (о) magnitude responses using о linear scale;
(Ь) magnitude-squored responses uslng а linear scoie; (с) log magnltude
responses uslng о normallzed dB scale,
Using Logor/thms to Determine Reiotive S/gnoi Power 491
! АА ..., ""","М., ц.
0,4
. .
0,2
o 2 5 8 Т ,....
(а) I I I I ,
2! 1 . 3 4 6 7 . 9 10 k
...(),4
. .... . "'(),48
",(),б
1,2
FIR filter magnitude responses
оп а linear scale
0,8
(Ь) О,б
0,4
0,2
О
О 2 4 б
8 10 12 14 1б 18 20 22 24 2б 28 30 Frequency
(т)
10
FIR filter magnitude responses
О ..."..? .. ..I<>9. .r.i! : .: ::( >".
1 (';; ;; 7 ш ' r ."ш ............ .шш..шш ..0
JI '1' , " ... ..', ....., ....,
.\шI. \ " ...... ii . ,,",' "'""""""
\ . .
......... ....
20
(с)
--30
--40
50
О 2 4 б
8 10 12 14 1б 18 20 22 24 26 28 30 Frequency
(т)
Flgure Е-3 FIR fllter mognltude responses: (о) FIR fllter tlme-domaln coefflcients:
(Ь) magnltude responses u51ng а Ilneor scole, I Н..().48(т) I Is shown Ьу
the white squores and I H..().s(m) I 15 shown Ьу the bIack squares; (с) log
mognltude responses using the dB scale,
492 Appendix Е: Dec/be/s (d8 ond d8m)
Е.2 Some Useful Decibel Numbers
If th.e reader finds him or herself using dB scales оп а regular basis, there
are а few constants worth committing to memory. А power difference of
3 dB corresponds to а power factor of 2; that is, if the magnitude--squared
ratio of two different frequency components is 2, then from Eq. (E 2),
power difference = 10 .10glO( т) = 10 .10glO(2) = 3.01 '" 3 dB . (E 8)
Likewise, if the magnitude--squared ratio of two different frequency com
ponents is 1/2, then the relative power difference is 3 dB because
power difference = 10 'loglO( ) = 10 .10g10(0.5) = 3.01 '" 3 dB , (E 9)
ТаЫе E 1lists several magnitude and power ratios vs. dB values worth
remembering, Кеер in mind that decibels indicate on1y relative power
ТаЫе Е-l Зоmе Useful Logarlthmic Relationships
Мagпitиde-
Magпitиde Sqиared Power Relative dB
Ratio (Р/Р2) Ratio (approxiтate)
Нт 1 / 2 НТ 1 10 +-- P l 1s one-tenth Р2
2 1 Z-2 = 1/4 --6 +-- р 1 iI one.fourth Р 2
2 1/2 z-1 = 1/2 3 +-- Рl iI one-half Р1
20 20 = 1 О +-- P 1 iI equa1 to Р'/.
21/2 21 =2 3 .... Рl iI twlce Р2
21 22 = 4 6 ....Рllsfоurtiш"Р2
101/2 101 = 10 10 .... Рl iI ten times Р2
101 102 = 100 20 .... р 1 iI one hundred timel Р 2
103/2 1()3 = 1000 30 +-- р 1 iI one thousand tlm" Р 2
Abso/ute Power Using Decibe/s 493
relationships. For example, if we're told that signal А is 6 dB above signa1
В, we know that the power of signal А is four times that of signal В, and
that the magnitude of signal А is twiee the magnitude of signal В. We тау
not know the absolute power of signals А and В in watts, but we do know
that the power ratio is Р А/ Р в = 4.
Е.3 Absolute Power Using Decibels
Let's diseuss another use of deeibels that the reader тау eneounter in the
literature. It's eonvenient for practitioners in the electronie eommuniea
tions field to measure continuous signal power levels refereneed to а spe
еШе absolute power level, In this way, they сап speak of absolute power
levels in watts while taking advantage of the eonvenienee of decibels. The
most eommon absolute power referenee level used is the milliwatt. For
example, if Р 2 in Eq. (E 2) is а referenee power level of оnе mil1iwatt, then
( Р 1 ) ( Рl in watts )
absolutepowerofP 1 =10.1og 1o =10.1og 1o .. dBm,(E 10)
Р 2 1 milllwatt
Тhe dBm unit of measure in Eq. (E 10) is read as "dB relative to а milli
watt," Тhus, if а eontinuous signal is speeified as having а power of 3
dBm, we know that the signal's absolute power level is 2 times оnе milli-
watt, or 2 mil1iwatts. Likewise, а 10 dBm signal has аn absolute power of
0.1 milliwatts,t
Тhe reader should take eare and not inadvertently use dB and dBm
interchangeably. Тhey mеan very different things. Again, dB is а relative
power level relationship, and dBm is an absolute power level in milliwatts.
t Other absolute reference power levels сan Ье used, People invo!ved with high power trans-
mitters sometimes ше а sing!e watt as their reference power leve!, Their unit of power using
decibels is the dBW, read as "dB re!ative to а watt," In this case, for example, 3 dBW is equa!
to а 2 watt power leve!,
I APPENDIX F I
Digital Filter Terminology
The first step in becoming familiar with d.igital filters is to learn to speak the
language used in the filter business. Fortunately, the vocabulary оЕ digital
filters corresponds very well with the mother tongue used for continuous
(analog) filters o we don't have to unleam anything that we already
know. This appendix is an introduction to the terminology оЕ digital filters.
Allpass filter an IIR filter whose magnitude response is unity over its
entire frequency range, but whose phase response is variable. Allpass
filters are typically appended in а cascade arrangement following а
standard IIR filter, Hj(z), as shown in Figure F-l.
х(п) OriginalllR Filter: Allpass IIR Filter: у(п)
H 1 (z) Hap(z)
Combined IIR Filter: Hcomblned(Z)
Flgure F-l Typlcol use of оп ollposs filter,
An allpass fНter, Ha/z), сап Ье designed so that its phase response
compensates for, or equalizes, the nonlinear phase response оЕ an orig
inal IIR filter [l З]. Thus, the phase response of the combined filter,
Hcombined (z), is more linear than the original Hj(z), and tbls is particu
larly desirable in communications systems. In tbls context, an al1pass
filter is sometimes called а phase equalizer.
494
Allpass filter Bandwidth 495
Attenuation ап amplitude 10ss, usually measured in dB, incurred Ьу
а signal after passing through а digital filter, Filter attenuation is the
ratio, at а given frequency, of the signal amplitude at the output of
the filter divided Ьу the signal amplitude at the input of the filter,
defined as
attenuation = 20 .10 g lO l ::1 ) dB ,
(p 1)
For а given frequency, if the output amplitude of the filter is smaller
than the input amplitude, the ratio in Eq. (P 1) is less than опе, and
the attenuation is а negative number,
Бапd Reject Filter а filter that rejects (attenuates) one frequency band
and passes both а lower and а higher frequency band. Figure Р-2(а)
depicts the frequency response of an ideal band reject filter. Тhis filter
type is sometimes called а notch fiIter,
nJ.1 I Xn J .
nJ.1 ! xnJ .
Band reject filler ,ввропвв
Bandpass flller response
......1
........1..
',/2
о
(а)
',/2 Frвq ',12
'o
о
(Ь)
'0
',12 Freq
Flgure F.2 Fllter symbols and frequency responses: (а) band reJect filter;
(Ь) bandpass filter,
Бandраss Filter а filter, as shown in Figure Р-2(Ь), that passes опе fre-
quency band and attenuates frequencies above and below that band.
Бandwidth the frequency width of the passband of а filter. For а 10w
pass filter, the bandwidth is equal to the cutoff frequency. For а band-
496 Appвndix F: Digitoi Fiiter Terтinoiogy
pass filter, the bandwidth is typically defined as the frequency differ
ence between the upper and lower 3 dB points.
Bessel function а mathematical fиnction used to produce the most lin-
ear phase response оЕ аН IIR filters with по consideration оЕ the fre-
quency magnitude response. Specifically, filter designs based оп
Bessel functions have maximal1y constant group delay.
Butterworth function а mathematical fиnction used to produce maxi-
таНу flat filter magnitude responses with по consideration оЕ phase
linearity or group delay variations, Filter designs based оп а
Butterworth fиnction have по amp1itude ripple in either the passband
or the stopband. Unfortunately, for а given filter order, Butterworth
designs have the widest transition region of the most popular filter
design fиnctions,
Cascaded fi1ters а filtering systeт where multiple individual filters are
connected in series; that is, the output of one filter drives the input of
the fol1owing filter as il1ustrated in Figures F-l and 6-38(а).
Center Frequency </0> the frequency lying at the midpoint оЕ а band
pass filter. Figure F-2(b) shows thelo center frequency of а bandpass
fil ter,
Chebyshev function а mathematical function used to produce pass-
band or stopband ripples constrained within fixed bounds. There are
families of Chebyshev fиnctions based оп the amount of ripple, such
as 1 dB, 2 dB, and 3 dB of ripple, Chebyshev filters сan Ье designed to
have а frequency response with ripples in the passband and flat pass
bands (Chebyshev Туре 1), or flat passbands and ripples in the stop
band (Chebyshev Туре П). Chebyshev filters cannot have ripples in
both the passband and the stopband, Digital filters based upon
Chebyshev fиnctions have steeper transition region rol1-off but more
nonlinear-phase response characteristics than, say, Butterworth filters,
Cutoff Frequency the upper passband frequency for low-pass filters,
and the lower passband frequency for highpass filters. А cutoff fre-
quency is determined Ьу the 3 dB point of а filter magnitude
response relative to а peak passband value. Figure F 3 il1ustrates the
Ic cutoff frequency of а low pass filter,
Bessel functlon Elliptic funcflon 497
Frequency magnitude response (dB)
о
20
Stopba"d
(.Fr Freq
. i
о
,
I
Passba"d 1fa"sitlo" r
, 'eglo"
'с
Flgure F-3 А low-pass dlgital ftlter frequency response, The stopband relative
amplltude is 20 dB,
Decibels (dB) а uлit of attenuation, or gain, used to express the relative
voltage or power between two sign.als, For filters, we иве decibels to indi
cate cutoff frequencies ( 3 dB) and stopband signallevels ( 20 dB) as
illustrated in Figure F 3, Appendix Е discusses decibels in тосе detail,
Decimation fi1ter а low pass digital FIR filter where the output sam
ple rate is less than the filter's input sample rate. As discussed in
Section 7,3, to avoid aliasing problems, the output sample rate must
not violate the Nyquist criterion,
Digital fШеr computational process, or algorithm., transforming а dis-
crete sequence of numbers (the input) into another discrete sequence
of numbers (the output) having а modified frequency domain spec-
trum. Digital filtering сап Ье in the form of а software routine operat-
ing оп data stored in computer memory or сan Ье implemented with
dedicated hardware,
Elliptic function а mathematical function used to produce the sharpest
roll off for а given number of fi1ter taps. However, fi1ters designed Ьу
using elliptic functions, also called Cauer filters, have the poorest
phase linearity of the most common IIR filter design functions, The
ripple in the passband and stopband are equal with еШрtiс filters,
498 Appendix F: Digita/ Filter Terтin%gy
Envelope delay see group delay.
Filter coefficients the set of constants, also called tap weights, used to
multiply against delayed signal sample values within а digital filter
structure. Digital filter design is аn exercise in determining the filter
coefficients that will yield the desired filter frequency response. For
аn FIR filter, Ьу definition, the filter coefficients are the impulse
response of the filter,
Fi1ter order а number describing the highest exponent in the numera
tor or denominator of the z domain transfer fиnction of а digital filter.
For FIR filters, there is по denominator in the transfer fиnction, and
the filter order is merely the number of taps used in the filter struc
ture. For IIR filters, the filter order is equal to the number of delay ele-
ments in the filter structure. Generally, the larger the filter order, the
better the frequency magnitude response perfonnance of the filter,
Finite impulse response (FIR) fi1ter а class of digital filters that has
only zeros оп the z-plane, Тhe key implications of this are that FIR fil
ters are always stable and have linear phase responses (as long as the
filter's coefficients are symmetrical), For а given filter order, FIR fil
ters have а тиш more gradual transition region roll off than digital
IIR filters,
Frequency magnitude response а frequency domain description of how
а filter interacts with input signals. Тhe frequency magnitude response
in Figure P 3 is а curve of filter attenuation (in dB) vs. frequency.
Associated with а filter's magnitude response is а phase response,
Group delay the derivative of а filter's phase with respect to frequency,
G = 110/11f, or the slope of а filter's Н,,(т) phase response curve, Тhe
concept of group delay deserves additional explanation beyond а
simple definition. For an ideal filter, the phase will Ье 1inear and the
group delay would Ье constant, Group delay, whose unit of measure
is time in seconds, сап also Ье thought of as the propagation time
delay of the envelope of аn amp1itude modulated signal as it passes
through а digital filter. (In this context, group delay is often called
eпvelope delay.) Group delay distortion occurs when signals at differ
ent frequencies take different amounts of time to pass through а filter.
If the group delay is denoted С, then the relationship between group
delay, а 110 increment of phase, and а N increment of frequency is
Enve/ope de/ay Groиp de/ay 49С;
G '" -Ll degrees /360 '" Ll ra(lians /2п seconds,
tJ.f tJ.f
(Р-2)
If we know а linear phase filter's phase shift ( ) in degrees/Hz, or
radians/Hz, we сan detennine the group delay in seconds using
G. д! = G.l = G = -Ll dеgrееsIНz = - Ll radianslHz seconds, (Р-3)
360 2п
То demonstrate Eq, (P 3) and illustrate the effect оЕ а nonlinear
phase filter, let's assume that we've digitized а continuous wavefonn
comprising four frequency components defined Ьу
х( t)=sin(21t.1. t)+sin(21t.з. t) /3+sin(21t.5. t) /5+sin(21t. 7. t) /7. (P 4)
The x(t) input comprises the sum of l Hz, 3-Hz, 5 Hz, and 7-Hz
sinewaves, and its discrete representation is shown in Figure Р-4(а). If
we applied the discrete sequence representing хи) to the input of an
ideal 4-tap linear-phase low-pass digital FIR filter with а cutoff fre
quency оЕ greater than 7 Hz, and whose phase shift is -0.25 radi-
ans/Hz, the filter's output sequence would Ье that shown in Figure
Р-4(Ь).
Because the filter's phase shift is O,25 radians/Hz, Eq, (P 3) tells
us that the filter's constant group delay G in seconds is
G -д radians/Hz = 0,25 = 0.04 seconds,
2п 2п
(Р-5)
With а constant group delay оЕ 0,04 seconds, the 1-Hz input sinewave
is delayed at the filter output Ьу 0,25 radians, the 3 Hz sinewave is
delayed Ьу 0.75 radians, the 5 Hz sinewave Ьу 1.25 radians, and the
7 Hz sinewave Ьу 1.75 radians. Notice how а linear-phase (relative to
frequency) filter results in an output that's merely а time shifted ver
sion of the input as seen in Figure Р-4(Ь). The amoиnt оЕ time shift is
the group delay оЕ 0.04 seconds, Figure P 4(c), оп the other hand,
shows the distorted output wavefonn if the fi1ter's phase was nonlin-
ear, for whatever reason, such that the phase shift was 3.5 radians
instead of the ideal1,75 radians at 7 Hz, Notice the distortion оЕ the
beginning оЕ the output waveform envelope in Figure Р-4(с) сот-
500 Appendix F: Digita/ Filter Terтin%gy
(а)
t Filter input
1 .. ..
.1!8 .8,. ..,. .,1'.
l ' I ' i ..: : I ;.. i i . '..! : I
: 1 i 1" '!!: 1\': '1 i' i i 1':\
0,5 ,! l i r ! I : 11 i I i i : ' : I i i : :.
О j l i I ! 11 i I i i! [: 1 1 1 1 : I!: I I ". ., . .
! \ ' : . I : , I , I i I : " I I i : I '1 I
: i: : '! I : 1: 1: 11' i /1 i 1 1 ' Time
) i i 1 < ; ! ,1 . ' I ! j ! : J j i : : i ,! !
-0,5 8 \ 1 i I i I \ ( '! i 1 i 1 [! 11 i ' , ' V I.
i I :io.i \, 1..:, II I :,!'
i.. .... .... .!
-1 ..
Linear phase filter output
0,5
(Ь) о
"'"
-0,5 0,04 sec,
-1
Nonlinear phase filter output
0,5
(с) О
-0,5
-1
Time
Time
Flgure F-4 Filter tiтe-doтain response exaтples: (а) filter /nput sequenee; (Ь) linear-
phase tilter output sequenee that's tiтe shifted Ьу 0,04 seeonds,
duplleating the input sequenee; (е) distorted output sequenee due to а
tilter wlth а nonlinear phase,
аоп is contained in the envelope of а signal that we' re passing through
а filter, we'd like that filter's passband phase to Ье as Jjnear as possi
Ые with respect to frequency, In other words, we'd prefer the filter's
.,.\1'&'-
Нo!f-8oпd "" '"""""h::
group delay to уату as little as possible in the passband, (А:АiIIJtAJ4'я-J . 'Щ
aspects of nonlinear phase filters are discussed in Section 5,8 !II!"
1!fI\'gI""
Half-Band fi1ter а type of FIR filter whose transition region is centered
at one quarter of the sampling rate, or 1./ 4. Specifically, the end оЕ the
passband and the beginning of the stopband are equally spaced about
f/4. Due to their frequency domain synunetry, half-band filters are
often used in decimation fi1tering schemes because half of their time-
domain coefficients are zero. Тhis reduces the number of necessary
filter multiplications, as described in Section 5.7,
Hlghpass filter response
'ш 1
Low-pass fitter response
ш1
'. /2
о
(а)
',12 Freq
'./2
о
',12 Freq
(Ь)
Flgure F-5 FJlter symbols and frequency responses: (а) highpass fIIter; (Ь) low pass
fIIter,
Highpass filter а fi1ter that passes high freqиencies and attenuates low
frequencies, as shown in Figиre Р-5(а). We've аН experienced а kind
of highpass filtering in our living rooms. Notice what happens when
we turn up the treble control (or turn down the bass control) оп our
home stereo systems. Тhe audio amplifier's normally flat frequency
response changes to а kind оЕ analog highpass fi1ter giving us that
sharp and tiппy sound as the high-frequency components of the
music are being accentuated,
Impulse response а digital fi1ter's time domain output sequence when
the input is а single иnity valued sample (impulse) preceded and fol
lowed Ьу zero valued samples, А digita1 filter' s freqиency domain
502 Append/x F: Dig/ta/ Filter Terтin%gy
response сап Ье calculated Ьу taking the discrete Fourier transform of
the fi1ter's time domain impulse response [4].
Infinite impulse response (IIR) filter а class of digital filters that тау
have both zeros and poles оп the z-plane, As such, IIR filters are not
guaranteed to Ье stable and almost always have nonlinear phase
responses, For а given filter order (number of IIR feedback taps), IIR
filters have а much steeper transition region roll-off than digital FlR
filters.
Linear-phase filter а filter that exhibits а constant change in phase
angle (degrees) as а function of frequency, Тhe resultant filter phase
plot vs. frequency is а straight line. As such, а linear-phase filter's
group delay is а constant, То preserve the integrity of their informa
tion-carrying signals, linear phase is an important criteria for filters
иsed in communication systems.
Low pass filter а filter that passes low frequencies and attenuates high
frequencies as shown in Figure F 5(b), Ву way of example, we experi
ence low-pass filtering when we turn ир the bass control (or turn
down the treble control) оп our home stereo systems giving us that
dull, muffled sound as the low frequency components of the music
are being intensified,
Notch filter see band reject filter,
Frequency magnitude response (dB)
1"
Stopband
attenuation
lш::1 Ш.
о
f ./2 Freq
Flgure F-6 low-pass d\g\ta\ fl\ter frequency response show\ng passband ripp\e
and stopband attenuation,
/nfinite iтpu/se response (I/R) fi/ter Re/otive ottenuation 503
Passband the frequency range over which а fi1ter passes signal energy.
Usually defined as the frequency range where the filter's frequency
response is equal to or greater than з dB, as depicted in Figure F-З,
Passband ripple fluctuations, or variations, in the frequency magni-
tude response within the passband of а filter, Passband ripple, теа-
sured in dB, is illustrated in Figure F-б.
Phase response the difference in phase, at а particular frequency,
between аn input sinewave and the output sinewave at that fre-
quency, The phase response, sometimes called phase de/ay, is usu-
аllу depicted Ьу а curve showing the filter's phase shift vs,
frequency, Section 5.8 discusses digital filter phase response in
more detail,
Phase wrapping аn artifact of arctangent software routines, used to
calculate phase angles, that causes apparent phase discontinuities,
When а true phase angle is in the range of 1800 to ЗБОО, some soft-
ware routines automatically convert those angles to their equiva-
lent positive angles in the range of 00 to + 1800. Section 5.8 illustrates
an example of phase wrapping when the phase of an FIR filter is
calculated.
Quadrature filter а dual-path digital filter operating оп complex
signals, as shown in Figure F-7. One filter operates оп the in-phase
i(n) data, and the other filter processes the quadrature-phase q(n)
signal data, Quadrature filtering is normally performed with
low-pass filters.
Input Х(п)
iXn)
Cosine sequence
q'(n)
Sine sequence
Flgure F-7 Two low-pass fi/ters used to implement quadrature filtering.
504 Appeпd/x F: D/g/to/ Fllter Term/n%gy
Relative attenuation attenuation measured relative to the largest mag
nitude value. The largest signallevel (minimum attenuation) is typi
cally assigned the reference level of zero dB, as depicted in Figures F 3
and F 6, making all other magnitude points оп а frequency response
curve negative dB values.
Ripple refers to fluctuations (measured in dB) in the passband, or stop
band, of а filter's frequency-response curve. Elliptic and Chebyshev
based filters have equiripple characteristics in that their ripple is
constant across their passbands. Вessel and Butterworth derived fil
ters have по ripple in their passband responses. Ripples in the stop
band response are sometimes called out of baпd ripple.
RoII off а term used to describe the steepness, or sIope, of the fiIter
response in the transition region from the passband to the stopband,
А particular digital filter тау Ье said to have а roH-оff of 12
dB/octave, meaning that the second-octave frequency would Ье
а ttenuated Ьу 24 dB, and the third-octave frequency wouId Ье atten
uated Ьу 36 dB, and so оп.
Shape factor а term used to indicate the steepness of а filter's roll off.
Shape factor is normally defined as the ratio of а filter's passband
width divided Ьу the passband width plus the transition region width.
Тhe smaller the shape factor value, the steeper the filter's roll off. For
an ideal filter with а transition region of zero width, the shape factor is
unity. The term shape factor is also used to describe analog filters.
Stopband that band of frequencies attenuated Ьу а digital filter. Figure
F 3 shows the stopband of а low-pass filter. Although the stopband
а ttenuation in Figure F 3 is 20 dB, not аН filters have stopband lobes
of equal amplitude. Figure F 6 shows that stopband attenuation is
measured between the peak passband amplitude and the largest stop
band lobe amplitude.
Structure refers to the block diagram showing how а digital filter is
implemented. А recursive filter structure is one in which feedback
takes place and past filter output samples are used, along with past
iлрut samples, in calculating the present filter output, ПR filters are
almost always implemented with recursive filter structures, А nonre
cursive filter structure is one in which onIy past input samples are
used in calculating the present filter output, FIR filters are almost
References 505
always implemented with nonrecursive fi1ter structures. See Chapter
6 for examples of various digital filter structures,
Тар weights see fi1ter coefficients.
Tchebyschev function see Chebyshev.
Transfer function а mathematical expression of the ratio of the output
of а digital filter divided Ьу the input of the filter, Given the transfer
function, we сап determine the filter's frequency magnitude and
phase responses,
Transition region the frequency range over which а filter transitions
from the passband to the stopband, Figure F-З illustrates the transi-
tion region of а low-pass fi1ter, The transition region is sometimes
called the traпsitioп baпd.
Transversal filter in the field of digital filtering, traпsversal filter is
another name for FIR fi1ters implemented with the nonrecursive
structures described in Chapter 5.
References
[1] Rabiner, L, R" and Gold, В, The Theory and Application о/ Digita/ Signa/
Processing, Prentice-Hall, Englewood Cliffs, New Jersey, 1975, рр, 206, 273,
and 288,
[2] Oppenheim, А. V" and Schafer, R. W, Discrete-Tiтe Signa/ Processing,
Prentice-Hall, Englewood C1iffs, New Jersey, 1989, рр, 236 and 441,
[3] Laakso, Т, 1" et аl, "Splitting the Unit Оеlау," IЕЕЕ Signa/ Processing Magazine,
January 1972, рр, 46.
[4] Pickerd, J, "Impulse-Response Testing Lets а Single Test 00 the Work оЕ
Тhousands," EDN, April27, 1995,
I'NDEX/
[AJ
Absolute value, 9
А/О converters, See Analog-to-
digital (А/О) converters
Addition process, 10
for соmрlех numbers, 44 47
in finite impulse response fi1ters,
158---63
Aliased sinc function, 101
Aliasing, 2 29
in infinite impulse response
fi1 ters, 25,5--.56
АН ones rectangu1ar function,
108---11, 11 15
Allpass fi1ter, 494
a.Max+ Мin algorithm, 400 404
Amplitude, defined, 8---9
Amplitude response, discrete Fourier
transform
соmрlех input, 119 23
real cosine input, 125
Analog, use оЕ term, 2
Analog fi1ters, use оЕ term, 253
Analog signal processing, examples
оЕ,2
Analog to-digital (А/О) converters
coherent sampling, 297n, 433
crest factor, 364n
quantization епоrs, 357---65
testing technique, 432 35
Anti aliasing filters, 32
Argand diagram, 444n
Attenuation
defined,495
relative, 503
stopband, 189
Averaging
applications, 319
coherent, 320 27
exponential, 341 48
filtering aspects оЕ time-domain,
340 41
in finite impulse response filters,
158---63
incoherent, 327 30
multiple fast Foиrier transforms,
133,330 4O,429 30
[IO
Band-lirnited signal, 29
Bandpass design, for finite impulse
response fi1ters, 191 93
Bandpass fi1ter, 495
BandpasssaD1pling,32 42
spectral inversion in, 4 46
summary оЕ, 46
Band reject fi1ter, 495
Bandwidth, 49,5--.96
Bartlett windows, 82
Bessel functions, 189,496
Bias, 356
Bilinear transform design, оЕ
infinite impulse response
fi1ters, 223, 272
Binary data formats
defined, 349
fixed-point, 349 56
fixed-point effects оп word length,
357 75
floating-point, 375....s2
fractional, 352 53
hexadecimal,351
octal, 351
offset, 356
рlасе value system for, 350
precision and dynarnic range,
35 57
sign-magnitude, 353
two's complement, 356, 36 68
Bits
defined,349 50
effective number оЕ, 365
Bit reversal, fast Fourier transform
input/outpиt data index, 145 46
507
508 /ndex
Вlackman-Harris windows, 409
Вlackman windows, 184....s7, 409
Вlock diagrams, use оЕ, 10, 161
Вlock f1oating-point, 381....s2
Boxcar windows, 82
Bиtterfly structures, radix-2 FFТ, 1 54
Bиtterworth function, 496
[g
Сапiеr freqиency, 34
Cascade filters, 290 92, 496
Caиer filters, 497
СаиБа! systems, 225
Center freqиency, 496
Central Limit Theorem, 438---39
Chebyshev function, 496
Chebyshev windows, 87, 187 91
Coefficient quantization, 28б....s7
Coefficients, for finite impulse
response filters, 166
constant, 173, 498
Coherent averaging, 320 27
Coherent sampling, 297n, 433
Commиtative property, 20
Complex conjиgate, 6
Complex demodulation, 297n
Complex down-conversion, 297n
Complex freqиency, 225
for Mth-order infinite impulse
response filter, 24,5--.46
Complex inpиt, discrete Foиrier
transform freqиency response
to,119-23
Соmр lex numbers
addition and subtraction оЕ, 44 47
аррliсаНоns оЕ, 45 S4
arithmetic representation оЕ, 444----46
conjиgation оЕ, 427, 448
division оЕ, 448---49
fast mиltiplication оЕ, 4l1 12
graphical representation оЕ, 44
inverse оЕ, 449
logarithm to the base 10 оЕ, 4Sl 52
mu1tiplication оЕ, 411 12, 447 48
naturallogarithms оЕ, 451
negative freqиency and, 458---74
raised to а power, 450
roots оЕ, 450 51
Complex plane, 460---61
Complex signals
qиadrature mixing and, 471 74
withoиt multiplication, 38,5--.400
Conditional stability, 236
Conjиgate/ conjиgation, 6
оЕ complex nиmbers, 427, 448
Constant-coefficient transversal finite
impulse response fi1ters, 173
Constant-geometry algorithm, 154
Continиous signal processing
freqиency in, 5---6
иse оЕ term, 2
Convolиtion, 20
discrete, 204----16
eqиation, 165
fast, 43,5--.36
in finite impulse response filters,
16 74
theorem, 209 16
in time-domain, 20 9
Cooley, J., 129
Cosines оЕ consecиtive ang1es, calcula-
tion оЕ, 436--38
Crest factor, 364n
Critical Nyquist, 28
Cиtoff freqиency, 496
[Q]
Data formats
binary precision/ dynamic range,
35 57
defined,349
fixed-point binary, 349 56
f1oating-point, 375....s2
dB, See Decibels
dBm, See Decibels
Deadband effects, 288
ОЕС floating-point, 378
Decibels
absolиte power using, 493
defined,486,497
magnitude and power ratios, 492 93
signal power determined Ьу loga-
rithms, 48 91
Decimation
сотЬinОО with interpolation, 314----16
filters, 306, 497
resampling Ьу, 304----9
иБе оЕ term, 304
Decimation-in-frequency algorithm,
148---53, 154
Decimation-in-time, 145, 149 53
Demodulation, I/Q or Weaver, 299 30l
Difference equation, 221, 222
Digital filters, See Finite impulse
response filters; Infinite impulse
response (IIR) fiIters
Digital mixing, quadrature sampling
and,301 3,389 3
Direct Form 1 structure, 244
Direct Form 11 structure, 252
Dirich1et, Peter, 101
Dirichlet kemel, 100 103
all ones, 108---11, 114
symmetrical, 105....s
Discrete convolution, 204----16
Discrete Fourier transform (OFТ), 21
defined,50
equation, description оЕ, 50---63
ехаmрlе using, 54---63
frequency axis, 67---68
frequency response to соmрlех
input, 119 23
frequency response to real СОБinе
input, 12 25
inverse, 70 71
leakage, 71....s0
lineari ty, 65
rnagnitudes, 66---67
processing gain, 9 97
purpose оЕ, 49
rectangu1ar functions, 97 119
relationship оЕ fast Fourier trans
form to, 130 31
scalloping 10ББ, 8в....s9
shifting theorem, 68---70
single-bin frequency response to
real СОБinе input, 12,5--.27
symmetry, 63---65
windows, 80....s7
zero stuffing, 89 93
Discrete linear systems, 1 18
Discrete time expression, 4
Discrete-time Fourier transform
(OТFТ), defined, 93
Discrete--time signal
/ndex 50!
connecting dots in, problem with, 3
ехаmрlе оЕ, 2 3
frequency in, 6
input and output shown, 5
use оЕ term, 2
Discrete-time waveform, methods for
describing, 8
Dither sequence, 288
Division оЕ соmрlех numbers, 448---49
Do1ph-Chebyshev windows, 187
Double-memory algorithm, 153 54
Down-conversion, соmрlех, 297n
Downsampling, resampling Ьу, 304----9
Dynamic range
fixed-point binary data format,
35 57
f1oating-point, 379 81
ш
Effective number оЕ bits (ENOB), 365
ЕШрНс function, 497
Envelope delay, 202, 204, 498
Euler, Leonhard, 460
Euler's equation, 50,100,249,267,275,
298,463,464
Even symmetry, 64
Exponential averaging, 341 48
ш
Fast convolution, 43,5--.36
Fast Fourier transform (FFТ)
averaging multiple, 133, 330 40,
429 30
calculating the inverse оЕ, using the
forward, 42,5--.29
constant-geometry algorithm, 154
decimation-in frequency, 148---53, 154
decimation-in-time, 145, 149 53
deve10pment оЕ, 129
double-memory algorithm, 15 54
fast finite impu1se response filtering
using, 43,5--.36
hints for using, 131 36
in-рlасе algorithm, 153
input/output data index bit rever-
Баl, 14,5--.46
interpreting results, 134----36
5tO /ndex
overf1ow errors, 369 70
performing two N-point real,
412 20
performing 2N-point real, 420 25
radix-2 FFТ algorithm, 13 45
radix-2 FFТ butterf1y structures,
14 54
relationship to discrete Fourier
transform, 130 31
software programs, 136
Fibonacci,459
Filtering/ filters
See a/so Finite impulse response
(FIR) filters; Infinite impulse
response (1m) filters
allpass, 494
aspects оЕ time--domain averaging,
340 41
background оЕ, 157
bandpass, 495
band reject, 495
cascade,290 92,496
Cauer, 497
coefficients, 498
decirnation,306,497
description of process, 157 58
interpolation, 310, 314
multirate, 314
nonrecursive, 158,253
notch, 495
order, 498
parallel, 290 92
polyphase, 313
prototype,254
recursive, 253
strиcture, 161
time-varying, 313
transversal, 162
Finite impulse response (FIR) fi1ters
averaging with, 158---63
bandpass design, 191 93
coefficients, 166
convolution in, 16 74
defu1.ed, 498
fast Fourier transform used for fast,
43,5--.36
5-tap, 162---63
Foиrier series design оЕ, 17,5--.91
frequency response in, 163, 181
half-band, 197 98
highpass design, 19 94
impulse response, 165---66
infinite impulse response fi1ters
compared to, 292 93
10w-pass design, 174----91
nonrecursive fi1ters, 158, 253
nonzero input values in, 158
optimal design method, 194
other names for, 158
Parks-МсСlеllan design, 194
phase response in, 199 204
Remez Exchange design, 194----97
structuresimplified,430 32
window design оЕ, 17,5--.91
First order sampling, 35
5-tap finite impulse response filters,
162---63
Fixed-point binary data formats
defined,349
effects оп word length, 357 75
fractional, 352 53
hexadecimal, 351
octal, 351
offset, 356
рlасе value system for, 350
precision and dynarnic range,
35 57
sign-magnitude, 353
two's complement, 356
Floating-point binary data format
block, 381....s2
ОЕС, 378
description of, 375....s2
dynamic range, 379....s1
fraction of, 376
hidden bit, 376
IВM,378
ШЕЕ Р754, 378
MIL-SТD 1750А, 378
overf1ow oscillations/1imit cycles
and,289
Folding frequency, 28, 30
Fourier series design оЕ finite impulse
response fi1ters, 17,5--.91
Fractional binary data format, 352 53
Frequency
center, 496
cutoff, 496
differences between continuous and
discrete systems, 5---6
Frequency axiБ, 67---68
in Hz, 112
normalized angle variable
and, 113
in radians/seconds, 113
rectangular function and, 111 13
Frequency domain
defined,6
listing оЕ sequences, 8
windowing in, 407 9
Frequency response, 21
discrete Fourier transform, to соm-
рlех input, 119 23
discrete Fourier transform, to real
СОБinе input, 12 25
in finite impulse response filters,
163, 181
for Mth-order infinite impulse
response fi1ter, 24,5--.46
for N stage infinite impulse
response filter, 251
singl bin, to real cosine input,
12,5--.27
Frequency translation
sampling and, 33
without multip1ication, 38,5--.400
Gain, See Processing gain or 10ss
Gaussian distribution, 325
General rectangu1ar function, 98---105
Geometric series, 99, 45,5--.57
Gibb's phenomenon, 184
Gradual underf1ow, 379
Group delay, 201 2, 204, 498---500
[н]
Half-band finite impulse response т-
ters, 197 98, 501
НаН Nyquist, 28
Hamming windows, 81, 82, 83, 84,
407...g
/ndex 51
Hanning/Hann windows, 81, 82, 83,
84....s6,407 9
Harmonic sampling, 33
Heaviside, 01iver, 223
Нет, 3
Hexadecimal binary data
format, 351
Hidden bit, f1oating-point, 376
НiфраББ design, for finite impulse
response fi1ters, 19 94
Homogeneity property, 13
ш
IВM f1oating-point, 378
!ЕЕЕ Р754, 378
IF sampling, 33
Imaginary numbers, development of,
460---62
Impulse invariance design, for infinite
impu1se response filters, 223,
254----55
Design Method 1 ехаmрlе, 262---65
Design Method 1 steps, 25 58
Design Method 2 ехаmрlе, 26,5--.72
Design Method 2 steps, 259---62
Impulse response
defined, 501
for finite impulse response filters,
165---66
Incoherent averaging, 327 30
Infinite impu1se response (IIR)
filters, 26
aliasing, 25,5--.56
bilinear transform design, 223,
272
cascade / parallel combinations,
290 92
defined, 502
Direct Form 1 structure, 244
Direct Form П structure, 252
finite impulse response fi1ters
C0D1paredto,292 93
impulse invariance design, 223,
254----72
Laplace transform, 22 38
nonzero input values in, 220 21
optirnized design, 223, 284....s6
512 /ndex
pitfalls in building, 28б....s9
reasons for using, 219 20
recursive filters, 253
stnucture,221 23,251 54
z-transform, 238--51
In-place algorithm, 153
Input/output data index bit reversal,
14,5--.46
Integration processing gain, 93, 97, 325,
331
coherent, 334----37
incoherent, 331 33
Intermodulation distortion, 18
Interpolation
combined with decimation, 314----16
filter, 310, 314
resampling Ьу, 309 13
Inverse discrete Fourier transform
(IDFТ), 70 71
Inverse fast Fourier transform, calculat-
ing the, 42,5--.29
Inverse оЕ соmрlех numbers, 449
Inverse оЕ general rectangu1ar function,
11,5--.17
Inverse оЕ symmetrical rectangu1ar
function, l17 19
I/Q demodulation, 299 301
Iterative optirnization, 284
QJ
j opera tor, 460---62
ею
Kaiser windows, 87, 187 91
Kelvin, Lord, 50
ш
Laplace transform
bilateral/two sided, 225
for continuous time-domain, 224----25
description оЕ, 22 30
development оЕ, 223
one--sided/causal,225
tables, 224
Laplace transform transfer function,
229 30
in bilinear transform design, 272 73,
280
in cascade filters, 290
in impulse invariance design
Method 1, 262---65
in impulse invariance design
Method 2, 265---66
in parallel filters, 290
second order, 233
used to determine stability and
frequency response in continu
ous systems, 230 38
Leakage
discrete Fourier transform, 71....s0
wraparound, 78--79
Least significant bit (lsb), 350
L'Hospital's ru1e, 101
Limit cycles, 288
Linear, use оЕ term, 13
Linearaveraging, 320
Linearity, discrete Fourier
transform, 65
Linear-phase filter, 502
Linear time-invariant (LТI) systems, 12
analyzing, 20 21
commutative property, 20
discrete, 1 18
ехатрlе оЕ а linear system, 14----15
ехаmрlе оЕ а non1inear system,
1,5--.18
homogeneity property, 13
time-invariant systems, 18--19
Loading factor, 361---63
Logarithms, signal power determined
Ьу, 48 91
Logarithms and соmрlех numbers
to base 10, 451 52
to base 10 using natural, 452
natural, 451
Loss, See Processing gain or 10ss
Low-pass finite impulse response fi1ter
design, 174----91,502
Low pass signals, sampling, 29 32
[м]
Magnitude(s)
approximation (vector), 400 406
оЕ С (modulus оЕ С), 445
defined,8,9
discrete Fourier transform, 66---67
Mantissa, use оЕ term, 375
Mean
defined,319,477
оЕ random functions, 481
МIL--SТD 1750А floating-point, 378
Мixing
соmрlех signals and quadrature,
471 74
соmрlех signals without
multiplication, 385...400
convolution and, 39 95
cosine mixing sequence, 389 91
magnitude and phase and, 395
quadrature-phase mixing sequence,
391
quadrature sampling and digital,
301--3,389 93
summary оЕ, 400
Modulus оЕ С, 445
Most significant bit (msb), 350
Mth-order infinite impulse response
filter
frequency response for, 24,5--.46
time domain expression for, 244
z-domain expression for, 244--45
z-domain transfer expression
for, 245
Multiplication
оЕ соmрlех numbers, 411 12, 447 48
frequency translation without,
385---400
process,12
Multirate filter, 314
[ffi
Negative frequency, 299, 300
соmрlех numbers and, 458---74
real signals represented Ьу, 467 71
Noise, 319
definition оЕ random, 477
Non1inear system, ехаmрlе оЕ, 1,5--.18
Nonrecursive filters, 158, 253
Normal distribution, 325
оЕ random data, 438---40
Normalization, 376
Normalized angle variable, 113
/ndex 51
Normal probability density function,
484----85
Notch filters, 495
N-stage infinite impulse response filter,
frequency response for, 251
Nyquist criterion, 30, 301
[Q]
Octal binary data format, 351
Odd symmpetry, 64
Offset binary data format, 356
One-sided/ causal system, 225
Optimal design, for finite impulse
response filters, 194
Optirnization design, for infinite
impulse response filters, 223,
284....s6
Out-of-band ripple, 504
Overflow епоrs, 286, 287....s8
fast Fourier transform, 369 70
fixed-point effects оп, 36,5--.70
rounding, 37 75
truncation, 370 73
two's complement, 356, 366---68
vector-magnitude approximation
and, 404---5
Overflow oscillations, 28в....s9
Overlapped windows, 41O 11
ш
Parallel filters, 290 92
Parks-МсСlеllan design, for finite
impu1se response fi1ters, 194
Parzen windows, 82
Passband, defined, 502
Passband phase angle resolution, 202 3
Passband ripple, 195, 503
Phase angle/argument оЕ С, 445
Рhзsе delay, 503
Phase equalizer, 494
Phase response
defined,503
in finite impulse response fi1ters,
199 204
Phase wrapping, 201, 503
Phasor outputs оЕ fast Fourier trans-
form, 337 38
514 /ndex
Phasors for representing real signals,
соmрlех, 462---67
Picket fence effect, 88
Рlасе value system, 350
Polyphase filters, 313
Postdetection averaging, 327
Power, signal
absolute power using decibels, 493
defined, 9 10
determined Ьу 10garithms, 91
Power spectrum, 54
Predetection averaging, 320
Prewarp, 283
РroЬаЪility density function (PDF)
А(О converter quantization епоrs
and, 360---61
mean and variance оЕ random func-
Ноns and, 481
normal, 484....s5
roиnding and, 37 74
truncation and, 370 71
Processing gain or 10ss
dis<rete Foиrier transform, 93 97
integration, 93, 97, 325, 331 37
window, 83, 409 11
Prototype filter, 254
Quadratic factorization formula, 234
Quadrature filter, 503
Quadrature heterodyning, 297n
Quadrature mixing, соmрlех signals
and, 471 74
Quadrature sampling
description оЕ, 297 301
digital mixing and, 30l 3, 389 93
other names for, 297n
Quantization, coefficient/ errors,
28б....s7, 292
fixed-point effects оп А/О
converter, 357---65
00
Radian, negative frequency and, 460
Radix 2 FFТ algorithm, derivation оЕ,
13 45
Radix-2 FFТ bиtterf1y structures,
14 54
Raised to а power, соmрlех
numbers, 450
Random data, normally distributed,
438---40
Random functions, mean and
variance оЕ, 481
Real cosine input, discrete Fourier
transform frequency response to,
123 25
single---bin, 12,5--.27
Real discrete Fourier transform inpиts,
63
Real numbers, graphical representation
оЕ, 44 44
Real sampling, 35
Real signals
complex phasors for representing,
462---67
negative frequencies for represent-
ing, 467 71
Rectangular functions
allones,108---11,11 15
definition оЕ general, 98---105
forms оЕ, 97
inverse оЕ general, 11,5--.17
inverse оЕ symmetrical, 1l7 19
sYD1rnetrical, 10s....s
Нmе and frequency axes and, 111 13
Rectangular windows, 80, 82, 83
Recursive filters, 253
Relative attenuation, 503
Remez Exchange design, for finite
impulse response filters, 194----97
Replications, spectral, 29 32
Resampling, 30 16
applications, 303
decimation, 304----9
decimation combined with
interpolation, 314----16
defined,304
interpolation, 309 13
Ripples
Вlackman window and, 184....s7
defined, 504
finite impulse response fi1ters and,
173
10w-pass finite impulse response fil-
ters and, 179 91
out-of-band, 504
passband,195
stopband, 195
rms averaging, 327
of а sinewave, 480....s1
Roll-off, 504
Roots of соmрlех numbers, 450 51
Roundoff errors, 286, 288
fixed-point effects оп, 37 75
Roundoff noise, 359
ш
Sample rate converter, 314
Sampling
coherent, 297n, 433
digital resampling, 30 16
quadrature, 297 303
Sampling, periodic
aliasing, 2 29
bandpass, 32 42
10w-pass, 29 32
spectral inversion in bandpass,
4 46
summary of bandpass, 46
translation, 33
under-, 30 31, 33
Scalar averaging, 327
Scalloping 10ss, 88 89
Shape factor, 504
Shark's tooth pattern, 28
Shifting theorem, discrete Fourier
transform, 68---70
Shift-invariant systems, 19
Sidelobes, 74, 103
Вlackman window and, 184....s7
finite impulse response filters and,
173
10w-pass finite impulse response fil-
ters and, 179 91
Signal processing
analog, 2
digital, 2
operational symbols, 10 12
use of term, 2
Signal-to-noise ratio (SNR)
/ndex 51!
coherent averaging, 324----25
incoherent averaging, 329 30
word length and А/О converters,
359---64
Sign-magnitude binary data format,
353
Sinc function, 7 74, 108---10, 123
Sines of consecutive angles, calculation
of, 43 38
Single-bin frequency response to real
cosine input, 12,5--.27
Software programs, fast Fourier trans-
form, 136
Spectral inversion in bandpass sam-
pling, 4 46
Spectral replications, 29 32
Square roots, obtaining, 400
Stability
conditional, 236
Laplace transfer function and,
230 38
z-transform and, 240 42
Standard deviation, 320, 479
of а sinewave, 480....s1
Starburst pattem, 145
Statisticalmeasures,47 79
Step response, 343
Stopband, defined, 504
Stopband attenuation, 189
Stopband ripple, 195
Structure
defined, 504
Direct Form 1, 244
Direct Form 11, 252
fi1tering, 161
infinite impulse response filter,
221 23, 251 54
simplified finite impulse response
filter, 430 32
Sub-Nyquist sampling, 33
Subtraction process, 10
for соmрlех numbers, 44 47
Summation process, 1O 12, 322 23
Symbols, in signal processing, 10 12
Symmetrical rectangular function,
105....s
Symmetry
discrete Fourier transform, 63---65
:1'1 О Iпавх
even, 64
odd, 64
ш
Тар weights, 498
Tchebyschev windows, 187
Tune delay, z transform and, 242 45
lime-domain averaging, filtering,
340 41
Time domain expression for Mth-order
infinite impulse response filter,
244
lime-domain signals, 5
discrete convolution in, 20 9
Laplace transform for continuous,
224----25
lime-invariant systems
analyzing, 20 21
commutative property, 20
ехатрlе of, 18 19
Time-varying filters, 313
Transfer function, 229 0
in bilinear transform design, 272 73,
280
in cascade filters, 290
defined,505
in impulse invariance design
Method 1, 262 65
in impulse invariance design
Method 2, 265---66
in parallel filters, 290
second-order, 233
stability and frequency response in
continuous systems using,
230 38
Transition region/band, 505
Translation without mиlНрliсаНоn, fre-
quency, 385 400
Transmultiplexing, 385
Transversal filter, 162, 505
Triangular windows, 80, 82, 83
Truncation errors, 370 73
vector-magnitude approximation
and, 405---6
Tukey, J" 129
Two N-point real FFТs algorithm,
412 20
2N point real FFТs algorithm, 420 25
Two's complement, 356, 366---68
ШJ
Underflow, gradual, 379
Undersampling, 30 31, 33
Unit circle, 239
Unit delay, 12
Unit impulse response, 20
[YJ
Variance, 3l9 20
detined,478
of random functions, 481
Vector, use of term, 444----45
Vector averaging, 320
Vector demodulation, 297n
Vector-magnitude approximation,
400 406
Video averaging, 327
[yt]
Warping, frequency, 27 77, 278
Weaver demodulation, 299 301
Weighting factor, 341, 343
Windows
Bartlett, 82
Bessel functions, 189
Вlackman, 184----87, 409
Вlасkman Напis, 409
boxcar, 82
Chebyshev, 87, 187 91
discrete Fourier transform, 80 87
Dolph-Chebyshev, 187
dual window functions, 409 10
for finite impulse response filters,
17,5--.91
frequency domain and, 407 9
Hamming,81,82,83,84,407 9
Hanning/Hann, 81, 82, 83, 84----86,
407 9
Kaiser, 87, 187 91
overlapped, 41O 1l
Parzen, 82
processing gain or 10ss, 83, 409 11
purpose of, 86
raised cosine, 82
rectangu1ar, 80, 82, 83
selection of, 86
Tchebyschev, 187
triangu1ar, 80, 82, 83
tricks for using, 40 11
Word length, defined, 350
Word length, fixed-point еНеш оп
A/D converter quantization епоrs,
357---65
overflow, 3 70
rounding, 37 75
truncation, 370 73
Word length errors, infinite impulse
response filters and finite, 286
Wrapping, phase, 201
/ndex 51'
[I]
z....<fomain expression for Mth-order
infinite impulse response filter,
244--45
z....<fomain transfer expression for Mth-
order infinite impulse response
filter, 245
Zero-frequency IF format, 300
Zero padding, 310
Zero stuffing, discrete Fourier trans-
form, 89 93
z-transform
background of, 238
defined,238
description of, 238---40
infinite impulse response filters and,
242 51
polar form, 239
stability and, 240 42
time de1ay, 242 45