/
Text
Transmission
of Information
by Orthogonal Functions
Henning F. Harmuth
With 110 Figures
2nd Printing Corrected
Springer-Ver!ag
New York· Heidelberg· Berlin 1970
OR HtN .. ING F. HAR'IU"ll
Consullong Engineer
0·7!>01 Loopoldshalen I Wos1crn Geomany
l h;ltl wmk Is 1n1bjocl lo copv11g111 All ' l!J h15 <HO rose1vea, wethor !!iv whOIO or pnfl of 11\Q
f"lntQdal Is conceu1ed 11p1,cih-.;1.l lly lhO$• ot ttnns1n110•1. H,:i>rlnllno, 10•1.110 ot lllustrat!on!.
l)tOSOtllStmg, 1t'r11oduc1lon b)' p hOl Ocop.,1n9 ·1mchinu or i;lmllur moan&. l'ln<I 111orage 1n d.11.11
bat'lk•
w"'"'''
Ul'ICIOr §Sot. al the Gcrnal"I Copyright l.4w
ccipf.os are made lor 01no1 lhn" prl'llllO uoo. a
ku l'l pri)l'bl o lo lhe p 1.1bl.S""-'· 1>·.11 t1moun.t of the toe to :ie d eto1111\ne<l b7 i.~'tOR"Ont witrt ttte
riuti 1SMt
try Spfll'l'!J:Cr·Vo1:aig:. Betl•n. u ..dolb"10 1U anc: 1910
QUl'U Cata•o; Cato H.:..mbec• n-111'81 TttlO·NO 1590
Pn.nted 1n Qo,,111any
LlbtatY of CO't·
To my Teacher
Eugen Skudrzyk
Preface
The 01"\.l101Joulllity or functions ltna been ex;iloi.ted in
co:n:nunic(ltJ.oJu;5
:.>iuce i t.s
very
Uep;in.niJlt~ .
Co11scious aJ1d
ext.enoiv<' u~e was made of H by KOTEJ,''HKO\' i ll tl1eoretical
work in 1< 1•'.7 . ~en year~ later· a consitle1·al lo nur.1l er of
people were wo ..·iti11;.: in tLis fi~lU ra.tt.er indcp~ndcnt!y .
Ho..,ever, liule ex~ erimem;al u:-;e conl<l lie 11ode of the tl:eoretical renu1-:i before 1:he nrrival of :;olid st.nLe operational am1 li fier~ and integra•r.d c~rcuit~ .
1
1
A thnory o~ co.truaunicaCioc.. batie<l on 01·l.bOf'.Onn1 f11!'lct1on!:
could hnve been pub:lahed m~ll.'f y~ttr:-: Ht!.:O . llowe•tr.l', t!:e on:y
useful exii1r.ples of orthogonal f11nc~ionn .it tllnt time were
sino- coo111e 1'u.nc~ions and bloc!-:. pulue ... , n11d t.lus made cl1e
thoory a 1met11' to be a coopJ.icnted wny Lo tlci·ive known re sults . It: wns ne;nin l..be advanc•~ o.f ar~1nicc11 1duct.01"' t~chno
logy 1,huL Pt'Oducetl tlle firsc i'eall;y n""'• unoJ'ul cxmiple
of ortbogonlll f1.u1cL ons : the liotlc - r.nowr. •fol i::i: fw:.ctions .
In this Uook cmphanis in placct orJ ::h~ 1•.;!'l.1..1Jh functions,
Si.net? n.mplo 1 itcravurc .:.s avai_a:.. 1c or: !line-cosiL.~ functions us well an on bloc~ pulge~ a::d pttls<' J.e1•iveC !'roe
the:n .
There are t.t:o s.ejor reaso~s ithy so re-...· ortr.ogona!. !'u.nctions arc of p1·acticsl Lnter'3"t in <'Oo;~1tr1 ic ac i on" . Fi r"t ,
a number Of m9the;i;atical i'eature.: Othnr l.~./ln o:-tboe;om:Jl~~y
are requi!'ed, .uuch as completeness. oi.• 'good' :nul tipl:.cnt.ion
and shi!L tltoo!·emG . One quietly leaL·n~ Lo ttppI·ec.wt" Lhe
uacfulnooo of mlJl tiplication and !lltift Luoorein: ol' slnecoaine func Uona for multi plexing on/I mo bi lo rndio transniisaion , whonover one tries to duplicatP thene UJ pl icaHon~
VI
ty
ot:heL~
!\met ions . T"ne second rca.zon is that t!le :'wictions
mu st be easy to produce . !I'h.e sevc!"ity of' this Gfl'cond requirement .::s r·cadily com1lrehe11dcd if or:e trl.e5 to think
or
syst~ms
o f .='.'unctions or whicJ1 a mi .... 1 ion O?'" r.iore can be
actually p~oduced .
P=io.r· to 1 960 is wa n mainly the o=-thogona.Lii;y feature
tr.at attracted attcn"tion in conneotion wit11 the t:ranGmission of di5it.a.l sigrwls in the p!'escncc o.r no;i se . But sooner
o!" laLer ti:e question hod to ·oe raised of why t r.e ortbc 5onal 6"'Stem oJ si!le and cosine !"unctions should be treated
di1'I erentl;y .:·rorr. otneI· systems of orthogonal funct ion s .
This questlon led to the
e;eneraliza~ioc
o:' r.Z1e
concep~
o.f
frequency and of suc!'l co.:1cep'ts der~ived £roi:r. i::; as :·requen<:y
power spect"rLuo or f.req uer1c.y rcspor:se of at"Ccauation and
phase sl1ifi; . T11e Wnlsh l"w1ctions :oade i t possiolc to design pr·actical J il Le:ri; and multiplex. equip:nr;m; based on
this gen.ertlizat;ion of frequency .
!my theol."J' in engineer tng ILUSt; offe!." not; 01~ly some uew
lllld.?rstf!.nding, bui:i rr.us"G lead t o net·.· e<1uiprnent ruld thi~:;
equipment must be econoaica.14.y competitive . A considerable
variet,y of equipment using orthogonal funct:-ionn has been
developed , but there is stil l muc:_ controversy about the
economic rotential . This is due to some e xtend to problems
of compatibili;;y, which always tend to favor previ.ousl;y
introduced equipment a.na met;hods . 1l.t ~he paL·~iculaL' case
of 1.'lalsh functions, tha r:cono:nic competitivenes~ is; intimately coru:ected to the stat.; of th!' art i!l bL"1ary digital
circuits . I-t is 1 e . i:, ., diri'icul-i: vo coe •,.1hy Waln!: functions
should uot be as in:.portaut 1'or die;ital filters as einecosine ruuct.ions are for linear.· , t ia:e-in·. .·a.:-iant net\,•orks .
'l'lae autltoL"S work in tbe area of orthogonal funccions
haR been sponso1'etl for many :yell!'s by Lhe Bu.'ldes1oi!llsle1'itun
der Verteitligun5 de1~ Reµublik De1.icsc!:la.nd ; lle \\1an'ts to
take thi$ opportunity to thank ProL•' . A. f!'tsCllER, Dr . E .
SCllW..ZE and Dr . M. SCHOLZ for their continued suppoJ·t . D1'.
TI . SCHl.lCKE of Allel"l-Br·adle:y Co . was among the Jil'sc i.;o
encourage and !3t.imulate work 011 the enginee:ring applica -
V.'. (
tio11s of :fall'h !'um:Lions ; ;;he au;;hor ir i:;rcntly L.del>ted
to him . Hr-1 r hns beeu reni!.ered furtncr i:a -cieutific as
well&:: adr.Lni!'t:ative vro:.leo:s by tho follo·~tr.g e;et.tle:i.en:
FroL ? . H. L/.1: ;E o! Rostoclc ;Jni••ers!ty , !'ror . G. LCCliS of
fonelrucr. UniYern ty , Lipl. !ng . II. ::11:;w,u Md Dr . H . HC1l!~
ot tt.e Deutnche Eu.r.Jespcs;; O'T7. - r'. D!>.rll:':;odt) , Ci~: . Pt.y2 .
Jf . bIL?:h~· of bo,·ch Gml>ll , the laLe llr . r. • Kl·:r•1•;:1 , or /iliG- 'fele funke 11 t.:i , I r·. f. K. VOrl SAliDJ::N and f !•oI. ,1 • ." ! ticm:n o!' Karlsruhe L1·1 LV<l':<Hy, 1-rof. G. ULJnc.m of •rechni~cl:e Uocbschule
llmenwi, C'wir . H. LUbv o.f 'l'echUische 'ioc lr achulc Aacl•en and
Prof. J . l\A!l1 o r t:t·.e Llr:_versi~y of SoutheL·n Cntifornia .
ThWlk" nr" J"llrticu:a=ly due to Pro!". K. KUPl'MOl,J,ER of Technischc Hocilac1Jule Darmstadt wl:;.o ~11oweU g.i:eat. iutere.st.. Wld
encournred Lhf" stut.!y o~ the applicatio:1s duscribetl in ti-hls
book .
Dr . F . ?IC!lLE:R or Lirr•· U~vers.!.:y , :>1 . l. . Tl!IK::iC:U,El'l' or
Mannheim l:uiversi>;" and Dr . P . \.:E_S!::of -«:'lsbruck U::iversi;;y
were o! gi·esc h"ll in improving n,e rr.eth<Htntic&l scctiom:
ol'thcbook . lrnr.~ . OLSON or St . Ol ar L:ollcg" , !'\rs . J . OLSC!;
and Mr . J . Ll::L or Ir.tcrn?..tiona.l 'I'nlophonf'o nnd J1.:1..,f"'r,rn.ph Co .
devoted much ti:rJC' to the ed i tinc or t.11f\ manu!lcrl ;rt , e
thankleS.!'l ur. well ae indispe!lSoblt' Lu8k . f':nny of t he pictux·os iu r~hic lJook We.l·e Ji t"Sl publisf1ed 1J1 Lh1;.; .itrchiv Je;
elekt1•inchC ll vuertragung ; Nr . l" . Rtl!IJ·lAI\~; Of S . iitr•.r l-V cro l ag
coui'teo1J.1ly 111J?rniit"ted ti ei1" use . l.c~t l,Jt not 11'.'ar.t , ;-~1!Jr:~s
1
arc due 1.o J-lr . . ii' . HAASE ~·or th<' tn•ir.~ nnt't to my ·11iT~ D:- .
E .!!AAMUTH-!10.;;:1;; ror ~he ro:-oo:-:-•·~clir:rr .
January 1969
llcldl.i.ne; F . l!:u·cutlt
Table of Contents
IN1'ROJJ UC'J 10:1. • • • . • . . . . . . • • . . . • . . . . . . • . . . . • . . . . . . . . . .
1
" . J-! A'l'liEr1ATlCAL FOUlf"ATIONS
1 • 1 ORTHOGON/o.L
Fm:c::-1m:s
1 . 11 Or~hogonalicy ou1d :.ir:er,: In<l<'p<:r.d"n;:e . . . . . •
5
1 . 1;> Series Expansion by Ortl:ogonnl l'unct~ons ... 10
1 . 1~ Znvnri~ceofOrtho~onallt.y to fr..urinr- 1rran!!-
fomation ................ . .. ..... . ............. 13
1 . 111 \.lnln
l''u nc1;ions .. . . . . . . . . . . . . . . . • • . . . . . . . . . ·1•:•
1 . 2 TfiE POURIER 'l'llAl'1$.FORl"l
,~o
n·s
; ; 1::1u m.H. lf.A'r:o n
l•'ou:r.·ieJ• SerJ.e~ l;o Fouric:•rranaform ....•.•...•.........•..........•..
Gel.le.t·u.li~ed Fourie1" ·Prau.sfor·m .. .............
InYuri..ance o~ Ort:~og;ona lvy r.o t,111.:;t Geuer~liz.e<l. .Fouri e::- Transfor.1. . . . . .. ... .. ....•....
;:Xampl~s of the Ger.oral i zc•d .~ou.riei· i:"·=;:!"o21l:
F~-1zt ···als!:- Fou!"icr Tra.n!"';·o:'ll . .. . . .............
Gcne~·aliLed La;>lnce ~rnn~;·orn .. •• • • .... .. ..
'l . 21 '11ransition Trom
1 . 22
1 . 23
1 . 2••
1 . 25
1 . 26
1 •3
26
5.3
37
3S
:.;.
"-')
ci::r>.m .u oa;:1 r'REQ;Jfil;cy
1 . 31 !'hyncal ln.ce.rpr'ecat.io.:i
J'req u~nc;y .
or
tliP G"n"ralized
. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .
1 . 32 Powor .Spectrum , A:upli1.u<le Sµec~1·wr. , FU tc ring
L~9
of S i gnals . . .... . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1 . 3'3 ExuruJ>les 01' \,'alsh .h'o urie L' '111.•n11 aJ'oJ:m~ lllld l'owe1~
Spec t r u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . '5'1
;x
2 . DlRECT 'l'llAHSMI!;SlOI\ OF SIGl>Al.S
2 . 1 (JRTHOGO!;AL
7li.t:~UEM'Y
orns:oi: AS GEl:EJi.!.LJ'.'.A~"O:l
01' Tl~::E At;r;
DIVISION
i! . 11 !lepL·e~euL,.vioL of Sogn~J ~ .. . . . . . . . . . . . . . . . .
;;{)
+,4
2 . 12 E~ci;cpl~tS c.ii' Si gnal~ - .......................
2 . ·1z. Am i 1 it..:.J.c.:e Sat!!pling ;ind Orr.no~onul Oecon.;loio1.1. 'ior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
? . 1:1 1Ji1·cu lt~ Lor Ort;.hogonaJ Diviul<;n ........ .. .
(';
?. . 1J 1'ra11f:ninnior: 01' Digi l. al S i e;t•n!u b;-y· Sino vnO
Conino l\J.l :;es ....... ...................... .
81
0
'i1
2 . 2 t.:!!ARA ;•rL.~IZArION Ol" COJ-D·iUN!GATJON Clli\:lllELS
2 . 21 1-'re-qUl'LCY Re-spon~e of .4ttenuction an<1 F"nast:
!:lbi!'t; o!' n Co:nriur.ica-ton CiJHl-:•el ...........
2 . 22 Charactcr.:zation a~ ~s Cono-.:nica.:..ion Chu.nne_
by
Cro~~talk
Farame,ers ...... .... ........ ..
.tlE-.
91
2 . 3 SZQl.:C:tlt.:¥ r' IL'rtR.:i BAS.ED o:: iolAr.s:r FmlCTlOllS
2 . 7 ,1 Ocq ueocy Lo·;tpnss Fil t C'!'::l • • • • • • • • • • • • • • • • • • • t:i•
2 . 32 Seque11cy bu.uui:;us::i :F'i lcerc ..... .............
t:.:7
2 . 33 1Ji1;Hul Sequenc.1 F.llte~·" ................... 1 04
3 . CARRlE!l 'l'RAJ<SMISSION
Q;;• SIGJ;.o\l.£
3 . 1 Ai'1Pl.IT~DE 110DULA1'ION(.:..:'i
3 . 11 i1o<lulacton !md s:incL!-ouou~ :J~:nodul:J.tion ... .
;: . 12 iiu lt 1 pl ex Systeas . ................ ........ .
3 . 13 ::>icit.U MuJ.tiplexir.g •.....••...............
;, . 1'1 Mr.:hods of Single Sideband Hodul ntiou . . . . . . 13··
3 . 15 Correction of Tiaw Differenc·~" 1.;.. Syuchrn nous Deoodulat i on . ................. .. ...... 1 1 ~7
3 . 2 Til'!E SASE, TINE POS1'1'10H lilffi CODE NODUL/L'l'lOlf
: .. ?1 'l'i lflt'• Bnso 11odula tion (TBM) •••....•.•.•••••• 1 _._.
0~
;s . 22 'l'ime Position J'iodulution ( 1'1 t'I) ••••••.•••••• ·157
3. 2;; C'ldO ModulaLion ( CN) ••.••••••.•••.• • •••••.• 159
x
T !c.llLE OF CONT Em's
3 . 3 NONSlNUSOIDAJ, ELECTROMAGNETIC WAVES
;\ . 31 Radi.ation of Walsh Waves by a Jler•t;ziwi. Dii,-ole 160
j . 52 Propagation , Antennas, Dopplel' Effect .. .. .. 167
.5.35 tnterferomei;r:y, Shape Recognition .......... 173
4- . S'!' AT ISTICAL VAR! l\Jll,ES
4 . 1 SDIGLE VA!ll.AllLES
11. 11
Dco'irlition~ .
.. .......... ... ... . . .. ......... 151
Density Function , b'\mction of a Random Variable , ~~atbematical E.>:pero~at;ion . ...........• 188
4 . 13 Moments and Cllarac ~eristlc FWlC~ion ... .. ... 191
11.12
4 . 2 COi'IBINt.TION OF VitRJ.ABLES
~ . 21
Addition of !ndependenc VariabJ eo .......... 19q
Inde~,cndont V;:u:-i.=::1b J. es 198
4 . 22 .Joint Di st:ributions of
I~ .
?• STAT lSTICAL DEPEh'DEKCE
1. 31 Covu.riance and Correlation . . .. . . .. . ... . ... . 21Cl
52 Cross- and AutocorreJ.ai;ion J•'unctj.on . . . .... . 214
1
~ .
5 . APPL:CATION OF ORTHOUO.NAL FUNCT roNS TO STATIS'rICAL
PROBLEMS
5 . 1 SERIES EXl'AllSION OF STOCHASTIC FinfCTIONS
5 . 11 Thei'mal Noise ... . . . ... . ... . .... ............ ?17
'.;> . 12 Statistical. ln<ieuendence of the Cocr.:-ionents
of 8l1 O:rthogo:r:u.i.l- Expan$ion . .. . . ... . . ~ ... ... 22~1
S . 21 Le.i:1st Mcac Sq11a.!""c Devint; Lo1.1 of s Signal from
Sao:ple fw;cdonn . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 . 22 Eimmples of Ci.rcllits . . .. .... . .. ..... ..... . .
'.> . 2;, 1'18.tched Jo'il.i;ern .. . ..... . ... .. .. ..... . . .. .. .
;. . 24 Compan<iors for Scquency Signals ... ... ......
5.3
~'.UM' ll?LlCA'!'I\IE
223
227
230
233
DISTURBAJ!CES
5 . 31 Inte1·ference !",,ding ... . ... • . . .... .. •.•..... 2:55
.5 . 32 Di '!Orsi ty Transmiof:lion Usinp; Many Copj es . . . 2113
e. 1
fiAr;s.':1~·:s
. .o::
Cl..f:.. :;111
€ . 11 Me:.mires of Ba..'lu~:.:.Ct~ ...................... ?J•'>
6 . 1.'.'.? •r-r:i nr-tti s~_on C:ipac: ty 'l ~ Co:r,m~.1.uli.:n:..ior.. t.:h~--
nel:i ..... • •........ . . .. .................... 25':
6 . 1,3Zir:;n.~1
Uclay :i::C s:gnn! !Jir't:nrti.r>Ol' •••••••• 2F1{i
6 . 2 ERllOR I'ROLl\ll I l"'Y
O~
SitH;AJ .$
6 . ?.1 i::.l'J'Or J•1·oharility or Simplo Sic:nnl11 dur-> to
TJier·mcil ~oise .. ... . .. .......... ..... • ...... 262
6 . 22 I'~uk t'ower L.bited S1i!irnl!l .....•..........• ;::r..a
6 . 23 Pulse-'l'ype
l.ii~tu.rbancn,; .....
.............. . .~71
6 . 3 CODilfG
6 . 31 Coduig nith Bin•u-y ~-..l!~r.te ................ 275
6 . ;;2 Orthogona I , Trar.so rl:t.og-ona l &.~d tlior~rogor.a:
Alplwbets . .... ·- ................. .......... 280
6 . 33 Coding for Error- lr'ree 'Ir.:utaL·11 ... iora •....•..• 2.55
6 . 3.1. Tc1·nnr,v Comliination Ali lt~betu ...... .. ... . .. 2t:/~•
6 . 35 Combinatior:. A1rhat-ctn of Ori!er• 21"'•1 •...•... 29')
llE~"EnENCES
ORDERED BY SECTIOKS ... .. .. ......... ...... ?05
ADDITIOI<AL REJ>'ERE;Jc;J,S FOR TliE
:>::corm rn J:1'!' Tl~G . • • • • • • 320
IlIDEX . . . . . • • . . • • . . . . . • • . . • . . • • • • . . • . . . • . . . . . • . . • . • . . 323
Equations ere numbe.red co!l~ecut.:.vel;: w1tl::1n P.::ic::. or..,,. of
1the 6 chaptc1·s . Rcre-:~ence to an ~quat.1uu of u diffe=e11t
chapter .i.:l mude by n·r-:..-:ing the n:unbe.:.· or t.he ch.z.ti. te= iu
i·ront Of tho nwr.ber of t!lr: cquat;ion, e . t-; . (4 . ~~) for• (;:~)
in chap~er 4 .
Introduction
Sine and cosine :u."lc-i.ons play " uniqu., i-ol,; in coc:~u.nications . The concei:- of frequancy, buSf!tl on t~ei , i£
defined by tl::e uarwtet~::: f ~the fmcttons 1 sin (2-rf~•c)
and II cos ( 2-r r<:~cr.) .
"!'here nrc muny i-easons for this •miqu'l i·ole . 1t was
hardly po11siblc L~ prodttce other funcL loll:- in th"' earl;days of comcuuicatious . f.lectron cube~ und cransisto:-s
made i t po.osible co produce s uch simple non-sin1111o i.dal
wave formo aa block p•t"l se,; or- :r-a:np vo ItAi;oc.. Hut i- i<as
not be.fore tne prrivaJ of th~ intPp;l'AtP.d drcuits tha;;
alm-oet
furtbe-.r
1'1llY
fi.Jnctionz cou:.J.d be p1.'t:·tluc1.:d econ.01:1.lca.l_y . A
f&CtOr
favoring
~i?:l.:fJOida.l
fU!JC!:iODS
·,;3.;J
tl:e
fact that linear tioe il:.7arit!n:: ci=cuit.s oi;.ly attenuate
and delny thca, the s.h!l?e- and frequr-r..cy rcauin. unchtW6ed .
Hence , tle a;ystett. o~ sine and co.sir:e .f~cl !.ons Lu.cl e. :remendou~ advantage over ci:her cooplure ::yncn:ns or ortho-
gonal functi.ona , as lc.ng as r(Hli.:;cor,- , N1pnci to ro =d
coils were the moot desi rable circui,; dcocnLS . The theory or linnnt-, ti>le i.uvarian.L netwol'kr ~cmonst.1·<1L<Js the
advantngea of ninusoidal .ru.nctions . Tllc ndvcnt or somiconductors hns brought a r.adic"l clHrnp;e . 'f'hei·c la uo particular reason why a digital filLer-, e . p; ., ·utuly,;.i11i:; the
£ine structure Of ll radar signal , nhoul d \,e Lt<Ol'd OJl si11e
and cosinn runctionc . :': turn" out tb•t. dil':itttl fll;ei-~
based on tbe socalled l·h;lsh fw:ctior.n ere n11f.pl er a...'"l.d
faster .
Sinusoidal .runctior.:3 ai.·e :es$ :.Oportn.n1 tor t;I:~ propagation of electrccagr.c,cic -...•aves 111 fr·e~ a1>aee or alo:if
conductors . ~·he solutioc of the wave C1U,.t1oi: t:r d ' fu..Er-lBERT o.nd the gcnoral :ooluti on of tl:o to} Legro pr.er' r, equation show , Lhnt n 1.arge class or fw:c11ion:: c:nn be t;r-an:;miti;cd fu.etortion- .free or can be r egenerated . Gimilnrly ,
a Hcrtzinn dipole can l.'adiatc non - sinuool r1<il waves . The
dominance of ainusoidal waves in radio co1nmw:dcation can
be purtin!ly O:>rJ lained by the invariancn of th~l 1' o:-~ho-
-l'l'l'RO:JUC':':Otl
2
gona.lii;y under varying time- delays . Ou:oles or open wire
lines t!:at could not , nor need net 1 tran:?-n.ii L sinusbj dal
fw1ction" have aJ.ways existed . The telegraph liJ.leS of be
19th cen-cury 1 using elect;romechanlcal r·eln,YS as ampli .:'ie:r·s, were sucJ1 lin-e s , and chey have recer1tly :no.de c.
Ct'tleback £,s digital ca ·o 1es .
One of the mor;t impoJ.·t;a.nt features o:- sine ar10 co::iine
functions iB tr.at almoE:t al J tirnc rw:ctions used in con.muoicatl.ons can to repre$ented by a superpo::;ltio11 cf
sine and cosine .functions 1 for wlli,cr .Fourier a.nal:ysis is
tbe
tr:wsition ~·=-·oru L:ae t.o frequency funcl;i.ono i.s ;.) re-nult of thia n.nal;ysl.s . This is
rnathcr.iat~ca!
too1 .
'It~e
often tar-e11 so muct foL' gL·u.nted b.y Lhe corr.munications
pnginr•or , that he ln::;tir~ct.i-..:el~f sees ~:i ::.uperposi tioD o:f
sine a.ad cor:)ioe ru.r1ctior:.s in tile otttput voltage oi· a microphone
01~
a i:.e:et.ype l;.ra.ns:ni tr; er . Actually, t!:n rcpre-
aentation Or a tiu.e function by sine- and ~osin0 functions
is only one
among
man~~
possible ones . CompleLe systems
of or1:hogona1 functions goncral ly pera:.it aer·ie1:i rd.:.:.pc-1nsions that correspo:id to l;b.e foui·ler_• seri e::; . For inr-'>tP. nc Q,
oxpansions into se.ries of Ec:::isel fw:tct:i ons :1rr- r.iuch used
in communications . The1•e a.r·e also trans.:·oT"ms cor 1·e.::ipon<liug to ~he Fow:'ier i;ra.n.sform Cor many sy~tema of l\mc·tions . Bence , one may see a sur,crposit;iou of JJegendre
po1:5'llOmiels , _parabolic cylinder .fu.ncLions , ecc . in the:
out1)ut vol tege of a microphone .
Gene-rel complete syst9mS of o.rt:togonal funcv~ons inAte£-H1 of vhi;i special system of slue ro1U co:;j no Iunc-cions
will be used in this book .ror che repJ~esenta1;ion of si g nals and £01· the C}).a!·act:eriza.tion of lines and net.•11orks .
A con sis~en ~ cheery must i nc l ude the application of orthogonal fu.:icticns as carriers, zince sine and cosl.nc a .re
not only used for theoret>cal analysis, but also as c<>.!'riers ; n a:ul ;;iplcx and i'a<.1.io systems . .Li; wil- be ehown
that ?no(lultttion mct}iods exist for them, which cOr!·espond
to nmplitude , frequency and pbase modulu~ion . l\utl1ermore, it wi 11 be sho·1rn that a.rrtenoas can be designed that
INTRODOCTJON
3
radiate noc-sinusoidal win-es efficiently .
Tbe i:rWlsitiou froo the system of si1.e-cosine !'unctions to gentH'&l syneos o!· orthogonal fur:c~:ior.~ brings
simplificotio:>s as well as co:ir;>lications to t!::.e :natne1111tical theOl',Y of co:nnu.Tticatior: . One may . c . e; . , avoid.
the tr•ou1:>le1101I1e facL that; any signal occu~leii <.>.n infinice
section ol' the timo-freQuency- docaln by sulJ~tit1iting a
time-funct;fon-domain . .lw:y i;:ime - lioited "ignlil compoGecl
of a liwHecl nLUlbor of orthogonal .rw1cLio11s occupies a
finite soc~J.o n o.r this time- .CUJlc',ion-dome.i11 .
The genereli:r.ation oi' ;;he conceµL of .!'r·equency """
been so fa1· the most satir.fyi:w LLeo1·eUcal result of
the tbeoi•y of COll'-'llunication oased on O!'t!.IOgor;l.ll !'unctions .
Frequenc:t is a paramete:.· or sine and coai11e functions
which can be 1ntc!"preted as nui':>cr 01· cycles rer unit. oi'
t:ime . !'!Ah'N (1), STUiiPE!lS (2] "'·-"'' '!OE!..CY.ER (3) )'IOint:ed out ,
&hat frequency may also b~ i.nterpretr.1 a" "one l:alf t!:e
number of 7.ero crossint;S peL' .lllit of time" . A aiue fwic tion with 100 cycles per second l:1'o "OO 7.nro cro:ictngs
or sign cbangee per second . One ho.lf tho numnor· o~ z,1,n•o
CL'oseing'3 ls 100 cycleG per second numc1·icnlly nncl dimensionally . Zero crossi.n6S ar e tlBfi:led .to1· £uncti ons in
which the torm cycle has no obvj ous :ueaulug . It is usefuJ
to introduce the more ge!.leral concep tc "ou u llalf l'.le i;verage numbeL· 01' ~ero crossing.9 per w1i t ot t...in~e " J..n order
t"O covei· non-periodic :ounct.ious . Tne new :cr1n 11 <.;>eq...teucy 11
is int;roduced for this generalizal;ion o!' frcquei~c.:_.r . Th:.is
sequency and frequency are ic.en- icA! Ior sint... and cooine
funct;ions . The term sequency a::ake!'l it ;>o,.siblc to replace
such important concepts as .:-requency J.I0'.'\·1·r SJ ectruo or
.frequency response of attenuation by ncquer.cy l'OWe<' spectrum e.nd sequcncy response of a&ter. uation .
The concepL" o.!' perioii 01' oocilla~lon T • 1/f anti
waveleng'Lh X v/ r are c-0nnected with f.-uqueucy . Zubsti lrution ol ' sequ oncy q> for froguency .r l~ttda ~o Lile Iollowing oroi·e gel10l'IJ.l uefinitions :
,.
IN::'llODUCT ~ON
4
uveru;;;e >er•iod of oscill1>tion T • 1/;;> (averai;e sepo.ratioc in lilr.e of the zero crossint:s aultivlied
b.Y 2)
average wavelengt~ X v/'I> (average r.craration in
n1><1cc of the zero cros,,ings r:iul t1pl \ed by 2, wher-e
v io :he velocity of propagutiou of a zero crossin!>)
'l'he ncid tost o~ any theory in eu1:)ineer1 n!l. arc its prnctioal app l icntions . Several suc11 llp]ll i.co ti on:; arc k:nawt:
and I-hey nre all intimately tie<.l ~o oe:oi conductor tochnolOL:>Y • 'ri10 little known liJ'S~ell' of f/i;luh func;t .i.ons ap1 ears to be as ideal for l~nea.t· 1 tic1t1-variabl e ci1'0.u 1 t" ,
iI bo.ooc on binary digital comroncnt:i, as the systen; of
sin~ and cosine fULctions i~ fOT' l 1near, ttia-.e- inva:'"i&.r:t.
circuits 1 based on resistor!l > co.racitors w.id coils . 1lery
siliple seqaenc.r fil~ers ba,,ed on Lhese Walsl:. funcciocs
have beer: cieve:oped .. Ftu-therl!>.01·~, M exµeriae:i~al ne quency 111".ltiplex sys~e= using Wdst: fwictions as ca.:-::-iers
hi..s be"u ueveloped ;;hat has a<lv&lLi<i:;es over rrcquency or
tiJae :nuHiplex systoett1> .:.r: cer~ain nr p.lica"ions . Digita.:.
f.Llle1·0 and digital a:ulLiplt>X <'lUi1.:nen~ arc li!llong tte
moot promising upplica Lions for coo ycuro "head . TLey
!Ira nimplor and f a steI· wheu ba~ed on \falnh funct.i.o n s i·athc•r than on sine and cosine l'uncr;ioau . Their pi·acticnl
npplicntion , however , will reqt1t1·c> con,,i<lerable progreno
in tbe <levelopJ:ient of largo scale intei:;«aced circru1'" .
Applic~t.ions of non- n.inuzoidal clec:..rotJagnetic ·.;aves
are strictly in tlle i;heo::-cc~cal scai:;e . Only vary recent1.Y ha\'e active ar:i;e=as been found LO te practical for
t.:~.. e 1•adiation of "°."al.st. Junctio::.s . Host prob ecs cor:cerning Wal!'lh waves can present.ly bo nnnwcrod .:.:i terD$ a:"
£OO:r1otric o_ptics only . since wavo opticu it:J a sine wave
optics . Ou the ocher hand, tLcirc is liLUe doubt that
non-s.i.nu~oidal elect::.--om.ognc"tOic wuvea ui·e a challengi.n5
l'ii.!lJ !'01· basic research . '!'he e:cut11'f.ltiion or non-sL"lusoidol l."&oio waves impl l es l:.hu.L a ncli w.o.vo:J can be genel\ated
1 n trhe rei;;ion of visil1le liglit: , and Chiu leuds ul ti.roetely
to ~h~ queo.tion of why white l.Lgl L ohould be decomposed
=
1 . 11 Ol!'l'llO:;oHA11'1'Y
5
uto !l!.!JUSOiJ.al Ltnctions .
TLt.; 'Nalsh rur.. ctior.s , C:l'JT!10J5iZ'J:C in this liook , are presently t!:e :oos:. importa::1; exar.:ple of non-sir.usoidt>.l func-
tions le communications . rLese fur.. c:.lons are hnrdly Y-..r:oifn
by cc:r..:nun1cnt1on Angineers al ~hcu~::1 they have been used
for rr.ort r-han bO .vea1~s f or t:..ti LL'tl.!lspo:::itior. of conduc torr inopon 11irl'! l ines . Rademacher ft.l.nCt.ions r11J , whicl:.
are n t>ub.;,yo l;cm or che 11/al s l1
func LJ.011 ~ ,
Wf l. 'e l.ltted l'or
this pu~· pono t owor·an the ena or Lhe 1)Lh century . The
complei;c syut<•Jll of '1ii;J <>h CRnctio11s se1Jrus ~o !;ave been
f ound aro 11 nr1 1 90G by J . A. l:\P.J!HB'rT' . Tt:e trano>posit iou of
conduct"orP l t"Cor:iin~ to ..BA:RBE:'·r ' :-- ~che:i1r. w~r :.tanJC:i.r~a
practic<J iu 1'12:; (l] ,[l], wlleu ." . L . WA1.3H [9) int roduced
the.m into ni~th{'matics . C.oct:t.un.:..cat.iun~ c-!.e-incr !":J and mothematic1o.nn ..-erl! no~ ah·are o! '!;hi~ .:o:r.:nou usagf': ll!'o!.i 1
very rt!cently (8~ .
1. Mathematical Foundations
U Orthogonal Functions
1.11 Orthogonalily and Linear Independence
A nyatcm ( f( j 1 x)} of i·eal and "1la:.ont <'Vrtry·,;h~re nonvanisbing functions r,O , x; , f(1,x), .. . i" called oF.;hogonel in the interval >: 0 ~ :x ~ x 1 if Ll.c 1·0110-.ti:.g ::ondition holds ~rue:
.,
S .C( j , x )i'( k, x)U.X
( 1)
••
0 for j
I
J
k.
. JOH!~ A. BARm:'rT is me ntioned by FOWLE ( ~] 1 n 190« as
~nven~or of ~he trani;pooi ti on of conductorll occoi·dlug to
nleh f1U1cLioJ1u ; oee particularly page 67~' of [>J .
1 . :t.;·rHEMA'l'lt;AL FOU1'!DA':'IONS
6
nre cal led 01~t11ogona1 and noroalizcu i.:·
the consLanL X J is equal 1. The two tcrcis nre usLtnlly
reduced to the single term orthonor·ma.L or 0J~thonormali2. ed .
A non- noI·.malized system of orthogo 11a.l functiorLs may
1
al.ways be normalized . For instance, ·t;he system {xj f(j 1 x)}
is normalized , i f X; of ( 1 ) is not equal 1 . Systems o:·
ortl1ogonal i'unctioos are special ca:::;e.s of system.s of linea.1:ly independent tu nctions . A system (I (j , x)) of m
functions is called linearly dependent, if the equai;ion
The
~unctions
I'll- I
2: c (j)f(j ,:x )
,.
o
(2)
j dl
is sa"tisfieC. i'Ol" all values or ;.: wit!1out all constants
c(j) being zero . I'he i'ullctions f(j , x) iire C'alloC. linearly
independenc, i .i: (2) is nou satisfied . Functions of ru1
orthogona I system are alwa~ys linearly -'independent , :;i nce
mu.1.tiplicai;ion of (2) ty f(j ,x) and im:;egration o f t!>e
pi·oducus in the intcrv!ll x 0 :i x :i ;: 1 yields c(j) = 0 for
eacb conste;~t c(j) .
A system fg(j , x ) ) of ir. linearly inde,;?enuent; funci;iona
can always be trans1·oz:mecl i:1co n system { f(j , x) ~ of m
Orthogonal functions . One may wri be the J'o Uo'.·tfr1g eqtw.-
tions :
f(O,x)
f(1,x)
.r (2 , :x)
C 00
g(C,x)
g(O,:x) + c
(3)
c 10
11 g(1,x)
c,.. g(O , x) + c ,, g('i ,x) ,, c 22 g(2 , x)
etc .
Su bstitution of the :'(j , x) into ( 1) yields just enough
equations for de l;ermination of the constaJ1ts c••
Xt
J f 2 (0 , x )dx =
Xo
J f 2 ( 1 , x)tx =
X, ,
••x,
'•x,
f
(4)
x,
Ji'(O , x }.i(1 , x ) cLx
Xo
f 2 (2,x)dx
'o
etc .
=0,
x,
=X 2 , f i'(O,x) f (2 ,x}dx =0 ,
••
x,
f f(1 , x)f(2,x)dx=0
••
1 • 11 OR"cli OGONAI.! 'l'Y
'/
The coefficie!lts X0 , x,, .. . "re arbitrai•;y . Tl:e;nu·e 1
for noi'l!lalized sy,.,tea.s . le Iollow,-, ;'roa (2) thbt (4) ac~ually yields values Ior t!ie coet"ficieuts c •• as only "
system fg(J ,x )} or li!learly indcpenQellt Cu.uctio~e coulc
satisry {4) i~entically .
l'igs . 1 to 3 :;how c xerr.;;>l e" of or·Ll:ogonal fw.:ct<.ions .
The indcrcnJl"nt van.able i s the normalize..: 1.ime e ~ 1./: .
The runctionr. of' i"ig . 1 a.re O!'thOllOL'nlt.ll lll ~l.e in~er..;al
-t ~ & lll ~; they will be referred co as sine ;.u1d cosi ne
ele111ent!'I . OnL' mn;y div i de t hem im;o even runc~ions r c(i,9) ,
ortd functioJL" r,(i , 8 ) ar.ct t he constant 1 or W(<l(O , 6) :
r{ j ,a )
fc(i . ~•
't2
c.os 2 nie
fs (i ,9 '
1
El!: 2ni5
•WS~0 , 9• =
1
r2
(5)
e
• undefi:.ec.
< -~. P > . ~
• d),61
ull 1.61!
0 llJ;ll
1-
I QClll
Ek.\1,81-=l=L..=
s.!lil 81~
u1u.a 1 ~
ial ll,81~
l 001!
30)11
1 a100
5 CIDI
"IO,OI ~ 6 0110
•'l•.lll ~ 7 1)111
"114,81 ~ 8 lOilil
..111,SJcrt.......f"lJL 3 '1;.')1
"'1~e1 =FlSLJLA:j=t ·o ·n
........ A=Fl::ft:A::R:f
CJlii,B' 1.ILfUl.R:=R:F
0
·-
112
11 11·1
11 llJI
ui17.8ll:fl:F=l:fl.llJ 'J l'V.
<,17,8, ~ I( 1110
>ollE,8) fl::Fl:fl:Fl::F 15 1111
- 112
0
6-i/f-
1/2
Fig .1 (loft) Orthogonal sine and conin~ clomnnts .
J.'ig . 2 (right) Orthogo nal Wal.sh e l eme ntn . •rh" I1um1.iel'$ on
the right 15Jvc. j in decimal and bi.nni'Y f orm , i£ the notab.l,on w~l(J,8) iR used . wal(2i,0)=caJ(i,9), wul(2i- 1 , e)
=aal (i , a ) .
8
The term elornent ' i a: used "t"O em;>hasi ze i;-h at a funct.iou is tle:"ined i11 a finite interval only c:ind in undefined out.si de . The terJI1 'pulse • ir.: u~ea to em_phasiz.e tl1at
a !'w1cti ou l s i dent i cal zero outsi\~C e finite inc:erval .
Continuation o i' the sine and. cosine eleojenva of l•'i15 . 1
~utside Of the interval -f ~ 9 ~ ~ by f(j , 9) : Q yiel ds
the sine and cosine p11J aes ; pe1"'iodic continuat;ion, on t11e
other band , yields the periodic sine and cosine runci:ions .
Ii; is easy no soe , tnat t he condii;ion 1 ·1) for ortho gonality is .satis.fi0Q. for sine arLd cosir..e elements :
1
112
J
-1
117
1\f2 sin 2niS d0
s
tZ
J
1'{2 c<is 2ni 9 aa =
o
- 1/ Z
l/ Z
I/ ?
J '{2 sin 2niB.>[2 sin 2nk0 de= Ji? cos 2ni8 ·lf:? cos 2Pk0 ae ~&;•
•112
- 11 ?
J'" 1{2 sin 2rri0 ·\f2 cos 2nk9 dB
= 0
.111
I"' 1 . ,,
ae =
1
- l/2
Fig . 2 shows th-0 orvhonol"Olal systeu of Walsh fu.ru.: t. iou.s or
- Jnore cxacLly - Walsn clements , consist int:> of Cl constant
wal(O , S ) , even l'unctiono c.el( i , e ) a11d odd .ruuctions
sal(i ,a) . ~'hese fuuctior:s jwnr bric!< a.oci ~ortll bor.ween +1
and - 1 . Conside1· tti_c product o f the first t\lio runctionn .
It i~ equal - 1 in tbe int.erval -~ ,; 8 < O ru.iti -1 in L!:e
inte1·val 0 ~ 0 < +~ . The i u t:eg:r·al of ·t:Ite~ (~ product..s he..s
the follo wing ·v alue :
0
f
112
I
< "1 )\ -'\ )d8 +
•
- II 1
c+1)(+1)aa
~ , : ~ o
k
'I1te product of the sccor:.d and third elf"1nent
in the
t.:lle
-t ~ 9 < -~
-t ;; 0 < 0 and
int.erv~alZJ
i.!.lt3l' Yalti
and
yields 11
0 -:; :J < ~~ ) and -1 in
+t ;: e
< ~t. Tht: iu1,,egral
of these prod.nets ugs.in yields ze r·o :
-i1r.
!
· 112
o
111.
in
(-1)(-1)ae + f( -1 )( +1 )ae + J( +1)(+1)ca + £(~1 )( -1)0& ~o
-l!f.
-0
llt
1 . 11 Ull'fllOc;;;11ALITY
One :oay e~sily veri~y :>hat :;he in~cgral of ~he rraduct of iw;v two Junctions is equal zero . A fur.ctior. cul t:iplied dtb itself yield.: t:no product:<
(•1)( -'1 ) or
( - 1)(-1j . F.cnce . tl:e~c ?!'oducts hnve ;he V9luc 1 in ;he
whole intervnl -t ~ & ~ -~ ar.d ;neir integral in 1 . The
Waleh fu.nctionz nrc tnus orttonormal .
Fig . 3
partic11larly !limp le nyr tom of orthogonal func Lion a . hvidently , .:;he product botwoon a.ny two
i"unctiouc vw1ishes and the integ:·alr 01' th(' rn·oducta m1Lst
vnuiah too . E'or noi·malization Ll:e ampli tudca of tl.o fun<;tionc rr.ur-t t n '{1;; .
s).OW'1 a
iB(x)
••
I
fill.81
fll,61
1(1.61
ffJ,91
I
fiO,vJ
n
'll.v)
1(1,vl
I
I
t r3....>
t 1 u n _ r = I _ H4.vl
fXl9)~ rxM
--·
•••
•
-~-
!
·I
"
P.O<J
9"1/1
V-•fl
1''ig. 3 Ortbogona1 ulock pulses
!(j,9) and f(j,v) .
Pig.'+ Bernoulli polyno:r.ia:s ---"''-'-*---T-...;;;._,__H...,...;l(top right) .
E'if? . 5 Legendre polyno1<iale
P.()()
(right) .
'
..,
P,W1
An ex=ple of a linearly independent. 1ut.
nal syotem of functions arc Bernoulli's
B 1 (x) (11),
ll 0 ( x)
c.1·Lllogopolynomial"
110 t.
( r; ):
1 1 B 1 (x) • x -
t1
B 1 (x)
B,(x) • x' - tx' + h, B,(x)
=
a
x' - x
x' - 2x' +
I
f
xl
-
To
1.
10
m
L:; cCUH J(xl
=
f':.oijll:IBMA!PIC.~L
FOUNDAT rot;s
o
jsO
ce.n be ;;a~isfiea for all vtllue~ cf x ::>nly :if c·~:nJ xm is
zero . L1hi.s implies c (m) = O. !'low c(m- '1 )jj •~·..1 ( x ) i s tte
highes t. ter m in the stuo and the :;8.f!!.e rcacci:.1n_g can :JC
it . Thia pr-oves t;lte linear l nder:endencc of
appli ed to
the Bernoulli polynomials .
On e m:i;y see fr.cu Fig . 11
3c-=noul l l
\~·ithout ca lc.1~la t:i
po l ;v nomials a!."'e not
01~t11op;on al .
?or
on
the~
tl!e
or tlro1~oua
liza:t i or. i.n ti:e i T'. -ccr·v·al -1 :5 .x ~ _,.4 :ine a:ny .sul:·tstit uL.e
them for g(j ,x ) in (}) :
1· 0 (x) = B 0 (x) = 1
P 1 (x) = c , 0 i:. 0 (x) + c 11 B 1 ( x ) , e:;c .
Using the cor.r;t;nnts XJ = 2/{2J+1 ) one o°Ol;t1 infi i'r·om (4) :
I
f 1cb:
x.
r' [ c ,.
+
_,
·I
=
2
' 10 +c 11 (x- 1/ ) ] dx = 0
J[c
2
_,
c ,, (x- i ) ] clx
The coefficie!lt:s c , 0 = I, c 1 , = 1 , et:c. a re z·ec.dily obtai.necL P1Je orthogonal PO-j'1lOmi1Jl6 Pi ( x ) ;:i"sw;ie the fol lo wing form :
?o ( x )
1 , F 1 ( ::-:. ) = x_,
? 1 (-x }
t (5x ' - ;ix ) ,
I- 1
- 1)
P 2 ( XJ' = - \;.X
P, (x) = ii(3'.:>x' - jCx' + 3)
These are tbe Legendre pol~;nominl!:! . P j ( ;..:) must be mul t i112
pl i ed wit h x·'n = (j + § )
f or normali7.ei;ion . Fig . 5
I
shows the fi.r·::;t r1 ...: e pol yno n:i als .
1.12 Series Expansion by Orthogonal Functions
Lei; a
function !'(x) bo expanded i!l a
series of the
orthonormal syst..em {!'(j 1 x ) } :
F(x) ~
==
L, "<nrcj ,x J
J= 0
(6)
1.12 SEFU:.S tXJA:JSlO!;
11
'!'he val t.<" o! ih.; coe~·sici9:;i:;~ a( J) :na:1 be obtained by
n.ultiply!ug (· ) 't>y f (k , x) &rod ir.toe~rntinc: tho rroduc;;"
in the
i~~e?"Val
J••F(x)f(~,x,dx
••
How we l
c: orttogor.ality x 1 1 x I x 1 :
=
(7)
a(Y.)
i>' l'\x: represenLe<l, lf ~:.e coo!!icienLn a(J)
are d<>L•• n.Ln,..rJ IJy (7)': .L<!t ua ''sswr1e" Hei• i <>a Lb(.) )f(j , x)
ttnv:i.ng m
t<'t L'lll tl
,y..t eltls a beLLe.r
r·e pi·esent~1Uion .
terio n •'01" ' b<'~ter ' ehall be th£: le!.-Jat. m&u.11
a.tion Q oJ' F(x1 from Hs !"~Pl'eeer:catior"
X1
L
-
>r:o
•f
X1
F
1
C.evi-
D\j):\j ,x ,J'ax
,.,
*"~
(
aqu.a1~e
m•1
r [l'(x )
•
Q =
'f'be cri-
x )dx -
X'1
Xt
"Lt-( j)[!'(:<) : , j, x )ex + .'
0
x 11
i•C
ic
0
l
-1
( L;u<. }1'(J , x ): dx
J•I
Using (?) and ;;he or~hoi;or:;;!~ty o!· t:1e l'w:cLious f(j,x )
yields ~ in the :~o:!..!.Ol.\°l.llfS ~or1 ;
..
f
Q
••
F 1 (x)u.x -
m-1
2:;a 1 {j) +
j
~o
~
L,['o(j1 - e(j)1 1
Tho lant t:e"m vani sheo for ·o( j) = n( j)
devintion n"11umQ3 it o minim= .
..
'..
(8)
i tO
.,
Mid
t.lie
miH•Jl
square
'rtie vocnlled Bac,;"l i11equal.ity follow11 from lHJ :
~ a 1 (~) ~
f
r'i(x)dx
••
T~e upper limit of 5lll!l!lation =.P..y hP.: x
since the intet;ral doee:
.::iot;
1nztood of ~ - 1 1
depen~ on t: tL:.ld J.l.. St t'r:s
bold for any volue o! o .
Tho syst.em (!(j,x)J i s called orthoi:;oni..l , nornali7.cd
and cor.iplet.e , if the mean square deviation Q conv.:::-gcs
to zero with increasing m :for> any function ;,•( >r) ~h"t i•
quadraticnll;y integrnble i.n the inl. ervol x 0 ~ x ;:; x 1 :
x,
.tim
m-oo
J [ f•'(x)
••
m•I
-
I;
j.iO
a(j)l'(j ,x))'dx = 0
(10)
12
1 . NA'1''1X:f·lA'IICil.L FOUN-:JATIONS
Tho equ ulitiy sign t1olds in
equcllLy (9) :
t.his case i!l
~ 1
I;o
(j) ~
J:-0
f"F 1 (x)dx
~Ce
Eesael in-
( 11)
i',J
Equation (11 ) is known as comple~eness Lbeor ea: or Parscval ' s t!1eo1·of.!!. . lr;:-; ph~,vsica.l meaning is as f o llows : Let
F( x ) reprcnent n voltage as funct.ion. o.r t i me acro.s!1 a
tmi:.. 1·eaiu-t;L11ce . The inr;ee;rsl of F 2 (x) r epresent:: Lbe11
tt".G' cncrr:y diss:i.pated in the i_·esi s t.or . Tt:is ene-rgy cqu-:i Lu 1
according tc ( 11 ) , the swn cf che energy of the tc.rms in
the san:L;o(j)i'(j )x) . Futi;ing it di.f.fe reot.;ly , the e!l<?rgy
it> the came wt:ethcr the vol tat;e is described b-y t 11e t i1ue
f unction F(x) or it,.; seri.e• expansion .
1
Tbe s ,ystorr.. f 1'(j , x) ) i s sai d to be clo se6 , if t!:crc
is no quafu·ati cally '"tcgratle functio n f(x) ,
.,
J Jo" (x )dx <
••
wllic~
for
~ ,
t ho
( 12)
~quality
.,
f
'•
F(x)f(j ,x)dx
c
0
(1 3 )
; s sat i sfied for· a.11 values of j .
l ncomp!ete s~1ster:ts of Ol"thogonnl f•.1.llct.ions do not pe.rmi·t a convergent serie s e x pans ion of :ill quadrntically
inLe5.r·able functions . r\evertheless, they a .L·e oI great
r,>r~ctical interest . l;'oT· inst:mce, 1.. he o utput vo ltage of
an i deal frequency lo••pass f i.1 i;er may be r·epresented
exac tly by ru1 expr.i.nnion in ~ serie s of che incompl ete
orthogonal ay,, tem of sin x functions .
.·.
• A Co3lpletc orthonormal s:,rs tem i s always closed . 'l'he in-
verse of this statement hole! s true, if tl:1e integr.als of
tb.is section are J.e be sgue rathei· t han Rielllann integrals .
'.l'he Ri eman n ititegral suffices for t;he Jnaj or· pa.et of this
boolt . H.ence , ' integrable ' wi.11 mean Riemaru1 intee;<·able
unless otlie-rwise s-cateJ_ .
1:;
Whet:her a ce;:tail: functior- F(x) cen f.(' c:xpa..'lded in c
se:ries of a fArticula:- ortbogor:o: s~·ate:L {J (j ,k )) cam.:.ot
be cold froa cuch Fiople !esturee of F(x) us its conLinuity oi· boundedness' ['.>] - UJ .
1.13 Invariance of Orthogonality to Fourier Transformation
A tioe function f(j , Ii ) may be t'epi·esonled u.ucier cercain cond1 Lion:;i b;y two fUllctions a(.) , \I) 1.uid b(J , \I ) by
metllla of Lbe fourieL' t 1'ans.!.'01·m :
f (j ,S ) •
Jfl'> (atj ,v )cos2nv a
r
..
--
+ ·o(j , v) sin2n"9 ]d v
( 111)
"'
a(j,v) •
f(j,6) cos 2 in1S d9
' ) ~.in
' 2 r.v 9
b (j , v ) • •' ...~c'
"' ;e
dC"J
(15)
a • t/T,
v • fT
It i~ ndvll.Dtogeou~ !'or ce.rta.iL Bt-·~llc·•tionPl Lo replace
tile Lwo .ruoc~lon.s u(j ,v) and h(j,v } bya ainslo fun~tion2:
g(j , v ) • u(J , v) + b(j , v)
V l<o)
Ic follows fi•om (15) thn L a( ,j , v) h •JJl even and b(j, 'J )
an ocld :ftmction or v :
b(j , v) = -·o{j ,-v)
E:quntionn ( 1 6) 9nd (17) yield ror g(j , -v) :
g(j, - v) • n(J ,-v) + b(j,- v ) = a(j , v) - b(J ,v )
a(j,v) and b(j ,v ) a:ay be regained rrorn e (j , v) by
Of (16) and (18) :
a(j,v) • 2[g(j,v) ~ g(j ,-v)J
b(j,v) • 6[g(j , v) - g(j , - v)J
llsing•t.e fu.nc;ior. g(j ,v ) on,- otay wrHt' (1'-) ~ml
in a more nymmotric for:r: :
a(j,v) • a(j ,-v ) ,
( 1'1)
( 18)
(19)
(1<.)
' ?or i.n.3tance, the .Pou.rier series of a. co:.. tluuou~ func-
·tion doos not; llave to c onverge in ovcry point . A theor901
due to BANACH stnte':', i;hat; there nre ~u·bit<·nri1:1 man:;
orthogonal cyutems wit h the feature , that Luo orthof!:Or.al
toei;ies of a continuous ly diff erontiablo function dlve!'~es o.lmoet; everywhere .
Real notaHon in u sed for the Fourie1· tl'DJ'.J.sl'o1•111 to facilitate comparison with the fo r:nulaG of tlte e;enerall;,ed
Fourier trDJlaform derived 1ater on.
1 . l'!ATEEl'IA1'ICJ\L FOUI'iDA: IO~lS
111
r
t'(j ,8 )
sin?n v6)cv
(20)
cos :?nve + sin 2nv9 )d.O
l21)
g(j,'J)( cos 2n ve •
-oo
""f f(j ,e
g(j, v) = _.,
)(
'l'he integr-al;:; of a\j 1 \l ) cos2'lv9 a:ld b(~ 1 v)sin2rv9 ir1
(20) V<1nisii sinc•i a(j,v) isandveL Md ·a(j,v) i.o an odd
Junct ion oJ v .
J.e" {f(j ,9 )j be a systan. ori;honorrn9l in the in~er·val
-~S ~ 0 l! +~8 anc zero ou;;side . '3 ma:; be l'inii;e or infinitr . The ±'unctions r(~ , 9) are i'O\ll~ier• tr•EIJ1Sforma.1Jle 1 •
Their orthc>gonnlity ini;egral ~
J"" i'(j ,e )n.k , e )as
= ~,,
{22)
,
- oo
""Jr(.j,S)[f5(k,v)(cos2,vS
""
- oo
1
sin2r.v9)dv]d8
6·•
1
sin crv~ Ae;a,, = c;,
-oo
J g( I<,,,)[ f
""
00
·' (j ,9 )(cos 2nvg
""
f g( ,j , v)g(~ , \IJdv
::- 5ili
(23)
-~
l-ie?1c~ ,
( f( j
the Jo'ou..l'ie:· t.1·ansfor·:u of
'e l l
01•t.liono1·ma:
01·~honorn1a: .~:y,;t ""' ( 15\ ;j, \I )1
:;i eld>J w:
S1io~titution
g(j , v) =
a1.1
:::;,y1:;:.L~w
.
of
~c~ , v)
• t(J,v) ,
K(k , 'J) • •(k ,v ) - L(M,v)
into (23) :;..telas i.t in terrcs or tt.e notation a(j ,v ) ,
1'(j,v) :
.,
.r..:<- .:i , v )e;{ k, v Ja"
<XI
_'[a(,j,v) •
-·~
b(j,'J,][n{k,vJ + b(k,v))dv
-00
=
"".s"fl (j,v)u{k,\i) + b(j ,v) b(k,v) ] dv
_..,
b1k
I OrttolJOL'ma li ty ioplies
~he existea.ce or the F'OU!'ier
t2·ansi'o1·m and tLe inve.rse Li:ansform (Pluncherel tL.eo1·en.) .
''l he i!1'tegrat1ons may be interchanged i since the integrand!."i are abnolutcly intcgr.:ible .
1
15
1 . 1 ;:i 11'/VAh IA.ll CJ:; OJ-' OR':HO;;o;r;.i..!.T'i
d •
\ \ x. /-!
·v ! \
\
'/I
of ~in f1:J~ coninc nulF'-g . 1 . a) wa:(o,e ) , iJ) f? "'-n "rr3,
;J) '[2sinu-,B, e) ·;2coai,r-e .
Fig . (> Fou1:iOP tra..."1£fo::'DS ,;s(j , v)
3eo
c)
occordini;
'{2cos~"l3,
to
Fig . r,
sine w1J
elemnntn
eide th.o
shews as 9Jl exa:nple t.he .7ourier t.1·a.r..sfo.r:t.E' of
co ... iuc t-JUlces . These pule':i~::J flt·e det·.t·1~d !'.tom. the
or rit:; . • l;v concinuin.- thPI i lentt,~ai ,·e:·o 011;; in Lo rval -t ~ e ;,; ~ :
g(O,v) •
J 1( enc Pnv9
112
ein rr"
r-"
+ sin 2n v9 )<JO
· Hl
Sc(i ,v)
[' 'f2 coG 2rr i9{ cos :?rrve
d11
(24)
:>nva 1 10
• 111
Ill
g«i,v) •
f
'{2sin2r-i3(cos2-Tv~ ..
eir_;.,vd/d3
-Ill
-
!""ii. '1
V+
I
r \\l•i)
\
I
Ii'ig . 'l ohowa the Fourie::- crar.sfo1~ccs o! 1l!ilsJ1 pUlee-3 J.erived by contlm<i.ne; the element" of Hg; . 2 lolcn •.i<:ol zero
outside thn in Lorval -t ti e ,,; •t :
g(O,v) • rwal(0 , 9)( cos 2rrv9 ,. ain2nv0)d9 •
... \/J
sin rr"
""
6s(' ,v • -
"'j ~al(1.9
- 11 l
One n:ay rf'ttdily see f1•om tbese ex Amp I C!'t:l tho. I; t:1ver. ti:11r"
J'LWct.illrlS t.J'a:l.n 1'nr:n into even
ti.me functionn
t. r~111r::foi·JJ
fJ.~aquenCJ'
•·
Jnct.ion~
:..tLd otl<l
into Odil fl•rqucncy J:.utc Lior.. .s .
,.•alu.eo Of t..l1e frequenc~• h&VO Ii ! r>r~cct.ly •.•:JliJ
ph:;si-:!al mear:!.ng . 1he oscilla~ior~ of fr<'qt:cncy v is u t:c-
(.e-u~iVe
0 . ii
Fig . 7 .?ourie.l' t.!':.U1Sf<,,rr·1~ £~~)" nr w, -"h
~"
Hg . 2 . ~ 1 .,,,, i <o ,a l , b
~,,_(1,e 1,
r.i) - s..1(2 , 9J, o) cnl\2 . 0 ) .
l~e F~~rie~
t'lllae~
c)
according
- c.:il(' , a),
.,.
:•ig . 8 Fo=·ier tr&nc. orme g( ~' ") or thf> block Plll!1CS .:( 1
r(2,8) nod f(;l,e) o.I' Fig . 3 .
'a ),
1.1' H:VAJUJulCZ OF ORTHO!;CN AL:TY
~!"aD!1fOrT.
r.•,:.;
oscilla~iC"'J,
lute
VOl~lC'
17
~hr; ::ar.ie ~:a.2..ue f0.1' -v ri1.d -vi
t!"rui~!ora
if Vhf; ?'ou.:"'ie.r
b lt
orr-c~i;;t'
Sit;:-:&
.:'O!~
it
LU!" ttc
i~q g
!"91H:
.aint"
~:..so
-t-V Dl.i -v.
Fi!!; . B she-•·~ tte Yourier tr:ir.<:fo=::: g(j, v) o;· t:.ree tloc"
putsa:: of Fif . ;'>. 1ltey nr-., no lcngcr nithcr ov-:o o~· ot!J 1 •
'\f,tl. >'1(0.)·:c 113-- r..~l
1
a-1/f-
!
Fig . q Ot•thogonv :-.,y~t:em of' si:ie ':t.i.u.1 :.:oh ....!!f· pL:lt:ies hav..:.:1g:
jumps or aquol. n i.€;!,i; aL e =
n.nd a
-t
•t .
li.. ig . 9 nhowu n oyr:te m of 0.1.'t.h.or.··onal tiln c: nJ1d coB .tne _p ulses . The;y a.re tune :or.if ted compared wH lt Ll1o<>e or Fie;.<,
eo that all fwict;iorrn ba·<e ~Ulll]).9 or ~q u al Mlfl..lL ude '1L
a = -i 'llld 9 • +i . Tl::.eir Fo·" rier tra.n~fOl'ID• g(j , v) are
f>hown in iig . 10:
k
=-tj for
ir
= t(~ +1l for od<l j .
e~en j
(2';)
The lo ourier tra.nsfora:s of the ~:ariou::~ "h ock i. tll.aoc are
di.fferent but ~heir frequency power:- npeCC!"' a:ro equal.
The Powe:r spectrum is the E'ourier trnnnror-cn of Cho a·.iLocorrelution fu.nction of a function 1 >tnd rLOt the Pourier
trn.nnrorm of t:he function itself (W1ener-Chin·tchin
t heo rem) . ~'he connectior1 between Pou1•ier tro.n::ifot·in , power
8peotrum nnd wn~litude spect-rum is diecuneetl in nection
1
1
1. 32. See also
[q) .
1r;
1.
~iA1'lll.~·JAI ICAl.
FOUlDATIOHS
;~x)1{2.0)
I
I
.
\
5
l"i i; . 10 fou.l'itH' t r11:>sfoo•ms
pulidC~ of Fic; . l; .
o.r the slr.e ~nd cc siJ1e
•;( ,j , v )
Tli~ function:; b; ( S } of the parubolic ~y-irnie:· .show11 in
li'ig . 11 and thel r li'ourier trru1nfo.rn:.s i;( j, \I ) htP:e l.b~ same
i;uapo
[SJ :
(2~)
f(j,S) • t (ll) ,
1
,1 • 0, 21' 2i.A.1 ; i
= ...
r'ig . 11 The function,-;
~
r.!il.>ol ic cylindei'.
~I (x) •
X
•
9
2 , ....
1
1 • t 1 (9) or
-•x'
e •
.
~de I (
1vJIV2n
01" 4"'f\I j
j
t1•
>,cl
:ie 1 (x) • •''
X) j
::. Q ! 2i
t
2i r1, i
a
q ("nv ) o:'the pa-
d
I
- ,\xi
( - -dx ) o •
11 2I
• • • •
~ (a)
decrease.G for large absolu~e valur·~ of e proportion•i1 l y to e 1 exp( -ie 2 ) untl t 1 (Lfnv) clocreases fo.r: large
1
a.lrnol uL !! values of v proportiom>l t:y to ( '"'") 1 exp( -! ( 4nv ) ] •
J'ulnoc: with the shape of para.bol ic cyl inder f unctions re1
quix·c " pa.t·ticulai•ly smal 1 pure 01' tho tJ.me- frequency-
1.14 W;.J,SH l'\JllCTIOliS
1.14 Walsh Functions
The Wai~· fur.ctio:ic •·a.:.(O,B ; , :-Rl(i,& l u.nd ctl(i,S )
are of connide1·a.OJ e iC!t:erest ir: corr..zu~icatiQ.OSJ . fh.ere is
8 elono connoctio:1 bct-•..:eon S8l and ~ir.e runc:tiior.v, nc wr:ll
as between cal at.al oos:.ne function.- . Tltt i .. ~~,_,.., " and c
in sal W'ld ell .. woro cho!ien
to
i.ndicul;a t..hlt:J COLJ1rction,
t.h~
• !:11 • f!T13. aor•i. ved f"i·om
while tbo I otbo.i:·a
nu.m'!I \,1n l. sh .
For eo1:ipu11ationa_ p urposes it. is aomet.imetJ moz•e
con-
venient to u.:.o oinc and cosine ftwct:ior1t1, while at otn~r
times tJ,o exponential fu.."lct:i.on fr more co11veoie11t . J\ si!:ri.ler duality of r.or-~tiQn i?X1Sts fo:- 'n'ttleh .rw.ict.iont:: . A
single !'unction wal(j,~) :i:n;; t,. 1cfiM~ in. tead of tte
&hree functions wal ( O,e), sal 1 i,3) and c~l( i,S ) :
wal(2i,9) • cal(i,ij), wal(2i- 1 , 8) = sal(i . a)
( 27)
i= 1 , 2 , ....
The functions >;nl( j , g) nay be defined \.l,v t11e follc:o.n!;
di.f.t'ercnce equaLion ' · ' :
wal (2j+:p , a) • (-1 J
P : 0 0~' 1
j
j
D
0
(i 11j •p{
I
1
wal(O,e) • o ro1· e <
>
,
wal [ j , ?~9+t)J +(-1 J
2I
-1,
...
e > +t·
I
I' >
wu
[j ,2(9 - tJ J}
for
-1·-e< t;
( 28)
1
Pulses of tho 3hape of pai·abolic cylindor f•mctions use
the ti!lle-frequoncy- don.ai::i ti::eorctically '\.lest'. l"LiE i;ooC.
~e hes not been of much p=aci;ical 11t.t.l ue so fai.· , .:inc:e
81.ne-cosine pulses and pulses derived t1·o :'j :iinc-co.:;inc
PUl.sec aro al.Dost as good , but. !!lu;;:_ e aisir:- to genera-;e
and detect .
~'2'!ie pt·obably olcest use of Wal!;_>i i·U-.,ccion G in co~.ciunica
'~ons is !or the transp<>siti.o!l of conducto1·s [1d] .
Walsh functions are usually defined by p«oductc of Rade-
~ache>r fWlctions . ThiG definiLior1 has oony ndvantsges but
aoes. not :yield the Walsh funct ion" ordnred li.Y Li.e nun.lie~
~f sign changes a s does the di.fferenco oq 110L.Loo . Tb.in oraer is lmportnnt for the generalization 0£ frequency io.
sec1;:1.on 1 . 31. lludemachor :functions ot'b tlii> J unctions
+sa1(1,e), aal(2 ,e), sal( 'f,9), .. in l'ig . 2 . Walch functions
~ 8:? also be> defined by Radanoard matriceo [ 1 •·1] .
(j/2) moo.no the largest integ13r smaller· or· oqual ~j .
,.
20
·1 . i'.ATJIEl"AT ! CAL FOlfKDJC IOliS
For explanation oi' r:l1is dirfercnco equat i or1 cun.::;i c.ier
tr.e runc~ion wal (j , &) . 'i'..'ie function •·iaJ.(j , 2~ } has the sruJ1.e
inco the interval - t ~ a < +t.
1<111[j , 2(0 ·ri)) is ot-tained by sh~ .f';;i ng wa l (j , 2 B) t o the
left inco t he im;erval -~ ;; a < CJ , a."ld wal[i , 2( 3 - * )J is
:ibtDi ned by shi l'LiJlg 1~al(j , 2~) to the right i nto tho :cnshape ) -out in
s~ueczed
tervril 0 ~ 0 <•I t.
As: !ll! example , conside.r.' r;he cases j
j
O, p - 1 end
[0/2) = 0 and [ 2/2 J = 1
= 2 , p ~ 1 . Using the values
o
o!1e obt.l?i.!!S :
wal.(1 , e) - {- ·1) 0 •1 [wal[0 , 2(9+ t ) l
7
( - 1)'•' ><al [ 0 , 2(3-t ) J}
wal(5 ,9 ) = (-1 ) 1 ' 1 l1·Jal[2 , 2(9+ t ) ] + ( - 1)" 1 ...-al [ 2 , 2(9 +t) l)
It D!11ybe verifi ed from Fi g . 2 tllat wa1(1 , e ) = ~al( 1, e) .i.s
obcained rrom wal (O , e ) 0:1 sqlieezing it to half i t s wi dth ,
muleipl .ying the function tlia.t i~ s!>iftod to the left by
- 1 , and the 1'tmction tha1; is shifi:;e<i to the r i ellt. by +1 .
;;al(5 , o) = sal(~ ,a) is obtnincd by sqiieedne; ..rnl(2 , 6) =
cal(1 ,e) to halr i ·t" widtn , :uU:.ti;ll,y iJl;; che functiol1 that
i!: !.1bi ftcd to the le.:'t b;r 1-1 ~a._ Lhe .function that is
shifted to the i·i;:;h~ by - 1 .
•r!1e pr·oducL. Of t;.~·.'O l-.'nl £.'h funci;iOC$
runclion :
~tields-
rt:iothor \·lal Sb.
wal(t , e)wal(k , e) • wal(r , a)
Thi ,; i·elutiou may readily be ;i~'aVed by w::-i ting bhe dif.i'e-
r ence equo.tior; for we.l ( h , 6) ~.nd wal ( k, 0) , ru:ui uul Ciplyizlg
i;he:n wi;;J1 e?-c~ other . l t t urns out that ~he prod uct
'(1al(t , 9)wal(k , 6) satisfies a cl.iffereuce equ"'tion o.r the.
S"'1Cle .form fis (28) .
The dete rmination oJ.' the val~1e of r f=on: the d.ifJ.'ere nce
equation is somewl1a.t cwnbersoue . The .r esul t is that r
equals the r.nc-Oulo 2 !'1um o.f h o.nd k :
wal(h , S)wal(k , 9) = ' "tl(hlBk , 0)
(29)
1'hc sign !!l stands for an addition modulo 2 . k a.nd h are
written as binary numbers and added according to tho rules
~ o = 1 , c e o = 1 <D 1 ~ O (no carry) . Adcli tion
o e 1 = 1
1 . 14 WALSll i"U!JC.!'r!OJ;s
21
n:odulo 2 is •hat i. lblf ad.:er does~ bica!',Y digital cocnutel'!' . As an exa:npl<? , consider the reul :!plication of
~al(6 ,9 ) s~.d wal(12,9 ) . !:sing lin!!.!'y mrnbe1·a !or u and 12
one obtnin:l 10 for
e
t~.e
su::i
·~ $
12 :
0110 ..... 6
1100 ••• .. 12
15'm ..... 10
It may be vei•i.fiod f::-om Fig . 2 t!:.nt Ll:e producL wal(o, e)x
waJ.( 1 2, G) oqua.1 11 wn 1( 1 0 , a ) .
Tlle pl'oduct of a Walsb fU!1cvion
W!~h
Hself yields
wal (O,a), since only the f:r>:id'.lcts (+1)(+1) =d ( - 1){ - 1)
occur .
wal(J,9)wal( ~ ,9 ) = wal(O ,a J
j
e
j
•
(30)
o
The product of wul(j ,o ) ·...i.tt. wtil(O , o) leave;; wnl(j , 9)
unchanged:
wal (j , a )wal (O , e )
j
eo •
w::tl(j ,e)
(31 )
j
Since the addi Lion modulo 2 .i. !> n~uocintivo , tho a:ultiplicat;ion of ~lalGh !unctions musL be aaaociat;lve too :
...
[wal.(h , 9 )wol( j ,a ) ] wal( k , e )=wal(b , a)( """l ( J , 9 )·• ~l(k , e )J(32)
Walsh functionG forrr. a group wit!" resroci; to multiplication . Equation (29) shows tha- i;JH> product o!" two functions yiolde again " Halsb i'unci;ion ; the inverse element
J.s defined by (30) and i s e<p•' to th<' ele:11ent itsel!';
•he unit element is waJ.(O ,o ) 11ccordi::g Lo (31); :te associative law is nhown ;;o hole by (;;2) . The gl'oup o!" Walsh
.f~ctions is an Abelian or comiutative gi-oup , since the
.factors in (29) , (30) and (31) aay be commuted . 1-Jat!tcmatically epeakiJlg , the group of Walsh !unction:; io isomorPhic to the di.sc.rete dyadic g roup .
To d1rtermlne the number oi elements 1.n u e;roup and its
Sube:roups 1 consideX' what nwnbers can occur, if two munbers
~ e.tld h, tho~ aJ.'e both smalJ.er· 01· equal 2• - 1, are added
11todulo 2 . It and. h are written ao binary n\llnbera :
1 . MA'i'rtl!1AT ICAI. Y.'OUNDA'l'IOHS
22
h
Ps- 1
k -
Q.s.I
2i;· I
2"'·i
The modulo 2
e
b
j
;
+ P:.. 1
-:.~· 1
~
() ,., 2
T
SUJl
(p,_, !!'
or
q.)• I
s-2
......
+ ... -..
+
>
1
p,2 +
0
·O
t --:
.....o ~...
q ,2 ' - q 0C
2' - 1
,...
c -1
03)
h and k y:i~lds :
)2"'
+
... .. .
-> °(p0 e qo
)2"
( 3'~)
The smallest J1umber occurs , i.f all ti he .f act..oi"S in ~~ro n t
of the power.:; of 2 ar.e zero . This num"oer is obtil.ined f or
h -:; j anci equals o. 'J1he larges-& nwnber i s obtain.:?d, if all
tt:ese .factors are 1; the resulting number,
i s obtaine<l for 11 = (2' -1 ) e j . This a:""""'• tl:at i n binary natat.ion j .has zeros where l1 has ones and vice v~rsu .
A group ·t hus contain s t he Walsh functions wal ( O,S ) to
wa1(2 ~-1,e ) , a total of 2:. fu.nct i ons . Subgrottpr:\ contain
l;he .l'unctio.r1s wal(0,0 ) to wal(2 ' - 1,0 ) 1 O tE t' < s . 'I·hese
are all the subgroups . Sinco a .subgroup contains 2 ' elemont.s i-c has 2' /2 ' = 2"· 1 cose·t;s . Evidently, powers of 2
play an irr.portant role f or Walsh funccions .
Using (27) one may rewri~e the mul ciplicat i o:i tlteor·em
( 29) of the Wal st functions as follows :
cal (i ,e )cal{~ , a)
cal(iek , a)
05)
sal( i, e)cal.(k , a) = Aal [ [ke(i -1 ) )1.1 ,e J
cal(i , 6 )sal(k , e) • aal ([i&(k- 1 ))+1 , 9)
sal( i,9 )sal(k , 9)
cal[ {i-1)~(k-1) , e]
cal(O,a) "wal(0 , 9)
'l'he s i ne e..nd cosine .rur1cvio1'1$ sin 2 n ia as1C. cos 2ni0
are oi'thogonal in the interval -; ,; e ~ +t . This is tile
s;; sterr. reguired for a Fcux·ie.r series expansi on . The .Fourier transi:orm requires the s:r-item [ sin 2 nv9, cos 2 n v0 )
which is Ol'thogonal. in tile wbole intel'Val - o:> < 9 < +co .
No te that i i s o.n integer and thus denumorablc , \;hile v
is a real number and thus non- denwnerable.
The system of Walah functiona orthogonal and complete
in the whole interval - CO < 8 < <X> is denoted by {Sal (u , 0),
1
?3
1 .14 WAJ Zl l FlfliCTIOHS
cal(µ 1 ti JI , where u i s a :real :mmbe::-. !t "'111 be shewn
lai;er on, ti.at thie syscem lll8Y te obtarnccl l.>y ' &Lretcl·ing '
sal(i, e) ar.d en l ( i, 3 i jus:; c.s •he "Y"t"a {sic:;.,, vo , cos 2nv9 l
cnn te obtained I y strPtC!1ing sin 2r.ia nr.:1 cos ?ri.e . nn1
other defi,,itior. rt·1r: to PICHL.E!! st•rt:· :·::-oc t~e periodicallyconti:iued functions sal 1,9 ) a.n<l ~ol(1 , 6) . Fi-::>:nthem
one may aof"ino tne G"bo" L oi: Che Walal1 Lllnc:t.i.o:.u known as
0
•'
C9l :
Ra<iemnchor rw1ctioos [ 8 ],
cal (;:t k 10) • col(1 , 2 'e ) ,
sa.l (2 ', B)
a
l$lll(1, .: '9 )
(3&)
k • ±1 , ,1;2 , .. . ; - m < B < +cc .
..
Let now µ bo written as bi:oai·y cwllicr ;
1.1 :.. "L..J
u.., 2·• • . . . u 2 2 2 +..i , ';:!
_
-uo 2°
-
•••
1-u . 1 c:,,
+u
1
.1 2·
·~-Clll)
u, is either 1 or O. u is called dyadic rai;ional , if the
sun ho.s a .finit~ nWlbe1~ of t(!'~c . i:-!'---ia .1neans , there a;:ust
be st most 11 finite nWtbcr o i l:inary digit>' to the righ;;
or &he bin11ry poin• . cal(u , S ) and Pat(u , e) ai·e then de-
fined bS
..
tol~ow~ :
cnl ( u, 9) • Ti cal(u.,?-',e J ,
••·Oo
a.el( u
,e )
( -cuJ.(u,e ) ,
•
+cnl(µ ,e ) ,
- <0 <
-=< a <
0 <
oal (µ , 8 ) . cal(g2· M,e)i;at(2-"
s. eve;J
e
a<
( ;:7)
,co
0
I' - uyrn.lic irrational
< ""'
,aJ ,
- co<
a
< co ,
nu.:tber ; u :: ( ~·1 )/~w • u~,adic ::'&\iio:al.!
ca l(u ,9 ) and t:al(u , 8 1 are s :,o>m ir. r'i:.s .1 .C and 1~ for i::-,c
'The non- denumerable syster. o: ;:9ld. rw.ction~ required
for the Walsh-Fourier transro= i~ d·1e to FJ:n; [12] , ·.;ho
also pointed out first the eY...lsteucc or nuch n transform .
The correct mnthrur.aticnl theory of Lhe Wol sh-fol.l!'ier trau:>torm unir\g sol and cat .fu;nctione, which nre ~o:newht:J:t <1i.f:"cl'ent l"rom tho ay.stem !lsed :Oy FINE , fr due Lo ?IC:i.LE!l [ 9 ].
A term liko Fine or Pichler transform nppont·n f ulr a~ well
;" eho:i:ter than the cumber.some term W11l~lt-J<'ou1·i ,..,, t r ruuiom.: Mnthematiciana use th_i.s l,erm , bcct1u oo t he \.Jnl!:lh~ouh ar LL·unuform i s a special caee of tho e;o nor n I 1''01,1ricr
rnnsforme on topologic groups , JlUbl iahed by VIJ,EllY.IN zwc
Y&nr• aner FIHE • n paper ( 22) .
1 . MA'::EU'.kl'lC.G FOillDU·IQNS
211
< 9 < • 3 . 3la~k ore as in:iica Le
B!'t~O.J. c 1C1 value -1. By drav:i::-:g a line
l>"r"llel to tu<> 0 -·U<l s on" obtain• cnl\1.L,bJ or sal(..1,0)
a!::i .funct.ion Of 9 J.'Of' ft <':f"rt&in V8l Ue Of j..! . 'lice \rersa) O
line p;u:allel to enr u-l'Xi& shown r.1·,e vli:ues of cal(u , \j')
or snl( µ.I fl '1 < !J functior or u for.. a cer·te.in V.s.l;.tt: of e.
intervals -'I <: µ < •'1
t:..ie '-'"Slue 11 1 whitti
llJl<l -,;
Fig . 12 (loi..L) Tll o rlUlc~lons CAl(µ , 0) in the interval
- 3 < 0 < +5t -'~ < u < +~ . A £unction, e . g . cal(1 . _5 ,e ) ,
~c obt~i ned by d t'JlWi llf-'' u llnc FJV µ =- 1 . 5 pF)T'.:1llel i;o the
0-axis . cn.L(1 . '.';> , 0) io +1 wl1e<·e ~hl,; l inn runs Llu•ough Ii
black :iriln and -1 wlle'"!"O 'l 1, J.'una through a whi li~ area . At
bordor!l 1.HJ Lween b Lri.ck und w~.i tc a1.. 1::a.::i use the '\~etlue holding Ior Lhc abooluLdy ll' r ger u . 'Iho j'unction cal(µ ,1 . 5)
i s obLain"d by drawing 11 line tiL 9 " 1 . ';> para2.lel to the
u - axio and pt·oc~r.d ing accordingly .
Fig . 1;; (l'if'.ht) 'l''1e l'u.nction,; ~"l(u , 9) in the interval
-3 < e < ... ,, _u_ < IJ < •'• . f hf· v.slues .. 1 and - 1 of the
ftmcLlon& nre ob~oino<i by <!r1<wing lin~" as explained ill
1'be Capcior. Of l'ig . 12 . ;,t bot·der>i between bl8.Ck and white
ar·ens uz;o the vnluo holding for the absoluce:y small er µ
or a . Tt.e!·e :<r~ no runction:: a!ll(0 ,6 ) or sal(1-,0) .
1 1
Yne !'ollowiJ'l> addHionol for:rn!.a~ n:?:e iapori:ant for
cocaputationn WJ. th t.:&1 :'ih function!".:
(38)
wal(., ,e ) • toal(0,9 ),
cal(u,9)
cal(i,9),
Eal(u,9) • eal(i,9),
1-1 < u ~ i
1 . 14 WAJ,SH .FUNC'.nOJIS
cal(u,94r8') • cnl( i;,c )cal( i.: ,a• )
sal(µ ,e,; e• } • sal(i.. ,b ) n,.l(u , a ' )
(.39)
Since 9 ll!ld 9 ' may be positive or negative one has to
ext end the defit.'-tioc of addition !llodnlo 2 to nega;:;ive
numbers. - a o.nd - b :
(-a) e (-b) • n • b
b
o ~ ( - b) -
(- a) e
(40)
-(a
0 b)
is oqunl co one half the averare numbe:- o f si15n <>hangee of col( u , 9) OL' s"1(µ , a ) in a ~ im e intuvttl of tlu.t'ation
1 . •rnia ma,y eas ; ty be veryfied for t:he J)eriodic ;"unctions
col (i ,9 ) and na l (i ,a ) oy counting thP. $j(;n cl1ange<1 in
Fig . 2 . cal(u ,9 ) oi.d oal(u,e) aro cot por~odic , U' u is not
dyadic r&:tionnl , but the i.c:terpretnt.ion of u as one ta!.:t
tile average DWllber of sign cban(l;eS pe:· ·.ime inter1'al of
duratioc 1 still holds true .
l i an arbitrarily szall section of a sir<e func;ion is
µ
kno""'!'.1 1 the t"unctioz: i:.:; L~Oh"l: e·1eryA~.~ra . '!'hi !! rca-art:! iz.
frequently cxpresced by ~a.ying t:hnt !>l.nuno1dRl functio:::s
transmiL iu.fo.:.·cotion at the ra-cc z1... ro . 11/n. Loh .:.·unctiont: n.i·e
quite di!!'o.c·cu.t; in this =cspec"i: . A.nr-iumc thot n mcn.:;urement
has yielded tlle value +1 f or a Wa l"~- function in t h e int e:-val -; :! 9 < 1 ~. It follo ws .from FigH . 1 2 and 1,:1 tt:at t his
must be a function cal (u ,a ) «ith µ iH 1,l1e iuLN·v,,;. O :; u < 1.
Lot an additiona l moa:;ur eme:nt in Ll.Je !ll' oi·vul ~ :; g < 1
Yield - 1; the valu e of u is thus restric• •.>d to the n:oel ler interval i 10 u < 1 according to Fig . 12. A ft.<rther :n<>a31.U'eMent yield8 1 e . 6 • -1 fo:r-C1lf:\ ir.t:~rVEl.1 1 ·;a< 1 . ~ .flnd
•1 ~or the interval 1.s ~a < 2 ; t~in rcetri c ~s u to tee
Still 11maller tnte-val 0. 5 ~ 1- < O. 75 . ;. daub -ng of t:::e
t:ime interval 69 required t:or measure:t.ent. ti.•" <;etit:iivel~;
halfs t >e interval 6u 1<ithii> whicn the M.iqu~uc.;· u !emailis
undeterinined . 1'he product 6. 96µ L'el!lnin:i con:itan t and may
be iutet•prE>ted o.s the uncertai.nty relation !01· Walsh !'u:-,ctiona . '!'bo tranamission rate o.r infox•mCltion ia not ~ero ,
0ince more in.Iox·mntion about th<: exact value o.C u i$ obtained with increasine; observation intorval AG .
26
1 . 11ATIIE!1A1'ICAL l'OUl'fDNl'lONS
A few wor·ds may be addeo for i:;he mathe:nai:;j,cally inc lined
reader about: the connection between the sysi;err.s [ wal(O,a),
Cal(i , 6), sal(i ,9 )] and (1,'(2sin2n i6,V'2cos 2niD} . 3otil
are orthono1'1llul systens i n Hilbert s;.i«ce L , (0,1) and one
:.nay base on both of them ver;; s imilar t!..eo::ies of the Fouriel' series and the Fourier transfor·Ul . The reason fo;r i;his
is that both may be derived fr'om eha.rac·ter groups. 'l'hc
system of circular .functions {cos Jr..x , sin kx) is derived
fro:n the group [ e'"Y ) , which is the character p·oup or
i;he topologic group of real numbers . The eyr;tea of Walsh
fWlctions may be derived frorr. tile character &I'oup of the
dyadic group ; tho dyadic group is the toµolog ic g<·<>up derived from the set oi' binary reprei;entations of the real
number•;; . The most striking difference between the f=ct ions - continuity of circular functions and discontinuity
of 'Walsh fUnct io11~ - is caused by the different topology
of the real numbers and the dyadic group (8,11, 12, 20] .
1. 2 The Fourier Transform and its Generali zation
I. 21 Transition from Fourier Series to Fourier Trans form
The }'ourier transforrr. belougs to the basi c knowledge
of every communication engineer . ! t s derivation from 'Che
F'ouriel" series ls shovm here in a special wa,y c-11at will
facilitate Wldersta.nding o.f the more general transi i;ion
f'roru o~tbogonal aer-ies to orthogonal trann.i'Ol"ms, .
Consider che orthonormal system (f( j ,9) ] of sine and
cosj,ne elements , the firHt few a.: which a.re sho•1m in Fig.1.
1!he elemem;s f ( j ' a) are div ided into even elements fc ( i) a)'
odd elements f•(i,0 ) and the conscant f(D , S):
1
The transition from the Fourier series to the Pourier
transform has mainly tutorial value . A mathematical correct transition without an additio nal assu.urption is not
possible, since the Fourier series uses a system of denwnerable fw1ccions bu·t blle Fourier tt·t1.0sform one of nondenwnerable .functions . ft corresponding remo..rk applies to
the transition from O!"thogonal series to t l1e general ized
Fourier transforms in section 1 . 22 .
1. 21
FOIJ!ln:R TRAl\SFORN
wal(O,~ )
.C(0 , 9)
rc<i ,a)
:
f(j,9) •
r.(i ,a )
9
=
27
'(2
CO<:
2ni6
'{2 sin
2m.e
undefined
tJT; 1 • 1, 2 , ...
-i
1
s
<
,. e <
-i'. a
>
·~
{41)
+t
sine nod coo.i.ne elemen ts may be con r.tnucd periodically
outside tho l ntorval -i lS & < +t to o'bt.1i n cho periodic
sine and cosine functions :
f(O , b) = 1
-oo < e < +co
.r( j , 9) •
.Cc(i,6)
\ f.(i,9)
=
'{2cos2ni9
'{2si n2nia
Periodic continuation of a function i~ o finite interval
is a special Wa:J to extend the intorvnl of definit.:.on .
Consider a !unction ?(a) de:'ined ; n the i.Ltervul - t ~ 6 < i.
An example i:i the trinngclar !'unctiou shown on top oi·
Pig.1 4e . U conditions requ.iredfor couveq,;ence are sat i s fied, ono mny expand F(S) into a set'ie" OJ' Lhe ort !:o normal system {f(j ,e )] being defined in chc same interval as
F(6 ) . The h•iOJJgular func~ion of Fig . 1 1~0 in Cl<J>unded inGo
a ae1•ies of sine rultl cosine element s . J J' the t <·i3.'1gule.r
!unction is con~inued outside i;;c interval of ,\efini ticn ,
OI!e muGt continue the sine and coaino elot11eut;H in the same
way; two 0£ tho possible ways are pa.rticulw:l;v importaz,;; :
Periodic continuation of ;;he ;:;rianr;ulnr fu.nceiou requires
P-eriodic cont1nuation of t~c sine n.nd cosine eleu.enLs .
Hence, the peoriodic triangul:u· function or Fig . 14a is exJlanded in a series o!: che periodic sL"le and cosine functions . Ir , on tile other hand , che i;ri911gular fll11ction is
coni:inued by F(e) " O outside the interval - i :< e < t, i t
has co be expanded in a serie~ of' cine And cozir1e pLLlses ,
which .nra zou·o outside that interval .
Let t~( 6) be expa.ilded in a aeries or a.Lue and cosine
eleman·~ s :
28
1.
~·LATI<;;.,r,
E'OlJHDAT:O!IS
..
...,,,
F(a) = a(O)r(O,a) • '!2 2::rac<i)cos2ni8' n,(i)sin2 nie]
111
f1"Ca ;f(0 , 9 )dP
alO) ~
• If]
Jl!'(e )de
•112
In
[2 JF(B) con 2nie de
1
ac<i)
( 43)
. ii 1
Os
('1. J\ =
··-'I" "'
j'F(3 ) Gfr. ,m.s
ue
- ti:
The coefficients a(o; Cl.Cd a c tiJ are plotted for tr.e
"r~
nngular fui:cciou of Hg . 1"" it! Eg . 1)a . Al! coefficient~
n,(i) a.r~ ze.ro , .;inco tne L1·iangu.L.n1.. f1Jnction .is ac even
l'uncti on .
J,ec the vurioblo G on tlie i·.ight huncl oide of (43) be
rep laced. b:y 1he now vs.r'io.tle a • :
81
F
9/S.,
s
> 'i .
This substitution 11 stretcr_es 11 the cle11.er..ts '{2 sin 2'~9 ,
'{2 con 2ni9 =d f(O , a ) b, a !"actcr ; . Th'l r.ew interval of
orthogonality is -H lli 9 < H . 'The orthoi;onal sy>Stem of
the stretched elemcntn '{2 sjn2m a ' , '{2 co~ 2ni e' &:id f (O , e ' )
is uot normalized , eince thcc;c ftl.Jlc Lions have the same
ampliLude a.; th<> orie::Lnal e ler.icnt.o but '11'" ~-tin.~;i as l·lide .
Phe ir..tegr a.l over thu uqun,re of che stJ:eL<~hed l'unctions
yield.s ~ ratl::.crtha::i 1 . Hence , c!le sL1·etclled functions nave
to be "ltllciplied by ~ •n to retain !lor!ll•lh... tion .
F(6 is not stretched, hit:'..:; ::ontinued ini;o :;he intervnl -;; ~ e < -i an<l i "5 < i~ l:y F(a) • O. Tiiis conti1.untion m· F(9\ nnd th~ cn::1'etciling of £(0 , 6),•f2cos 2ni6
n.nd \f2 :.:in 2 ni0 l a eho',ffl for ; = 2 o.nd ~ a 1~ in Fig!3 . 111b
1
r.wd c .
The expan.,ion ol' i;>(a) ir. a ser'ie~ or the sLret ched elementn has the following form :
P(e) -
~(ac~.
'"(o,e•)
00
T
1[2
2;Ca,Cs,i) coa 2niS'
~
"1
+ a 5 (~ 1 i ) sin2ni9'J)
(45)
1 • 21 l''O JR l .'::h 'l'RJtJIEFO RI·:
"tubo6'
17 ...
'"'8'
1'1u18n9'
b
-:_j======i:~;~~====;·:_:• - FIO!
wal (0.0)
wol (0.0') :.
V1 Mn 2n!T .,_.-·-=~====--~...;=--::o--""-="::i-:---:=="""-...... ii/"• 8i7
YI"'' 2n6' =:-r===~. . ==!=====:f----=~=:::::=ci__ V2"u G/l
''""'"o· ::.,,-"""':::::=::......._~~.;;;---"':::::=::::,-=:::-~ ,'2.... b
l?c .. i.O" --
--
i':laln~ne' -. • ..........,...,...---...,,~ ~ ,,..---..,, ,_
l')c,.6n9' ._c;:>
i?=•a
l'z,.,Jna'Z
c=:>.......__,,. .c=>.G~ l'la.3n9i1
- i"l~nz,a
l'icln6n9'
Fig.14 Expansion 01' a funct:ic:: F(~) ~n a series of Sillt conino ele"1ents bavinr various intervals of 01 th::>f,OUltl1 ~:: .
a~ -i - a< i,
o,e) '{2cos2ni6, 1{2sin~·,,1QJ
b -1 ~ e < 1,
1<nl O, t 9~ , \'2cos2n(ti}e,1{2sin2dfi)5}
c -2 a 9 < 2 ,
>ml 0, t e ~ , 1{2 cos 2"( * i )e ,•[2 sin 21" (ti )SJ
l"al
1 . r·:A1'HE11Al:LCAL FOU:fDA'flO:m
i; · ' contBir.ed ~ ll '
'l'h" factor
Ila,:' ·:.e ~o:nbine.l .:ith t~.e
:.n the llrg'Jl>Cnt 2ni8 ' . rhi,- lG trivitil for zine
and cosine func~i o;.:i \>ut it may be urcd a,1 n roi"i:; of deput·Lu.:·e ro!" the 1.teut::1·~1l lz.ation of the fu~i·j ~!' ,..ra!'lsfor•o :
fflCtOl' i
co"2ni0' ~ cos2n i(9/~)
c o s2n(i/i;)a
sin 2n i 0 ' = sin 2n1(9/0 : nin 2n(i/i; )fJ
.!'(o ,e · ) = r{o , e/~)
• '(o/i;,e)
The nozation
f(O/~
,e) is !!-;rictly
on<'I in o: no con-
!01~:\l
:-Pquer.:.c~ .
Ti:e seriez e>-::>a."1~ior. of F( 9) assun. .. s tl:e ~·ol lowin.; .:'om::
1/2
IT
~ll
..
SF(9) cin ?-.7a
., •(It
n(i; , o) :
~
dS
:i
in
f F(6)Ll9
-!,11
lrtLL'O<.luctio:n o"" 'lCW COH~tnnt::; ,
1
"cl
--,·
...
1
='I!!
'
..
a (: ( •";) '
1'
J
D~
<">
s
(i
( • 1 .!.') ,n ':"
y , a ·'{;&s.;
,
ld~
i
""'
°"'
.
.
.
.
I/' 11(
-:-
':t
-'
.-
t•'(3) • }[3(!;. 'f1~ 1 9);•,i;.
"
i:;
:i
l:;[sc<i •coo 2nja 1 a,(~·)~LJ: 2:i~9])
(49)
~c( ~ .i r1ro plo l.t;erl for ~ ~ 2 and ~ • 11 ifl Fii;; . 15b
'
s
nnd c ; che;)r hold f"o'r l. I e expansion of 1'(8 J iu a .se.ries of
0 ( T) and
1
t.h~ ~int;.
anrJ .::ot!inc clcml'Jn"'ts of Fig .. 111-- 0 9n<1 c ..
4
Lee ; inc.rease to
- oo
l/~
=
i/S
ti~l
rn:nnin constant:
(50)
li:r.
•.t
1nt"i.ni~y ;
v :
£T
1.21 FOLIRllR J:h.i.:lS5'0R.'1
OS
I ~:1
I
01
~ 01
- 02
'j 0,
·~
•
O>
~
~CJ
0
0
1
i-
t/4-
D.&
OS
l
01
I
502
-s-
OJ
DI
O•
Q)
J O.l
"j 0.t
O.•
a
tn-
l
d
•
o·o
L~...1.....L-£
, ,.........,:;::;;~.-
v-
Fig . 15 Cooi·fi cients of che expansion of c!:e Lrlsngular
f unctior. i•'(a) in a. serien or siue ana cosine t-lemen~G ncoor<.l.Lng to l"ii;. 1 4 . oc(v' <!"•notes the l imit CUl'V& Ior tho
elc1nonta ott·etcl1ea by n l'nc~oJ: ~ - oo.
i
may be any integeT'
nUl'lbc~· .
i as wel; a"
i/~
u1·~
dem1-
merable. v t on i;J:e ot_jcr ha.n<l , m"J.st ·oc allo·.1eO t.o b~ any
non-negati vc real nU!!lber 1-1.n1:S ttus be non-denumcrg'!;)l ro, or
some ot the follo,.ing integTnls ·doulJ be zero . Hence, the
Fourier seriee: contain!" c!cnua:.crably :nany orthogonal functions, but; the Fourier trone.fo1·m contains non-J.enumero.bly
mony .
'Phe limits a,(v)
and (47) :
ac(v) •
n:,( v ) foll ow !'eAdily t1·out (4&)
( 11
i
lia: (2 .' F(:l) cos 2rts9 d9 , 1{2
~-oo
-l/2
~ 11
a,(v) •
and
li;n '(2
~-~
f
·(ll
()(I
J F(&)cos 2'1v9
cd
-oo
.
..
(~1)
F( 9) sin 2n~9 d9 ~ V2 J• F( 9 ) $ill 2nv9 d9
0
_..,
In order t o find an integral repL·eaencati.on £01· F( 9 ) ,
Co11 side1• 11 certain value e • 9 0 • Equation (11Ci) ,yieltlE
Jl'(g o) ae a SUJ:l of denwnerallly many t erms , w11i ch r.my be
Plotted along the numbern aJ<is at t;he points i/~ a3 :;t;o;m
in Pig .16. The disi;ance between the plotted ter210 is equal
to 1/~ . Bence, tte sw:i of the terms multiplied by 1/~ as
given by ('•9) is equal to the area under the st.. p fu;ictior.
32
1.
l•~ig . 16
_f~;
Tr·ar_sitloc frorr. Fouri e1~ series
Lo !+'ourier. t~an$':;:"'orrn.
: ~Vil
i \
L
Jo<J/f)
><41!)
)((2/J) \
r· -~ \
·1 \
L _J
• - ••
X(O) •
A(gJr(g,e.,
J
.;.
s '
X(i/s)
iva c( ~ .i,1 r.:
+
0
!~i<THEf·'.P.TJC.U, FOll'WA'l':O~IS
lif
21! 3/j
COS
~ i 0
r.n;u
a,(~)>[2 sin 2r ~e
4/j
of Fig . 1&. Using (49) , one Illa;)• aprro>=imnto ;;ilis 1;1.rea lU'bitrarily close :'or sutficient:l;y lu1·e;e vFJlu<:r; of i hy the
followine; integral :
~
;•( s ) • \1'2
j[a, ('1>) coo ?l!vO + a 5 ('1>) sin 2nv8]dv
•
(52)
':'he l ower l:relL c.f the intcgPal is z.eJ.~o , because the lo «er limii; o.r ·che sun: in (L9) .ap:;iroaches zer·o . '!'he ii!·st
term of ~he swi. (1iQ) m:i~t be nF.el.ec-~ca , :;ince it contr·i·~utes arbit<'arily litt le .for large valuos or ; . [he va~·i
able v in (52) must .assu..me the values o:' al I rea:.. poGitive
ntunbe.r·::; and nor, or.ly o.f'
de-11ume1~ably
mar:y o.f thorn, ox· i;tbe
integral could :iot i;1e i.-nt~rµrec;:ed i.!S
t;quat i on ( 51) shows tr.at tl.c ( v) i>i
Riernan..'1 ini;cgral .
a 5 ( v ) is
an oda rune tion of v. Renee , .( '.>2) rne.y be i'ew.l"it ~en Into
c"he following lo.rm :
;;1.
= evon a.'>d
~
1"(0) • i"[A(") coi; ~'.Tv9 ; f:l('V) oin 2nv9)cv
_;;.,
(53)
as(v} io identically zero -=-or "ti'le t"ri.a.'"lgular function oi'
ro·i~ . 14; ac<v) is plotted in Fig . 1 5d accordini; to the following formula :
)/ !I
ac( v ) • 2'{2
J'
0
(1 -
8
~)cos2nv9
d8
,1r2 ( sin 3n v/8 ) '
B'
.3n\J/8
1.22 GE:ef'ulJ.lZJ:.IJ FCL1UER 3AIISFOH/''.
35
1.22 Generafized Fourier Transform'
;onsi der a syntc:o or functions [ r(o ,a J ,r, ( i,0 >, r~( 1,e )l
-~li:l l! 9 < ~EL
l'h,. nubscript c il)dicaten on even function anrl the •ubacript ::>
e.n odd function . 9 r.iay be ~ini te or i n!'ini te . He nett , t!"•c
results will be applicable to fw:cUous ilaving ~n infir.ite
interval or or·Ll ..ogonal ivy' such !lt~ Lt..e r·:..."lc,.; i Or:.6 Qf t,he
parabolic cylinder . t.et all funct ior11! f cC i , e) be non-ner,ati ve ror e = O, n:id let al- fu!'.ct::iou!J r 5 ( i , e ) cro"s fro:n
n ogative to po s il ive val11es '1t e • o. 'J•he funct ions do not
have to be cor.tinuoue or nir:arcnti ~bl~ . A func,i on F( S )
d"fined i!l 7 l:e inte:'.'Val -;e> l! 9 < i3 i'1 C>rpr1nd1'd in 8
serie s :
o.rthOOOL'l!ISli 7-Cd ir. ~be i nterva1
-,.,
F(B)•a(O).r(o ,a ) 1L:(at(i ) fr.(i,9 ) 1 u,(i)::, (i ,a1]
~12
S/2
J F( 3 ) ~c(i,9 )tl9
n,(i )
-9/2
3
n(O)
·= [
;;. 5
(i) • J, F(3 )f 5 (i
(~4 )
,a )c!9
-'31:!
<
Y< S f(O,&)d~
-<9/2
0 fr replaced' by S ' in thr rur.c~ .Lorcs r (O , e), tc<i , 0 )
and r 5 (i ,0 ) :
B'
~
9 /y ,
y • y( ~) > 1 ,
J.i.rn y( 0
i-~
• cc
The expm:isior_ of F(9) ir. a :::e=·iei: of the
tions is oboair:.etl in analogy cc ('I' ) :
v? a( s ,o) r(o, e • ) + ~ [ ac( ~ ,
~
F( B) •
+
( ;!>)
s~rctc!-.l!d
i) r 0 (i , a •)
,
f:.tnc-
( S6)
"s<s,i)r5 u,a ' l :l
;For other generalizations see (1,2] .
Thtl method used BFPlien to " lnrge closs of ~y::i~c:ns of
f unct;ions . E.xact muthernacica- proo.re car. be obtained with01.1t; excessive mnthomnt::ical requirnmc-nts ~or .Lmli vidun.L
syot;ems or runct'Lona only . 1'' or instance , the l'e~uH n of
tihis nection soom to upply .for dynclic rational v cd ues of
I~ - 1.1 only in c l10 cnse or Walsh functions; in !-CWlHy
they apply to all rnal values of i/~ .
3
~""'·
f'**"'$fotOll of tn~
Oft
'I:he sti·eLcLed funcLions arc o.r-thonormnJ in the interval
-b'EI ~ e < ~ye . F(9 ) is contbued by F(a) " 0 into the
intervals -tye ~ e < -~<a a.'ld t e " a < ~ye .
I:he f()ctor 1/y is combined w;i t b i so that a i nstea,d or
e ' mnybe written on the right hand s:i.de l)f {7b) . 2 ni ( 0/s )
hr;d been 1'Cp ta ced trivi ally by 2 n ( i/S)0 i n ('~6) ; since i
and 9 are not nece~!ia.r:ily connected as proauct in fc (i ,a )
a11d f s { i 1 e ) tl 1e follo\'1ing f$UbstitutioJ1s munt Oe con rii Oered
plll'el:.r f o:rmal until proved otho rwi~e . Jn particule...r i i / S
sl1ould be c.:ouside1·ed a symbol l"ath.er than a fraction :
c< i / g, a )
rc< i ,e •)
r ,Ci ,e · )
fc (i,9/y)
f
r , (i ,e/y)
r 5 (i/~ , e )
f(0 , 9')
J (i , 0 /y)
f{O/; ,& )
(57)
Tr_c seri es e xpansion of F( 9) assumes the i'ollowine:i; :'orm:
( 58)
l
rry
'
yS/2.
f
"
- ·y~/2
·1
F(9 )f 5 ( ,, 6) tl9
'-
~.-S/2
;f-- f
F\6Jci9
• y - yS/2
:<Iew coefficie11t.s c..!.'e int.r·o<luced :
ac<~> = ~·yac U,,i ), n 5
lJ1
(!)
= '[yn 5 (; , iJ,
;i(~) = YY"( s ,0)(59)
cede r to uake (58) artd (59) "'""'e tlu.w " .fu1·me.1 no t edon ,
one munt demrui.d t h at t he coefficients ac( ~ ) or a 5 ( ~ ) have
either the same \r-':il tte foi' all valu.es or i and ; , ag long
as i/i; = µ is corlstant , or tllal; t hey coni.rerge 1 towal"'d a
l.irr.it ror I a:re;e valu.-s oi i and ~:
The lef t hand limit
hand linit d iffer .
1
shall bo token, if laf·t and riglrt
35
1 . 22 GENEP.A.:..iZED FOURIER TRA1'1SFOR.'1
,\ge.i.L , or:n !ms to pos:ulate tl'.:a~ .,: i" " non-noga~ive real
numbe:- e.nd t.hun ir ::on--dcn·.ui.cratle , \<·cilc i or i/ ~ •~ de -
numernbl<' .
The limiLs (uOJ exist , L'
lilllit funcUo:i:• fcl µ , ·l J and
.rol lo•,. ,• n 1
lin
1,1. -
>11t<l
's(f , 9) approact
tl'IE:t t .:-11"1"' dc rinr.d
az
:
yS/2
.
-:ra12
,
r F(S)f;(i,e)Je
y • y( ~)
Tlie func:ioi::.s fc(~ , a) o.::.d. !' 5 ( ~,S J com••,:.-ge in ;ne intc::-,.al - be ;11 e < t ys to -1.e 1 i!!:it r·mctlon:: f c< µ , e) :>-=:a
-f 5 (µ, e ) . 'rht.o typn nf co:r~·,...rgt:"ncc: !.!l c 1ll~d ' weak conve.r·gonce' (;) .
It follo ws f:·oa: (:i 1 ) t o (61) :
'j~/2
.
t-"' - yr'::! I 2
s
l iw
S J(0)fcl ~ . a )d9
Y'3/2
.
J F(a)r~r~ . a)ae
1-00 -yf'j/2
s
11%
Let J-'(9) be n funct~o:i tl:s'.: v=i"hcs out.side a fii.ite
l.nte1'Val. Equutione ·2) reduce to he followin"' siit;•li !ied form :
..I
-oo
F(e )!"c(µ , s )de,
..J
F(9)fs\u , 9)d8 (65)
-~
In order to find a.n integral ropranontAtion !'or l'( e) ,
~~he inLogl'Lllu sboll represenL Cmteh.Y ' s pJ·i 11ulpal value .
1'(:).mus~
3·
hold for all quadratically i Hte!';J"»blo rwicLions
1 . r·:Nf'iii:J'!A'P J(;AJ, f'O\JliDP.'.: IOt;s
36
consider a
F(6 0 ) n.:o. a
cel'tnir: value
a=
.Equfttion (58) yields
eLun o.r rlenwoe1-ebl.y many terms ;;hic!1 :oay be
00
•
plottec tlong che nw:1be1·s axis ai; the poini;s i/y " i./.y( i;)
instead o.:- i/S ae ir1 Fig . 16 . The tlistaJl<;e 0etween tJle
pJottcd tr:::ms is 1/y. Hence,. 1..he eum of. l..}te ter·ms mul'tiplie<l by 1/y es giver. by (.)8) is equal to L!.te a!'ea under
a step function . This arc:;:i may be rep.rese.nt..ecl by an lnteg r al , if ~ und L!tus ;;( ;; ) t;row ·oeyana ell botmds :
00
j'
F(e ) -
•
[ac( u )fc(µ , e); "s(u)fs(µ,a) ) da
nnd a,( ;. ) are cal:.ed bhe ~ener«liZ>Jd ••ou:•ier
~i·ar1sform o.r F(9) .for che fwictions fc(µ,e; a.nd rs(µ , 6) .
Equallon {64) is 0.c integral rep,..esentation :.f F(B) or its
gene:ra.Li7..ed tnv~rse Fou:.. ier tran!::fO!"l:!. 11!:-:othe:.· ";;hese integrals e.c"tual ly e.>:ist ca.."Ulot- be stated w.:_t.lLottt.. specifying
the f u.:ict i o11s fc{u )e) ancl f 5 (u , 9) JJt<:re :::lcsE:l y . Tne va:•i1.;tble µ plt-i,ys the same role a~ the vari able v in the
u:-;ue1 Fourier t-ran~form . P..o:lco ~ u is culled u fJt:Le1··alized
- and :!orn:a.Lizc-d - frr:qu~ncy .
ft<i , 8) r,na f 5 (i,S) a r e defi ned f Ol' posit;ive integers
i only . Hen ce , :',(u,a ) and 1' 5 (u , ~ ) are 6.e;-inQll for nonOt?gatiVE;' J..·eal nwol.leJ.·s v onl:i - Or~e may extend thQ definii;iOn$ to negative real nurr.ber~ :
o.c(u)
(65)
!' r. (µ , 6)
ie a.n even £wict io!1 of 9
1:1.s ...,.ell
as well at1
Equations (•~2) Md (ro;.) !'!how that a c ( ;.. ~
a 5 (µ) io rill odd f·.wcLion of '- · llence , (64.;
into Lhe form o;.. ( 5 3) :
: 9 {µ,0 )
F(a)
ic an odd ftul.ctiou of
..
_S"' [A(u lf c(µ,e)
e
+ B(µ)~s (u , a ) ]d;..
-9~ of u , and
of µ .
is aJ'J. even and
may ::.e brought
(66)
1.23 INVARIAHCE Of ORTHOGONAl!'I'Y
t.23 Invariance of Orthogonality to Jhe Generalized Fourier Transform
V<>nAider the function G(u ) :
s ince A(µ) ts even anc ~(u) is odd , one obtair.s for G\ - u; :
G( - µ) • '{2[Ai-u) > B(-u) ]
= \f?[!,(u ) -
B( u )]
A( u ) and B(µ) m~.Y b'.! r egai.ne<.i f rom G( u ) :
A( µ )
a
~1{2[<i(u) + G( - u ) ] ,
B(u) = * \[2( <0( u ) - G( - u) ] (b8)
U9il1Jl: G( u) ono may r~wri te ( r;3) •n~ ( f>~ ) i nt"o ;hc> r or:n
O! (20) Md (?1 ):
F(B)
• t'f2
..f
-oo
•(l c u, a J
Gu
- i·, (u ,a ))du
(6~·)
~
(70)
00
G(µ)
-
• ;y2 J F(o )~fc(., , 9
Use is mudu iu
(70J of
i ·,,J , 0))d6
lle f3cl , lhat ~hu inl"l'r'-'ls of
A(µ )f5 (µ,6) nn1 B(u)f 0 (µ , 6)
vani~h .
Coneid.ot' a ~yntcm ( r ( j , e) l of orthonormal fu:ictione
tha t vanish outi;ldc n rinite i nt<::-vnl :
..s
f(j , 9)f(k , 9)d6
( 71 )
-00
Lot g(j ,u) danoto the generalizeJ
....f
i<'ourier L1•ann.form of
.f(j , 6) . It "ollows froc (70) :
g(~,u) • if2
oO
f(j,9)[~,c~ , e)
..
_l .f(j,9)(t1{2 ;g(k. , u)[fc< u . a )
r g(k, u )! t f2_..,Jf(j,9)[fc( u , a)
oO
6
:,(u,a l]d3
(72)
~ <5 ( µ ,a ]c!uJd& • 6,.
..
+ fs( u, 9) ] d9)du.
b,,
~
J e;(k ,µ )g(,j,µ)du • 6;,
-~
(/3)
1
58
MATf!EM .4~' I CAL
FOJJH),'.TICl'/S
A-n o~thogonal !ij•stc-;a {f(j ,e )1 tha~ vanishes ouLs~dt a fi 11ite i nt ervf.!1 i r.: t-ransformeC b;i the tsone:·~lized Fouriel;'
t .ransfo.rm into a n ortnogonel sy!lcca: ( g(j ,u ) l.
1.24 Examples of the Generalized Fourier Transform
t h e ~ene.l'al i zeC. Jo'ouri er 1~\raru; forrr. of -che vri=gula.r fu.nc ti on of Fig . 1? ror Legend l'C !'O L;ynominls [ 1 ) :
Conside1~
J- 0 (xj = 1 , T,(x) = x , P 2 {x) = -(3x' - 1) , ecc .
The interval of
OL·t hop;o11al ity 1s - 1 t< x < +1 . x = 29 is
sub!::it. i LuCer! an d
tile 1'ol.Lo-.·1i ng c r ansl'or:oai::i cn.s are rnode :
i
=
( - 1}' (4i • 1
j"p ' ' (2 9 )
( - 1); ("-i - 1
)"P,,_, (25 )
1, 2 i
~'he syst en (f (O,a) , F c (i ,a ) , P5 ( i,G ) ) is or·t hono.rmal
i.n the interv:il -~ " e ;; +i . ;.:11 l'uuctions Pc(i , E) are
positive ror 9 = O, ar:d all func·t ion s P 5 (i , 0) have a po-
s itive d ifrereutial
<;1.<0 ~ient .
explicitely,
'.-i~·: tten
the
l'irst l' e" polyrwlllials rea<l as .follows :
P(0 , 9)=1 , ? 5 (1 ,0 ) = 2'{~8, P,(1 , 6)= - { '.[5( 1 2e' - 1)
Fs(2 , 0)
=
(75)
- '{'/(20&l _ 3a, , Pt (2 , 9 ) =Wt\(5609'- 1 209 1 + 3)
The co.,l"f i cients acCl.J and a ;Ci ) f or Fie; .17a may be
1·ea.tlil;; computed :
ncZi)
'"
J P(;J )Pc(i , 6)d9
• Ill
~s(i)
= "'
J P(S}P5 (i , 9 )da
- !I'
a c~
'"
J (1
•
-
~9)Fc(i,8)d6
0 , a(O) = 2
,,.J
•
(1
(76)
- 83e )d0
ac(i) a.nd OL(O) are plotted ir> ?ig . 18u .
r.e~ a 1 n (75)be replaced by 9 ' = 9/y , where ya y(~)=
a g - :> . I'r.<i ,0 ) and P (i,a; are stretcl>ed<:Prerdo uble ~b.e
5
inteI·•:al as sbo•,m i.n Pig . 1 7b . The functions (?5) nre rsplE;ced by the e creched functions Pc(i/2 , 0) and F 5 (i/2 ,9 ) :
1. 211 £Xl.:1P!,::s Ol"
1'RAt<Sl"OR~'.S
~
--<:::-t.--- --L-=- ---<lt.--~
-----.... -+--_..-... _
. - - - - - - -... - - · f(9)
P (20)-::.:.-:-- ------ ~
0
P1 {28)
-P1 (Z9) ----..:;.,...--
-
--:_;:----- wal(0.8)
--:.---- P, (1,9)
'J/----.;::;-= P,
"<::V-
(1.0)
,., lzel -:----~-- -- -·~ - -~ P, (2.91
r~
~~
-t
C
'
..
I'\.
9 -.,,........ ____ _ _ _ _ . - - - - . _ _____,,.,,... .... F(8)
:c------ wal(O,Bl
Po (29') _____
r1 (£8') .....-,,.---L j----..
~------ P5(11Wl
~-- · - - - P0 !VWl
-P2 (29') - - - - - ... -J.?> -
0
-=--r---...;'·
I
"' ·------ ,.("2,9)
_fol
p, (28">------~'-PcWl.B>
----- '"'...;t>.. -.......__-J
P (28°)
-3
Ps (2e') - - - -P&(ZS")o
w
P,M.9>
-
-- --~------ - Pcll'l.9)
-P7 (28"/----.---+,-.
b
...J........";, ·-
·"p
.
1
-1
G
.,
-1h
c
- l=""'',.....-- •->: Ps14'Z.Bl
.
I
t
9
-
~
>ii ·a· · - ;-
·-=::===E-----....
---==~~1~:::::::::===1··r· 11oi
wal (O.el
Po (ZB'J ~L
P1 (ta') :.:1f====~~---=9f===---=[: P5(V4.Bl
-P2 (20')
·P1 (28")
• P0(1/4,8)
f>--
-._J...: Ps!Zl4.0l
__.-1' PefZi4,8)
P, (28') 1'._...
Ps (28') •
·Ps (29") .
"' Ps(J/4.0l
r- P0 (l>,9)
-P7 (29') ""'.J'~----=--==-~~--..------.J~ P.!4A.Ol
~
(29") A=-='"'-~=l---~~=_..,=--..-;../A:·- Pc(4ABl
- . P5{~!.9)
·P1o(ZO'J '•/... =
==~
P,(5/;,91
-:
-1
o
,
c
•112
-1/4
0
ii:9'· .:..: .- ,;1
li'ig . 17 ~pt11\eion of a. !unction F(0) iu n uo!'l1-u of L"gcndl>n P1lyno111J.cils J.iuviL1g various inLervaJ. s o.J' o1'thogonulity .
Ps (29") /
=
a
b
c
-i
'<"
e -- :
""9 < !, { wul (O ,e ) , Pc(i , 9), Pq(l~8) 1
-1 ~ 9 < 1 , (wal(O ,e ) , Pc(i/2 ,0 ) , I's i/? , O)J
-2 s e < 2 , (wal(O ,e ) , Po(i/11,a) , I's i/~ , a)J
1 . NATliEf·IATICAL FOUNDAr lO:JS
"-0
(77)
P 5 ( 1/2 ,0) = Ps(1 , 0/2)
<\f 3(i d)
1\ ( 1 /2 ' O') = Pe(1 , 9/2)
-+,15r1 2ael - 13
P 5 (2/2 ,0 )
P 5 (2 , 9/2)
- >{7[20( te }' - 3( ta)J
Fc(2/2 , 0)
Pc (2 , 9/2)
~'he
::ito \ ! e)'
= i\1'9(c·o'·
~Ile
''c ( i /?) have
coeffic i ents
~Je
·I
acC i /2)=j'F(8)Pc (i/2 ,9 )d9
_,
=2 f
- 1 20''
\ '! e ') '
~
5]
follo wing value :
8
(1 - 3e)?c<112, a}d8
(78)
0
Values 01' ac(i/2) are plotted in Fi~ . 1 8b . Tl'.!ey do not i.lave
oxactly the same value:o as the coefficients a.c (i) of Fig . 1 8a
since , e . g . , Pc (2/2, e ) .i s not eq_ual Pc(1,9) .
func~ion:o
Let che
;:;~1e
i nte rval 'try -che
= y( ~ )
5
=
=
4
as
(75) be st.I.'etched ever four times
$-ubst.:i t.u t.io11 9 '
=
0 /y 1
shown i n Fi g . 17c :
(79)
?,(1/4 , 9)
P 5 (1 , 0/~ )
Pc (1/4 ,8 )
Pc (1 ,0 /i<)
-lt\f5[12Cte l' - 1J
i's ( 2/4 , ij)
l\(2 , 0 /4) =
-\'"7( 20cte )' - 3( ta ) l
P, {2 ,0 /'·)
l;•f<J [ 5C.C (t9
Pc<21~
,a )
where y =
= 2'f 3( ta)
J'· -
'1 20(~9 ) ' +
31
Some coefficient~ a,(i/4) are plotted in Fi5. 1oc :
·2
1:11
·l
0
e 0 (i/4JgJ F(e)Pc(i/4,6)d0 = 2
J (1
8
- )'9) l' 0 (i/"- ,0 )d 9 (80)
In orde1' to comput;e the l i.mit a c ( i/~) J'or lai'ge values
of i and ;;, one needs
.f' 0 (i/s , O
Pc< i, 9/$) ::or large val ue s of i .an.:i soall valuns of e/; . 11.n as:ymi;i;o-.;ic series for
:.ei;;onci r e polynomials F 1 (x) is k nown <;hat hol(ls fol.' la1'ge
=
values of j n...'ld .fo r Small values of x :
,'; 2
~"J tj1-x'
U,;~ae;
((1 - ,;'Jsin[(j•t)cos· 1 x +
x
/1- x2
8j 1
cos
( 711) one obtains :
re j +Pcos ·'
~r ]+
x ; ~· ) )
(81)
1 . 24 i:.Y.Al1.:w.hS 0}' T!UJ:SFO?J".S
I
I"'
n'
OJ
- O.l
.:!,
~01
0
"'. Ol
~:)1
.
-:: 0 l
=
•
c
0
I c"
-
2
J
t,p -
•
b 0
2
{f:,p -
I
{I)
"
~OJ-
-t.
==;; 0.'
•
0
I
I
(/•.µ-
<
Fig.18 Coefiic -~n ts of t!:e expSlH:1<"P, or the t:riang·..1-nr
functl.Oll r(8) in ~ !led e:i ot• l~f'•lldr0 polyno:niuls !lCCOrclin~; to Fig . 1'/ . l'\c(IJi is ~ue t1nit, ~u1";"' J:or tt" polynomials strctcllad b,y a factor ~ -· o: .
(%)
..
~
ac<1.1) •
J F(0 )Pc ~u ,9 )d8
-4- l/41
'° R
J (1
0
-
S.e)
cos 11µa ue
;i
(83)
ac"u' is the t.~1. 0 1rali::ed ?o:zrier t1~a::.::form "f tho !;:-iangular funclion of Fi,; . 17 for J.egcndrc !='olyno:r.ial:c . It
io plotted in }'is . '18:i to c . One n•Y ~e,,dily see >.ow tl.~
coc.C!icients a,(i.), nc(i/2/ and a,(i/io)
converg~
to "c'u '·
n,(v) in li'i g . 1;i and a c(µ) fo :!'.tf. 1'3 at•e cqunl ~xcept
for <'Cale :·actors . One may see from the d:LJ'ferent_nl ecpaiion of Legend~c polynoir.ials er, gt tl:in is gc10e:·al ly r.o:
(1 - X 1=" - 2xz' + j(j ~ 'l)z
a
O;
J
=
0 , ~ . 2 ,. .. (81-)
Thi.a equatio!l reduces .for small value~ oI x antl 1n!'go valuo11 of j to thl1 dlfferential equ$tionol' sine and cosine
functions :
2"
j'z
o
(e5)
--.......__ __ __ F(9)
=sal (1i'.,9 l
~=
sal( l/2.8l
- sal (3/4,fJ\
: sal (l,ill
·: sal (5/4 .8)
....---i_., Sul ('.J/2,8)
·: sal (7/4 .8)
Fit:; -1' E-,rpa.osion o[ a function F(9) i.n a series or Walsh
elerne!1CS- havinF" '-'arious int.erval s of orthogonality .
ai -ti;~< i, {wal(O ,e) , cal~i,9), snl(i 9) ]
o) -1,; e < 1, ( "1al(O,e), ca~ i/2,e), sa1/~/2 ,e ) J
c) - 2 ~ 8 < 2 ,
[wal(0,9) , ca- 1/4,a) , sa1(1/4,e)
1 . 2'1- EXA!i1-:.ES ct·
RAN~CR!'IS
Hence 1 t;!'le r,eueral ir.atior1 of the Fo-:.iric-r trn.nsfortl i::: moiuly or i nteret1l for syat(lmd or ortlioi:;on(ll rum.:~ion:: ) t h nt
are not dnfis.td by :;uch Uif.ferential ~<]uatioc.s , ·11liic:.n a.re
reduced by etrieo:.clling t:o t:ne one o.: sine and cosir.t• fur..ctions . Sinct' "A'nlsL fur.ctioc.u ll'e defined by a di~'fcrc::ce
rather tilan a c!if!"erec:tial equnt i oL , tr.ey CLay be ex1 l)c:tcc'.
to yield !< tr.oro t·e« tu'i:ii.nG 1·c-uu. t than J.ep;e 11or·c poly:iom 1 al ti .
Tb«> gen er• lioo Uou of th" l'o·Jrie:: t.r·Br."J'o!":a to lh.-, I.Jn I chPo~icr tran~!'n.:i:i is t!»P r:o Fl13 . Ho·.-1evor, FI~~E ~id uot
disnnguinn b!:t'lfeer. even r:.nd odd func-,or;.:s . T!:.is !~::tinc
tion is importnnt fo1' th~ npp:icat on .. of Walsh-rOUt'ie1·
1.malysi.s to COCL!llUJ1.1.catio 1u,, . The u1nt,he:unt.ica.ll;y ri("ot·ous
tbeor.·.y for ·,.,'nlsl... runctior.a separau~d into even ru:.d. odd
function~ that is cal a.'"ld sal Lmctlons - is due :o
PICHLEP. [ 2} •
Let tne functions f(0 , 9) , rc(i .~) tind f s(i , A) r"f-""-
ncnt Wal sh
func~ .i. ons :
!(0 , 9) • i<&l(O , a ) fc(j,9 ·-·~<1Hi,BJ, r,(i,e) =Se.1.(1,0) (l\C)
J~,a
u2
a(O) =
f
l'(9Jwnl(O,o)d6 • ::·
J
o
-m
""
ac(i) = 2 J'
1
"
1 - ~e)cal(i,e
)r!~,
0
0
Pig . 20.s. show!'I ooae vultter. ot: n(O; an<l ~c(i) .
Wit:~ y(~) • ; one olltninB ca.l(i/s , O) = c~:i.(i , 0/0 =d
ial(i/S,a) ~ t.1al(i , 9/~) - li1speccior:of?ie; . 19a to c clio•···~
tl11.+- ca1{2i/2,9) a?Od ctl( ·li/n,a) "r" equal to th<' func tion
co.l(i , il) couLnued periodic ...lly over dou.u!e or fou« t.i:te•
the original inte::-valofdei"in1tior:. . Thi rezu.l: :r.o,y ul.!'o
be ini·e1'red roudil:y from the dilruence oo.uatio" ( 2tl) .
Hence , it liolcl;:; in the inie,.val -~ '"' O < ~ :
co.l(i,e) = ""l(~i,a/~). c .. ir;:.;~,3
!n;;pcc•io,-, of ; ig . 19n to c sh0'4S furL!le1· L!:nt the fol:owing relations ho!d in Lhc j m;erval -t :i 9 < f:
cn l li ,o ) = cal(2i/? , e • calf(2i+1)/2 , B]
(87)
1
= cal("-i/'+,B) • cnl(('li+i)/1+,3] • cul.(( 1i"2)/a,3]
• C<tl[{'li +))/11 ,~J
n
csl((~i111)/~,9]
u,
••llu ,0 )
'l'f
i
~
I:;
0 , 1 1···~ -1;; =
2 '.
u < 1+1, or.(" o.ttai!'l:":;:
ca:.(i,9),
C· ~l u, 6; = wa l(0 ,9 ) ,
-I
<
o
< t
<.:orl'0$pOndi Ill!, ; ·elui,ion!J ll.l'" obi;a i ne'l fol' Su l ( J, g):
s.g_(i,e),
aul(µ , 0)
-1 < u
~
l, l • ·1, 21 ..
(89)
-c ~S< t
'!nc liElit ~wictions cal(1- , 3) and !191(u,9) h~ve ceea
(!C>r veC. here in an nourisvic JllUUler for ~he :nteI"\fal
-i ~ a < ! . PICilLER l1n!I obtained cal( µ ,e) ~nd cal(., , e) in
e II.ttthernat ically l·igo!'Ou3 wo:; :·or the .-11tola iLtei•val
-co< a <co, bu" hi~ pi·ool'n requii'e " vex·y r;ood command
o J' tnli Lhe c1at ic a . fig . 1 2 !ind 13 Mhow a vr-ry i n,;enious re;ir<>senl.atio n 01' the rune Lions cal( 1- , a) •nd rnl(u ,e) fou."'ld
by ici (2] ,
hL'lct.ior.e thnc are ident:icfil i.c the intterva! -; ~ 8 < t
:;ield c!::e s=e expan!liO!l -,oefricie::cs fo;- F(5 ) . licr.ce , one
ohtnin~ for sc(u) lllld n~( ... J :
Ill
a,(i) =
JF(e )cni(; ,a )de
i
~
u < 1+1
-111
I (1
Jr'(& )s:il(~ ,a )de
i-1 < u ~ i
-Ill
'"jl'( 3 )de
-111.
-f <3< i; "
1, 2,
0"µ<1
(90)
.,.
0 ,.
j
I
Ol
OJ
_01
-
~01
•
t
0
n
fl·
-01
0
O•
0
,_
!
~
c
oL
I
Ol
0.1
.fj=o.2
3 0.2
-=-0,1
tS 0 I
..
b
0
0
. l
' 121'\l/l
-
)
4
00
d
µ-
Fig . 20 Cceffic .. e.nt~ of rhe expAn~ion of .. J1~ trinn&"ular
!'unction ?(e) i11t.c a .6t>:"'ies of \,'r:1lof, clnntf>r;t!: 'lCco::-dir:c;
to ?ig . 1 ' · Ac\u) i.:.tbe licit cm:·ve !"01•tl1e ele:Zler.ts stretched by a foct~1· ; - :o.
Gc({2i•11)/:-), nc(( 11i~!]J/"• ) c.!id the liait a,(u, r.re z!lolm
in .:'ig . 2Qb to <l fOl' tt t? <;ri8flEr'll'lt'" function Of fig . -1'-" .
Tl1e con.~utucio:.i of t;1e :qnc:t::.o!'l!: n~(µ.;1 rtnJ r1~{u) is very
Sim{.lt! .f o:• V.1UlfiL. fur..c;;ions , ;;i!'F""I"' Or.-:! !t'1 t<J COJ:.q_:ut,e the
coo1't'ic11.. r.tt~ ~(CJ , Rt(1) A..Tld ;-~ 5 (i) onl,Y u1H.. plot thc::e
valuer ft'Om Ci to 1 , fr om 1 to i+1, ul' from i to i. - 1 tc
ob'tai.n nc(u) and n 5 t µ ) i u the irltOl',lf• r 0 ~ u < 1,
i ~ u < 1 11 or 1 -1 < 11 ~ i .
1.25 Fasl W alsh·Fourier Transform
~f' ti:r:,.. f""'quir~d ::o ob:.!li.:l t~o:: fo•..:.rir-t::" :.rAr~!.ifor:o aa.:~·
be dr&svically =educec b:: =ca;:s of a aethod kl.own a:? f9~t
Fow:·!.cr tl·anr!'ora . A co!"ree_;:o!!ding l"neL '111'olsi~- F:iu.rier
tra."ls!"orm war found oy GREZ!: (1} and g~nl!!"Alu"d \:y •.;;;:.Cii
(2,3] . KANE , ANDREWS ,-..nd P?_l.T·r :-.Rvr u~1'd n cwo-JL:i~usional
fast ~·inlP.h-.Fourier t.t·ansl'or~ for tne CO!:lp:-e~tiiCr.;. O! iJ:;.for-
lllation or pic~urea ( 4) . "•TIELGBEL and rnntrn have use d i t
fo r signal ctansiLi.cai;ion (5) . The form 1 i ·esent~..1 t ere d1stingulslle>s b<'t1<een evon and odd functionn nnt1 liati: the:n
aocortl.lng ~o tbe nwnbex· or sign chanp;ea "" Jt1 F'ig . 2 .
Coneid<?r o function F( a ) in come interval. L<'t chis
ir.tor\·'ll be tii\'i~ed into 2n
P-<;~a.lly
i11•1r:~r:-tc::_on 1 t;he siiecia\ case 2
l'he t: 1·Jerage: vn~ues
ot
1
w_c!e
subictryr·:a~:: .
3ox-
=-a. will be Cir.c:i::s.:-d .
F(tJ j in ~t:.e B ir.tervt.il11 src rler:.c"t;ed
\l,y A, 1• , ... , fl. F(0) l.s th1.10 rcprceer.tc>J ·u,y a cter '"-"·C-
Liuu Lhat is a J ~~st ttil'ttiJ.I s quar e ;... i-c O!' F( a I !OJ.' t;bi s n:unbcJ.• o~ int&rval .s: . 1l'l1e- '.-.'ulol:.- l•'our:.er t!·ruiw .to.L'e1r a , Cu ) and
n)(u
of tr.(?~1' !JtCJ.• fu.Hct.ion:3 may be oblal.ut:d f ro:n t:JJe
nvera~e
·;alt:c= A, B , .. .
-.A-I:+...;• D-1-E-1-F- :i- tt
•J\. 1f_ - ; _ r .. £+1''- !.i- H
&
a, ( 1)
D
ac(1)
•
'1!j ( ~)
a,, i.J. J ,
as(µ),
nc (2)
•c(u) ,
. ·•c0)
a,( u ) ,
I
•
•
ll•C- D+E- F- G+ H . "sD > . a , ( i.. ) ,
• A-ll- C+ D+E- F- G+H
-/11
ac{µ),
a, (,J) ,
a(O)
.. A. Et~i.t-E-i.?-;; I H
-A-H-C-t -E+~·G•~
H xi th -;be
t
-At fi- t:+ I:·+.:.- F+G-H
• ,·,_ p +t:-D+J:i- P+<;- H
•
~,<Al
e
a 5 ( u ),
~elp
c
~
0 <
oJ Fi.g . 2 :
u < 1
1
1
u < ;;
·1 < u ~ ;:
,
2
u < ,
2 < u 1 ,,
l. i
'
'
u <
;· < u ::;:
?•
I
~
"
T?".,,.re n.re ~'('"'>- 1) • '-6 or ger.···:-1)~ly 2"{;:" - 1} :l.C.di~lona ne"essa~·:·
to obtain t!':<? ?' coeffi"i··r.ts "c (a.:) and
'!l( .,. ) . rt.~ fas· Wal !th- fouri0:- ~:r·o.r:s for· 1·e p1irr:.-; 2nn ndtlit.lono on:..y . Note thnt: ~h~ ~·,',qlsh-Fou1:ier Lr.:Anr-foro does
fJOt 1•eql;irc mu4- tiplicnbiou.:;; 1 •11h i ch art:- r tme r.nnsuming i n
LLo c:ano of thr: i'11nt Fourie.r· t x·ansf o r m 1.
.li'nr :in e x plrin:iL_oJI OJ.' th9 f!:lst 'r,'91 :;t-Fou:·icl: L;ra.nsfo.!l!l
rt:frP to ~n!°:lle ,. . ColtuW 0 lists l;hc'. 8 runpli~ude sa..irples
A, B, ... , 1:: ::oget!..er wl~L a niore gcnc-rql :-:otntl.on s!:: .
Colu.:w: 1 list:: !::'..J!J:? l.Ul..c! difference~ of two eac:.. o;- t;he
~:i.mples , again t.ot:"'th~!''AitC. 9 mo:-e gt~?:r-rnl !l.O~ut.ion . SUit.sn.1_u! difference:~ of C·:l I umn 1 arf!' s.:iO\\'L. in col ~1rr.r. 2 , \-Jh.ile
.-..:L l u.:n1:~ : sho·1:s :.111r.i nntl <l iJl'"! reuceP Jf i~O I lu:i.n 2 . The gentn~t.1 l notation n l. n t$hows --11 e~ch caoo , which -corms of
r:h,., rircvi::uo col:11!U1 t"-J:e iu.i.i.l9d :>r :;uttJ:•a.ct.ed . 1Phe third
colUJIW y i e l <is LJ>e Wd~n -bouricr coe:'ficien;;r n(O) , ac(i)
""'
fu~t; Eaar- Fourior t.ra..u3form Ci.a~· be d~rivcd for the COJDorth.ogo""~ systl!m
llaa:_onn [&) . '!'tis cransfom 1:1:iy te ~~·~n be~t.er !lait;ed f<!r digi col coaputa-;io~s
L.han the : ast ~:alet'.-Fourior tran3..1oro (perno!"'.al cotut.un1cution .:."rom H . C . ~1Dl!EWS USCLA) .
'.k.
plete
or
r=ct
47
2
1
0
0,t>)
o,c • .,.(Dc,otso.oj
-o,e 8 1,0 9 a, 1
e, ~ 1.1
50,D-=A ._~ D .O -=+ (er 0,0 .
•••
··c,1
•
= tA · 3
-•.• -- - ( .,•.•
- e•·•)
s~:! ;3 .:;0.l
'"'tO
1,()
: - r'. ..-!l
9 11.1
0.1 ..
-
0
· =C !!i.•.•s 0z.o
lP - + ( s'-"
1,3
I
•
so.o =-D
1.0
+.._· •.•
1.0
)
1.0
CJ
= - ..!t:l
SIJ,tl
t,O
=-"P
.,..CJ! _
-... _. ?,1 -
•
i.( .,.:t.o . _ ... o.u) "o.o
.... , ,0
.... li.O
;!
s.o - -
,,. !1,1
.:.:Z,I
:
00
• -- G
s 6,0
=
s
6, 0
-
~,.1•
•
t"l,O
't .2 •
•
r r,;1 Ii
00
• -H :::0, 1=-( 6 0,0
7,0-
l, l
, ,,1
1
- E-1 F
.0,0 -= i.( f,0,0 .... ..,o. o ·1
e l,1
6,0 "' 1,0
• 0.1
"11,:
Q
J.J
.:>~•'
"'··
- ="') )
··1,1
~o.~
_ n o, o) ~'·'
1, <I
1,2
- Ci+li
·
(
1.1
<I ,l
)
• 11,l )
._!;i
1.1
J, l
.1
; +A - H - G ~ V~E - f - G+ E
. 2 ,I a
~O,l
" 1 l, I
1
( , 1,0
I, 0 )
uli,1 - !:JI , 1
~ - A+ H+ C- D+E- F -G+H
I
• l,O
t·o,,
=-
( J::ll ,1
0.2
~·' •I \
- ~ ,. 2
~ -A~ k- ~+D+E-F+u-H
- E+l"+G-11
• - ( 0 n. 1 1 UO,I )
1.1
J'\
--1 1 1
u~ ·' =- (:.:>' .o J.- 'l;J' .r:)
.,e.a _ ..o.o )
+.( 6 0 ,1 - ~0, 1 )
2 ,1
J ,I
.....
t- A+ ;-i - <..:. - 1.L--J:. t-~·-G - !-:
-
-l::- i" 1 G~·H
.yo.z
,• .O.J
A- lhC- lJ
l .I
Gt.!.J
., 1, 1
.. _., _.,...
Eo.i .,. _ .. 0 0.1
~r.: .,. l••G+H
_,
l'-4-_E., 1''1
c - A- ;,...c~ rµ~ :- G -.:'
( eo.1 +a~:~ )
4•
I
'o)
:~
--·--~
= - A-~
- ~ - ~+ L• ~~ ~ +~
,;;;so.0
uJ,o)
I ,' 1 J ,1
•
~ -( ~"···~'·"
1., 0 ..,
~ .o }
!':IJ.I _ " t,O} ~3-'·- ( l'0.0-"0,0)
·•ltt:Bt
""'l,l • ._ •
= + r.+!·
so,o -
(
~+ D
- A.. S.-C- D
0,l · -
'-'1 .1
A• S1
( <1'!'1:1. 1
_ -e-o.o ) s'-'
... l,O
._ .. o.o
t
•T
•
+t:.D
"'0,1 __ 1 &
,o
... ,) - "' 2 2.t
c 0,) _._. ( ,oO/J
0 ,>
0,2
• - A- &oc ";4-t
6
.'
-
,_.J,1 c -t (!J l,I 1,,;_-,1 ! )
.... II, )
• •E- F •G- 11
O,Z
""'. 2
••Ji..-H•C-D+E-F +G-H
and",(!. ) .
The Su~t Wul ~h - Fou•·i<:!' L1•w1 .' lfOJ:ia cun ll e r epre s ent eC. 1.;y
a r ecurrer1ce .fOJ':tul:. or
f ro:t t r.Rt
di~fer1:ncc
equr;.Uon Lbat 1'o::ows
Of tbP '.-/"' Cl i f1.mr.tio11 C (.".') :
(91)
(.1/2) = larges:: ini;eger s11ulle1· or· eqaal , ,l
JC = 0 ;--or j • ever .. , x • 1 fo1.. J • old
k_ .C 0 , 1 1 ... , 2"·• - 1 ; C l • 0 1 1, .. 1 llj p D 0 0!" 1 ; j a 0 .... C: \
2" = nuzber or a::iplii;•.1de samples
-~ " ru: exn:r.ple conaid<'r the term for j • 5 , p • 1, l< = O ,
w • 3 . JI; l'ollow" with ( J/2) = [3/2 ] • 1 and x = 1 :
l,1
"•.•
( -1 J
2(
1 I
s 0·,
'
(
-1
)'
1, l
"•.,
J
'1'his i!: .idvntical With the tr: rm in vhc lower Pit:,;l~~ corno1•
of ra.0 1 0 1 .
_:_:.e qu ui:"ltics [,1/C:] nnd x n!l~~ be "rod-.1ced in a ·cinary
coinrnter 01:1 fol Law ... : Le~ / ::i•.: rcpr"!'seutnd by '1t1nnr7; nUJ:Lbi•J.~ - DivirilOn b)' 2 .shi.f t s t.ho bio.nr~/ !1oint li,y nno ple.ca .
:One nl.;.l!Jbet' tn t;.... lt. r~ a :' me o:.nru:;v 11<>ir.t i~ [j/?], thi>
rn1iber :;:o t ~e r:.-~4t. le;): . .Exa:.nrle:
,j ·23 =10111 ,
'
'!' ,·
(j/2J
= 11
>: - 1
The computatior. tttartz \\'i 11 tt..:- 2 " t.e1~ZI:s s ~ :: , ~: = 0 ..
. . 2• - 1. !t !allows !re::: [j/2) • [0/2) • o o:.:1t 'h" cer o
$~'. i 1 \.: = 0 .... 2 11 • 1 -1 , can 'n~ corr,µur:nct . 1'11i?l'lf"I n...!'e tho
Liarmn- in LJ1c ni::>coud cr:lU!llll of 1J''lble 1 . Fu1·th1'l"' te:'t:lS with
[ j/2) = [ 1 'J = O (:111wo.: be :O!LpUte~, ~i;.ce .,.hi. w<>.1lj requir~
ten ... ti~:~ w:1ile only cct'il.Q •,fi;r. x - O arc cvuilab.Le .
~le t0:'W6 gO.P reJ..'Oit t:-.e i:!Omp11t at...iOl1 Of tht:." t,;.n- I
terms
k, I
s~:~ and tue 2""'
1..t:.:.'Jt;"; :.~:~
, Gince x mny b~ '."',oro or 1 ,
ar.d j :r.ny t.l&us ;-,., 0 01· 1, tot1'i va:.u-,e yi P~<L!l,. ~ j/2) = 0 .
'11t:.e f u.e r inve!"'Sl" vinl sL-r··our _er tra.n:::fo.:.· o t ... otit;aint:d
by computi11g the co.,t'1'icienc:; A, B , ... , E !1•oni the co<>fficieat~ q( 0), a, 1 ', .. .. . As l ~) . l'hi~ :oa;; :.~ dot.e by in. Or.e obta1ne
vcrtiL~ "the =ecur~io:... fo1--:L·:1n
frott. Lhe f':tLi1 u.nd rli!'l'er'encc o.f
:;11e fol lowi.ni;
-wo recur-!1ion forr.tula.;;:
(11 '··.
II•', mo!
"''"]·
.,,,,-1 '
• i{ - 'IJ' · {111J f\
• H - 1>i10J
~ 1.0
- It,"'
j
'·' )
' lic,m
(,,iO_
,,I.•
)
>,m
>,m
(CJ2)
p.
o
or 1i x = O
~01
j
=even, x
= 1 for
j
(J/2] = 1 "''''est in~eger smt11le1• 01· equsl ~j .
•odd;
1.26 Generalized Laplace Transform
l'he
arm
~I L1c• ~ra.ucJ<>r"'
:<t:o,•,) '" '
ti:nc fuuctic:. F(oJ
1 "t !: inV~l'tle m.a.Y be \o:rl. &L:e!l a:- :·o ~-Ott."~:
00
7.{c ,v )
Ff 8 )
J F(e Jc·•• .t
'
1 ''
..r xc
..
•
1
•••
de
(95)
o t v Jt·i z:r~~de
H in t•J')lllt'Cll~ t hriL Gue J,ui.·lr.tce tran~ 1'01•:n OJ' E'( 9) may
be coIJ!JldOt'Ocl l,o Le a Fouz·lcr tra:1Sfor10 o!' J•(G )o''' . The
f act.01· c· 0 1
tt.nketi fu.nc:;io.!l:"; 1•'(~) Jo11ri~P t.:·tUJcl'ori:1a~le
that ~r" not qu:idr:itica:l-;y i11ti:"v-r-c.it.1 r . l hr- gi"'ur.:--n!.i.:.ed
Laplace trnnr..:.'orc J.L .:'t:al no'tai;ic~· fol low~ .1'1-0U1 t.his !"a!f.!ll'k froa. ( 9}) and (';<.'l) :
1 1
a 5 {o,v) •
ac<a,v) •
r F(6)e·
•":'
lo( e •e ..'
•
111 •
J
f~{v 1 B)C5
(95)
re< v ,a: )cte
00
r(e) •
!
-oo
n• •
r nc(o , ~)r,c~ . e)
- n,Ca ,v ;r.cv , e))dv(9h)
Th0 into.gt·o.1.t1 (95) do not have Ll':e lo i<e1• limit -co as
\io the i1H.ee;n1lr o r
Si!lCO t:i1r1
rflCtOr r.o• al
th" genen1li2ed
f?ouz·ie1• cronsform ,
Et'( Q) l!lil~t
Wig}Jt.. LJake thnm C. iV Ol'j''Ct:. l. .
vanish su!"!.. l.cieuLl;y !l::l.st for lal,"'bl'"' =-nent.i•1t! v~luc~ or a .
= O for d < 0 is used hc:-n .
'!'he usual oasu:i.p~J.OJJ f(ll)
1.3 Generalized Frequency
1.31 Physical Interpretation of the Generalized Frequency
Frequency is a pa=aaete= that dintingu1:: .. r u t.l~ ir.di viaual fULlctiona o! tt'.e c;yste!f:!:i ( co~ ~nft.. l o.z:· { sin C'nf-:;~.
Its usual ~ilysical tr1terp:::-etation i.: "nwi;b<ii· of cycle•
pe1• unit; o r ti mo'' . 'L'hc normalized !"r~qu0nc;y v • l''r i c. 1nterpr·ot o(I nri 1'numbe1· of cycles in a t..11111..• ln ~ t:J.'Vts.l o.!' riuvation 1" .
'fhe ge ue.rali,ad .rrcquency may be l m:erpJ'ete1l a" "t<Ve-
.-·
50
1 . i·IA'i'Ht::;·JA'.fICAL FOU:r:lATIONS
r~ge nu:r.~er
by 2 '' or as
of zero crossings pe r
LU1it of
titr.e di Vi:iea
'' aveT·;:1ge nu:nber of ::J i gn chnn,gea µer tuli
time diviaec: by 2 " . ':'11e normali::ed ,
u is
in~er1.;.retea
Lim~
in Le:-val of
e;ener·ali~ed
t ol'
frequency
as "averago num"nor of zero r.:.ro ssins;::: per
C:u.r· a~ i.on
1
Civid~d
by 2" .
Tl~e gen~l'al i zea
fL"equenc:1 :1.as tne o;Lmonsi on ( s "] :
(97)
1
Pt.e defir:it i or: of the e;en e:'al i eed .fl'.'e-quency !--.as been ct'.o-
se:i >io tha~ H coincide@ with that of :':-cqu,,.i:cy, i f applie,;i
t<> sine o_"lC. cosine fun-ctionz . Fc-r inst£:.!1Ce ~ a sir:e oscillaUon witil t'requency 100 R" ha" 100 cycles pei' second or
200 ze.ro crossings por !lecond . One halI tne n\JJ:1ber of zero
c:•ossings per second e qua I s i OO , wl1ict-. iB l. h.e sa.:oe ntunber
and d:in.en sior. as that o:: ::he fr·equei:1cy' . 'l'!le zero c ror;sin~s cf sine and cos ine f lmctio!!:::. :i.re equall,y spac ed bu"&
·enc Ccf'inii;icn ol the genflr.nlizeu fx·eq Ltency tl&kes l \; a;>l)li-
cab"' e t o t'nr~ct-i·:1ns who.Sf' zero c:i:·ossine;:s are no\:' equally
spaced a:ld which need nc l: ever: be pe:::-iodic .
1-;; is :ise:·ui to .:.Stt .t·~luce LLe new t:e r:?r~ ' 00·111oncy ' f or
tb-e g~neraliz ed i·r.,sq~enc:1 ;i . One rea~o n i:: :;nat r.lle te.l"'m
e;~r1errrliz.eC._ !'t·~que::c:y
i.s aJ Pea.C:y ue.ed in co1wect.ion with
damped oscillavions • a.r..ot'her is that there ar·e r;rar.s\~er
sal waY9.G i n thT'P•'"" dimensional space which h(lVf: .a fre q_uency
as
.,•eJ l
1
an
a
::iequcncy.
"ever.ag.;i :r.imbcr a .:· z e r v
for 1·:nic:l one may
i.::~e
':~e a::eat:iul'e
c~ossi.:!5s
of :::equ ency i s
pe r s eco::id d ivitled ·oy 2 11 ,
l.11e ab·c.eeviati on •z.ps 1
•
Co=ider Ute Walsh fWlCtio'1s cal(i,9 ) and sal(i , 0) in
Fig . 2 . 1 .:qua.ls c~1e half th~ nllllber of Gi gn cl1anges in
tl1e bt.e1:val ~ ~ 8 < §
and
::;> = i /T
is.
th.:: seq uency 01"
;:;j1e periodical lj" conti nued funcl ionz . If Ll1e f1u1cti ons
are scr2t:cr.;ic ~ya ~actor ; they 't!i1 1 hnve 2i sign changes L'1 "thc inticrval -~S
=: e
< ~S i
i/S = µ will be one hell'
1 The !1Uiljbe1• o : sig r.. chtLt1t;e2 per unit of t i:oe 11as been used
t o define an in::stant;a.neous frequency o.r fl"equenc:v 11odula~ed sinu so idal oscillations [ 1 , 2 , 3 ] .
1 . 32 FOWER ;;J '}.(;'II/Un 1 ~ :LT!:R um
the
.
nu.:rit:er·
sv~f'!lf;e I
~.1""
51
'
• ar: i n-:orvri1 o f d:.zs1fn
c t ~ri::gce 1-:l
rst1on 1 .
Coneid~r
.,, a .:.·urt:!".er extt:J:?le
~nc
pcrio:11crill;; e:c.::ti-
nued r.egenrt::-e pclyno:n.:s:.s ;;, (i , 9) "nd !s0 , 9 , •. :ig. 17a .
Tbe,7 lH1Ve 2~ llit;;u c!1a.n;.;es in i:he ir;:;crval -ii < i < ., .
S-eretchir.i:; ;-t.f'ID t::: a l'nl:l..or· s fil:J..~~ the Ji;r~tl ""U or thiE
intervfll E>'l'll'll ~ nnd i/s = µ becoa:e<" one hn ( the 'lve:-.~ge
nwnbe1~ o! r i1:t. chani:;e~ pe r· tir.1e inC ·..:'rvnl or "1ur:-i.tic!l 1 .
LeL Lhe nnrrnn.lizcd v~riables v a."'ld 9 j 11 r.ln 2nvtl be
repl.~ceu 1.Jy tLo uon-Jl•)t'Jllal i ZE<d ya-,.i"b I •rn
f
v/r cm<l
t
a
0']':
(95)
The tir'Je bl"l ... o '1 d!."VJ..S ... u - . s~nc and coeir.e .fw.ct..lo;.s co:: -
taiu tho three pa:-ar.iet,,~·s a!!lr · i tuie, f :-~qJency !illd pha"e
angle . 'fhla is no-; so !'or coarlete ~yet-au or ort::o;:;or:"1
.functions, wn1 ch do r.ot
h~·:e sequer1e;.
ti.!ld t ico bar.f' con-
nected l:y :iml tiplic'lticu . '.iaC.sh :''-'."lC;;io:w :-9 I ( 1-, a
or Le-
gendre poLY!J.oa:.inlu P&(µ . 9) !10.vt= a co:ri:na bt?tW('·ru u 9.U:..i. e .
Hence, th.c-: aubntitutioJl,;j co = u/'":. and r • sr yiP.ld :
altl(u,e) • nul\<1>'1' , t/Tj ,
ThnAe fu.nctionki ConL.ain l u tl1ei1' 1.renL:r1.;11 l'o!lm the .fo ur1 para.meter~
'Ullplit.;Ude V 1 sequeuCy cp , C.elay to
V sal (
t-t.o
:J.'
m
c+I • ,
1
l'nd lil:I"" b:i.;:;e 'i1:
J '
1.32 Power Spectrum, Amplitude Spectrum. Filtering of Signals
OriP tl .y deri v• ._he frequency 1-'.lllCt-Oli c~ \I I ,_ 9; ( v
f:t-oo the l·ouri<'r tran~fon1:; ac(v)
t<s(v) or (<1) F.n<!
inte:i-pret it. u.s f1·e::quency power· !:JJ.;e:cc::wn . le hn·11ou , one
may interpret the neqtte=>cy funct~ori n!(u) + 11~(1-; ccriveC.
from the geueL·a.l.izcd Four:.et· t:·r,n".lorcic nc (µ) um! a 5 ( 1- •
0
"":!
-------
1
The ser1uency o.r n peri odic i·unction "'ll.lDlo one ltnlf Lhc
numt.er o.t sign changes pel' period . Tho· noquC'nc;; of a non-
Periodic £unction equals one half th" nwnbor of oie;:> changes per wU.t or time, if this·limi~ exis~a .
••
1 . 11A'l1l l E:'lf\'r WAL FOLHllJA'l' i.ONS
52
of (62) ·and (63) a~ ·a scquenc;y po•...•er spectrU£l .
1.et {53) be squnred and integrated
of. (59) for the coe1·1'icient~ :
T!le
o.r t!1e c:t•oss- p:-oducts of
integ1~a1s
Ul:.i.lng
Lr!:e notation
difJ.'eren~
t't.t~ci:;iocs
vru:iislt <lLte Lo Lile orcbogonali ty of the rw1ccions . 'i'he i11be-
srols or r' {O/; , 6) , r~ C:./s ;B) l.l.lld. fiCi/i; , e) mulGJ.plied 1iy
;1 · J
y i eld 1 :
The stllll !!as ~r~e SaJfle f orm as thl~t cf ( ~8) . Hence , ii; may
be interpr-eted. as the area u nc1D:r· a :JVCP
swn rn.a:y ·oe re pl
~cetl
funct~o11
and tbe
by an i r.tel;re 1 :'or I argP. va-1 ucs of i;.
ano :; = y(O :
00
¢(1
J
F ' {6)d9
-r..o
00
I)
Using
""
J
')
=; [ri~(µ) ~ i:; ~ ( u ) ]du =~ j' [a~(uJ ~ a~( µ , J rlr; {100)
-(JO
non -no~mali7.ed
F 2 (L/'.i')dL = T
-~
'l'[ a~(\.I)
1
notation one
~b~uin~~
~
J [ "'(oi'l')
c .
•
.a~( u )]C.u
~
is th~
a 2 ( cpI)]<i(cp'!')
(101 )
'
~ne:rgy
c l'
l.he
components
a , Cr.. ):'c(µ , ;i) to "-c(u+du .lfc(u- dµ ,a ) nnd a,(µ ) f;(u,e ) i;o
a 5 (lL+du)fs(u+du >e ) 1 i:· the i!ttet;ral of ii' 1 ( C/'f') i.s inter-
P''eteii ""the energy oo· ti'.c si;:;nal F(O) . flence , a~(u) •
+ B ~(µ) ·... an tho dir:io-n!;ian o.f rowei· and may be inl;erpreted
a!"'; r.>equcncy power ~pectriirr. .c:: seq uenc ,y po 1t1er den sit~; s_pecl;riln: .
Li sing the fu.nc:.tion G( 'J ) ~
G( v ) : A~v) 'i1(vj: t'{2[acCv) + n 5 (v)] ,
one may re~·rri Je the i' r•eqt.i.enc:y J)awer spect-I·u:n a~ ( v )
into the following -form:
.a.
a~ (")
a~(v) + e{(v). 2CA'(v• + B 2 '\i)] = :;'(v)
c'( - v)
I
(1C2)
use hs!". t.~cr. :nnc!e of ''lE), 19) aLd (!,.2) . TlH? se1u<?nc::
po,,.er !"pdc:ru.c oay be i~e\\'.!"it Le~ a.ls fo!.lowa :
el (u) + u~\µ
1
• '•~A (!-') • 3
~.e ~Q..J~J;e .!'OOt
[ ~' ( 'J )
....
1
( ..ll:
=
G'(~) • G (-.;J
8..s' ( V) J'''
1
n~:,~
be
(1C?:)
.
l.nLe!'p.r~l..ed i:I.~
frequc .ry 'J.!r.>d i't\l(! C" G.pcct.t'a:n . Such u.:J iace.rpretation iE
not posnibll.' .rortLc square roo~ [a~ ( µ ) 1 u~( µ ) 1111 , .. th("
sequcncy power Gpcct J"u:n , s i nce n specific reat•tl'e or si.r.e
and cosine r'wJct.i.o::.s i s i ·eqLLire<i for- i. t 1 • IJ:;ing thr r C'le.tion
A sin x + 9 cos x • (A
1
+ a ' ) coo (x - t1;;"' ~)
117
(10")
one may 1·e•ritc (;..2) as follows:
..
F(9 ) •'f2f(ef(v)-
a~\v)J'"co>"[21'Tv3
- cg·'
~;~~~]dv(105)
0
The !actor (n~(v)
quency runplit..ude spectr·wc. , since it rr:f'r'4•Ct.r.tu t.he wr.;.1litude ot t.-he o.ecill1;1Lion h'ii;h frccr.lcncy v w.it.~11Jut J.'E±bard
to the ph11.Se "I.i-;le Li-;·1 ,. 5 ( -v )/a , (v) . Syntctr." of !'1.<.nctiono
tl\a.t do noc 1111.ve 11.n <1ddi~ion tbcoriom lik1: ( 10'·) 110 :iot
pern1it; thi:i ir.torprot·ation of t11e .SCUtll.'e J"OOt ra.~(u ) •
+ n~(u)J"' . ll o>:cvor, ac< u > ::u::d ·,$(u) .i·e J•.:i1; lil<c :ic(v)
and n 5( v) che o.:nrliLucie .;r:ec ~~·" of tie ev~n OJ1,.; odu p:u't
or the !unction ?(a) .
l'iltoro or , mor., generally ,
sy~i;c::~
ti.ai oliwwe ar. in -
ir.-o ar: outpu;: ei-:_u~ F 1 (9) mny =·" deby operato=.li. . :r.:ie coZ1. ... ept. o~ 11ncsr orcrat..ors
descr_-ibi. ng linear .:ij'Gte-G is Of r~r!"ticulnr i;upa.:.•:Oa;..1-'.:e in
connection "•ith complete !iy~ter:t:;. of or:!.loe;ounl fw1ct.!.ons ~
Let n d~notc a.n opero.to1· and [l'(j , B ) ) o con.;.,le;e z,yotf':t
pur; signal F(8)
scribed
''l'he addHlon theorem!J of sine ar.d conino n:-o •·cquil:·e;l
.for tbe do1•1vation of cbe Wtener- Clli ncch in theorem i n
~i"l nato.~ion . fie nce , other "ystiomo o! runntiono ~""'c n:>
f l:'ect auol0(5UG to the Wiencr-Chintchin thoorom . Wolsh
.unctions havo a.11 abstract analogue bnaad on the dyadic
correlation t'unci;ion J E'( 6 )G( aer )de .
5u_
1 . f·!A'l't!EMA'l1:CA.L FOUKDA'!'IOlfS
:Jf orthogonal f unctions . Appl icstior. of n to ~ _p.sr t ic1.2-ar·
f\u:tction or input sie;nal f( 5 , 8 ) gcnern Les (ill cutpu·~ signal 5(,) , e) :
= g(j,e)
nf (J ,S J
1'he Off"ratol." 0 l s called li!1f::C1T' if -:;hr,- :p:topo:·t.ior..alit;t
law a.nd t;he supe r
. µositicn law i1old for 0-ll fttc.c~lons of
1
the syst om ( f (j >8 )J :
pro;.ortio:-;a:;. i ty l aw ( 1 07)
= a(j)nf(j,e )
na(j)f{j , 9 )
00
I: ·:10.( j )£( j
00
ii L:a(j ) f(j , 8 ) =
, ,o
1,0
n fil<.;.Y be
1
SJ ~u:perpae.=. t: icr. law
( 108 J
function of ,j a."'tt.l 3 . l f D <leff!nd!i on 0 , the
opeJ'ator n::id the !::iJ"S tom it descri Oes ar.e i. ineAr a:H!. t ime
vari;;.bl9 ; otherwise they are l inen:.- and t:l.:oe-i.D.VB.l."'ianl:. .
P.J1 oxaople o.r a linea!' , vime -vari ablc .s;v·st:e1n is the a.:.'llpliLude moclulator . Let an L'>put sie:nal P'( 6 ) 1Je repJ' 0 s en~ed byi;be SUIJl lj(j)f(j,8) &'1d the cai:i'ier b.Y h(k , O) ~ n .
h(k , e) may be , e . g ., ;:. sin<-> cru·1·ie~ '{2 si.~1 2r kB or n Halsb
carrier •·: al (Jc , e } . /._'ll;>li ~ ttde ttoclular.J er. ·.·1i tl: supp1'essea
'1
carrier yielas :
¢0
~
!"(9)h(k , B) = nF(S) = 0 IaC')f(j , 6) = L:;a<j )g(j , S )
1sO
(109)
pO
g(j , B) = r.(1< , e) r(j , D)
Ir. is oest to use ', .'al sh functions wal(j , 0 ) for :- (j , 9 )
is a Walsi:J carrier ·.-ml(k , 0) . O:ue obtciir:s fo r
i:' h(k , 9)
e;( j
' e) '
g(j , 8 )
~
wa l(k , 9 ) wa'l(j ,S ; = 1·1a l (k.J,.j , 6)
If h( k , a ) is a .sine ca1"'rier 1t2 sin 2ttk0 or:e shot.tld use tl1e
func~ ions f ( O,S ) , '{? sin2ni0 and 1{2co<; 2111S for the sys-
te:i: { f( j , 0 ) !
. 'l'l'.e
f 1mctio i:t; g( J 'e ) P..r<· then
g(0 , 9) : ·t[2 s in 2n k0
g(2i,B) = cos2n(:.t- i)9 - cos2:n (k+ i)9
g(2i-1 , 9) = sin2n ( k-i)9 • sin2n(k+j.)9
j
=0,
2i' 2i- 1
j
i
::: 1 , 2) ...
The d"fiuHlou of lineacit~· lia< cilwi;~t!d du.:·ing developitent of cowmnica;;ion ;;heor:; . H:-s; i t ·•~:i restricted
i;o di.!':·e::-em.ial o;>e:·ators with conr.canc coe!Jicien;<; , chen
to time i1w~i·iable but co;; nec.a~,oarily Jil'fere~-ial ope rators . The p::-e~e.:.l defi.!litio.a clots noi ~<tqui:-e n to be
a dif!·e:-<'!n• .io.l o .t· l..iu;.e invaL~i.s.t!.o cµero.tvr . t r..as b:eerr
us eti by mn t.h.eo.:.tt.lci.au.:::. f or· a :oc.g C.l!rHE' ; i I..$ •,,•10.e~;r:·end
introduceio11 into co.nw:.u..'1.icaL.:011~ l ..; ofl..l':t Cl'e<.lit~d t;o a
bQok by WIJlfSCH ( 1 J .
if u nyotor.i is de.icrlt>ed by a lltH:lH' ?pei·!:<~ Ol' (l ~.r.d i i
one is frno to cltoo,;e ~lte s;;st:am of £ 1;r1ccton~ ( l' (j , 0 )l,
one- may cl101,,;::o the syote:a o f eire1u"tu;r~Llon.t1 or n . EquaTiort
(1Cc) ar.cumc~ tue l'ollow'...ng foro. fr, thin c•1r":
nf(J.~).
l:\.l)l'(j,e)
(110)
is conven1.-.n: :o- ca .. l .:"(~,a) :i11 ~igt-u..fucctlon of n even
,e) on ~:.e :i;;ht ~.ar:d ,;i·~ .. or (1~:>) uas to be replaced 'cy n." ti:ne oLir'te~ :'WJCl;iO<. f( j , 8-~1 j )J .
!ti
i ! f(j
In the l'1·eqJ.~nc;:; tlLeor;; of cor.trr.ur.i·~a-:icn ::h(" elect.ri...:al
t;haractel'liPtics of l'ilters are r!.e::c!":.Ccd by bhe 1'.l'equt!ncy
response of otcenuar.ion and pl:;:isr sr.i t' t . '::1.i..'!i deacri;Jtinn
assumea t hnt. n volvage V cos ~~ft. is ap;.1lied 1; 0 the i.:tpu t.
of n fill..el' . The s t eudy .sL.al:e voltt1.i;e '•/c{f)c<'lr.(iJ, f t ~o. c(f;j
apJ>< are at ~It~ outpu; . 'rhe r'requoncy ftwctioHe -2!ogVc(r )/V
• -21og Vt<v)/V aJ>cl ac (J') n ct civi 1>re C!>. lf:d f1·oq ucney !·esponae Of ottenuati on and pr.ase "nl!'L . le~ "" input <lign al Y( 8) hnvo tbc ...ourier i;ransforms n, ( v) n.ud o, ( v . Ti.e
output signal F 0 (0) :'ollo~s froi.. , <.~ :
~. (e) •
i2
..J
l•,'v'Kc v'cos[2T'.vS + "c'" ]-
0
+ s
Kc(v ) • v,(v)/'1 ; v .
5(
rr,
v)K,(vj~in(2T"v~ - ac<' l] ]d•
e • t/r .
The dosei·ipLion o r filters by IDC'1ll0 of !'roquency =~
sponee or at ~enitation and phase shift i a eminently ~·.• i~~J
f or telephony filters . l''.ntched filters, 0 11 t;he othr.r hond ,
arQ ueua.lly deeorlbed by roeanr; of the pul ~c i·e:;pou~e . A
5G
1.
r•'.AT!.l~!ArICAL
FOUNDA'.UONS
vo l t9.ge pulse o;" tl;e shape of t he ll i r9 c functton
~ { 9)
appl i ed to 1.; he lnput. and tLe s.:1C1.pe of Lhe
v:)lt.nge
o ut. µt~ t.
i.s
D(0) l s detei·miu e<.l . No !.'"e f e r euee to sine atld cosine l'Wlc~.io n s i s .r e qui.red . \.1!1loh tiy::;teci of ..:.'~c Lions is ~petl f er
rlesc1·iption o:' a .filLe.r i s sci•ict-l y a mutte.!' o f cor:v-enience .
Let the volt~ges Vfc \ u , a ) ~nd Vf 5 (u ,B ) ·be applied to
t~.e
i npuL of U filte~· insGead of 'f cos er ft . The f•.lllctious:
Jc ( u, 0) and fs(i.: , B} are t he same ;;hat occu:r i n the ge:ieralized Fourie:· transf o r m ( (,~} . rlle s t. ea<ly stati> •ml teges
Vc(u;fc ( u , e- ac( ;; )] ana
'1 5 ( µ )is(·~ , e - a 5 ( µ)J
sne lJ nccur at
the filte:- outp·.>~ . Lei:; -2logV, ( u )/V =d-?1og1/~(u)/V be
c.alled attenuation . s,(µ ) an:! &, ( µ ) arc cal!e(l oelay,
5 i.ncP. the term ' phtisP. shift; ' canno:; 1',le .;:ipp l ied to f u nction13 othe::- tha~ oir.e .;nd, cosine . 'Ihe~e si.:n:plo ro.la1;io11s
bc:twocn i r.pu:: and o utput vol tagr: ex i ;:;t f o r i'.:.1 tors consi~ting of coils a nd capa c i(;ors if i'c (1.1 , a} a!1C i' 5 ( u, 9)
n r c: sine and co sine funct:.ons . :iowr:vr-:r ~ o ne; may desi gn
fi lters that cor. t a in mu_ tlpliers , intee;roto!"s , s·l;or~es ,
i~e!f..ist.:o.r.·s b.!lci S\\'it;ches 1 •1.1hic~ '" Jill a LL enuate and C.elay
:.'alsh f"!.UtcLious , but. \,·111 d:sLo.rL sine a :ie c or.ine .functions . Suell f iHers are better de~cri'>ec by '1ial;o;h .runc -
ciorts
. •et-
c~a11
8
·D-;;
sin~-cosine
;""'uncvion:; .
i;.i_gnal I•'{ 0) have tl:o gene"!'alir:oC Fou rier t.:eansand a. 5 (µ ) . Lot ·t-~e s·l;e.'ld:y stat..e at~enue.t i on
1'01-:r.;:-; ,1,(µ)
P.ncdelP.ybe -? l og'lc\u }/V , -?logV;{µ )/V aud er~ µ ) , :!.(u ).
T:ic nut put signal fo:..lo NS .f.r·ca (6L.) :
F0 ( a ) =
'~
J( "/u )Kc< u )r,( u, e-acC u )J ~as< u )K 5( u ) r, [µ , e- o, ( u)) J<lu
'
( 112)
Gou:p;;~·ison or ( '11 ) '1lld (1"2) shows cl:a~ only K ,(v)
o.c<">occursiu
and
(1 11 ) , ::>ui;r.ot Ks(u ) and n s(") . SLtch terms
';·1 ou Lt! occur i f frequenc:;~ filtei'S NOUld distinguish ·bet·w een
nine and cosl.ne !'unctions o l' the Gamo frequency . The input voltage V sin 2nf t woulc! then produce the o u tput voltage V5 ( o') $in [CllJ't+a 5 (f)] rather thun Yc(f) sin ( 2rr.rt "o..<rll ·
1. ;P £X1'J'lf'I
SUch
:;s
OJ' l·OWER SPEC'l'RA
diu;,..inct..:oo ':>etwee:: :;i?:C F.nd COSine !'t!QUi:.""eB OOJ.ie
ti
-ti?te·varinl 1,... c1rcuii e l)Jilent: a!ld ctu.: t.!1us not occt:.r
i-
f!'r-~uency fil te:·s •hicb a~e :1r.ea1· ru.d ti:te-i.J1;r:;:·iant .
filters bssed on sine a.."lC cosir.e p~~es t•nt?-.cr than c::
the periorti.c Gine and cos~n !'uncr.ionn ttiztine;uish ·~e
t;wcen sine anl'I co~ine . An exa:nple o~ !'n.tch ~. !iltf!r 111il l be
given nLet· on .
1.33 Examples o f Walsh Fourier Transforms and Power Spectra
Ji'ig ..'!1 .;ho'dG ~:c1~ .r1ulotioJ1~ f(6 ) , tllei!· Wnh:..-rou.:-ie1·
transl'O!'Ul:il G( u ) 1 at<u) ,
spectr1.t ~i( u } • b.i(u) :
as(u)
am! Lr.ei1· ij•qu.. cc:r po>ier
..,
i {2;
G(u\
...
r(ll)~•'lll(u , B)
; sb.:(i.; , c· ]dil
ac<1-1> • f \';?('i{u) • ~(-u)) , 9slJ
af(u) • ,.~(ul • G 1 (u) , Gt( - .o,
fl9)
l
I w•(qQ>
2 z..:tll.m
3 6(D)
'CJ
CJ
1
l
1
I
i::=:r=i- --d"-5 tal(l,6) c::Fb
----i¥-
4 &Al(l,9)
---'1....J1._
-9...JL..r
5 031(1,9)
. nw,.,,_a
1..-J
1m1118)
8Ml(~l
a:r.uJ .. a!<µJ
' CJ
.,11'1
a,11'1
'{j,G(p.)
iflA:::fl-
.w-9o-"Vi
.l
-
-L4A-
u
a
I
' 0
p-
___n...n__
n
~
U'n
a
--.J:]_
.JL....J1...
n
u
~
D
p-
0
Jl
n
4 ·4
0
' '
0
µ-
n
' ...
a
0
"-
4
Fig . 21 Sou.e t.ia.t. !unctior:s F(9), their ~''n I !'l -Zou ... er t::'~nz
forrr.a Cl(µ), o,(µ), n 5 (µ) and -;;heir ol'quoncy rower ~:-<:>cT.ra
a~(µ)• o;C 1a) • G2 (µ) • G 1 ( - µ) .
One lllfj,y see t hnt compl:·ession 01' the l'i1·r.t 11lock ;lulse
by a PO>Hlr 0 f 2 in the time -domi;ai!> jll'Oducco !I pro~ot·~ioual
1 . l·lATE.EftA'I' ' CAL POID4DATIOHS
stt·etcilingo: ;;he transforo G( .1 ) . The ci:lta fu.ur.:t ion .s (e)
is obtained in L.he limit; . Lt!=: -crans~·orrn G(u) ht\S a COJlSt;ant va_ue in t.b.e w_hol!.1 inte:-val - oo< u <co.
One may further ~ee , tllat
tihC transfo:"c1 G(u) of -chc
Wulsh pulses in lines 1 >4 1 5>•• )8 aL"e
'lronis i s
ir~
'sequeucy- limi-ced ' .
cont:·ast; t,;o Gl'.1.e \·:e ll known r·csul t
o.f
:E'ou1.. ier
analysis > t!1a t a :ime:- lirnited. ru.nction cannot; have a fre -
quency- l imited Fourier "tX'flnsforn: . ':i;.c Fni.;,r.te.r· ~ransforms
shown i.n Pig . Er for t:-.e sirJe Ct.lld co9:I n(~ pu_se.:s acco1~ding
to 5'ig .1 ,;o on co i::ifinity. WaLsh-f'our·ier crru:sform avoids
the .;roublesome iru'inite tir.ie- bar.cl wictlt
p·adt..c~s
of
~he
a11al~1sis ; btind\..•idt<n reiers or co1.trse to
scq-..iency band1.·:ldtl1 in c:'1e case of '1/a-l~lL-FouL·ie!" -crans.rorm .
o-rdinar.y Fourie1
A class of time J'u.nctions. that a:r·e t.imi:= anC. s equencyliui tetl may be
c:il(i ,e
)
a nd
inTerred
frare F:.g . 21 . 'rhe
sal ( i , 9 J var1ish
'tlbl~h
p1,1.l.sen
oi.:tside cLe t ime iJl~erYal
-~ § e ~ ' · Thei, r' '1i;;ll5h -F·ourier.~ t..rr-1.n s;~.. Ol ..Q:!'.> VP.!""~sb outside
-1;he seque:1cy intervals - { i-r~I ) t:? u ~ +( j +1 ) or -i ~ u ~ 1·i .
!ienc1J , any timr:- .ru.ncG i on f.( 8 ) con~iating o.:' a finite
!1UJ!t-
ber o:..- \·/alsh. pulses is tirnCT ar.O sequc1i<;;y-li1:Lited :
I
l•' (D): a(O)wal(O,&)+
2:; [u,( i
) c al( i ,S ) + as(i)>ml(i,9)] (1111.)
i* I
weJ!0 , 8) • c'1l(i ,9 ) a sal( i , 9) - 0 1'nr 10 1
Let F( ;J
J hc.ve t<'>e
'f" I =
>;
- Fou:·ier Lc.·ane.fcrM G( u ) . It holds:
(115)
P( 8J • 0 for 191 > •
G(u) • 0 f or 1u1 > 1+1
The 01·\ihogonnlity o ~
Ci
::!:ysten al' .:·IJ.llct.ionn is invariant
i;o r.hc 13enerullzed FourieJ" tran~l'oL':JJ and tt-,a~ includ.es che
'v,'elsh- Fourier t.r·a11sf orm . l-l er.ce , one mu;,· t·:""i-c !'! G( µ. ) e~'-:"pli
citly , i.t: Lne 1~oef'Ji.ciem;" a(O) , a.c(i) and o. 5(i) o f the
cxpimsio11 (11'+) arc know:i . Let e;(O ,u ) , &c (i ,µ ) filld g5(i ,µ )
denot.e tne W&ls;i - Yourier tr.·auofo.L·ms of wa.: ( o , e) , cal ( i . EI)
and oaJ(i , 0) . One obtaiu s tlle tran~foL'm G( µ ) ol' F(S) :
I
G(µ ) : a(O)g(O ,u )
I: [ ac(ile;c(i ,µ )+ a 5 (i)g 5 ( i,u ) ]
i:.O
(116)
1 . 33 EXAMPLES OF rC',JER S?EC'IRA
The J°un<.:Uon~ i;(C , uJ, g,(1 ,L ) , e:cC1 ,u ) , ...... gs(»,u) 3::-c
sho'dn lr.o ~j~ . 21 , rrconc co~:1i:t.1., 1!.ue ... 1 , 4 , ; , ... ~=> . O::f?
!!.tJY ren<lil.V infer ttr. ,-1rnpe of E'cCi,u) und ~,(i,J) :or
larger
v~.i.1e
of \ .
"" 11'1
_._~,_._._~...__..-!
oi•'19
- , '
1 ~-~r
_,, . . ~~~._._~. '. _.l~j.T_._
.. 1
11
_,_··.._
.' ..
_ __,_
..,,I '
•
I
-:.,...--.=..--..•
--.~-~,--,~
. -~..
JJ- -
-
tf.totft4Jf
".g.-- :o.ijr--:. t-t--t· -·· s
,
lb
l- -
Fig . 22 (lal't) Walch- Fourier trar.sfo1·:t1'< C( µ ) or r.hr c;i;:e
W:ld co~,ina pulr;es de:J.~ived from P~.o ol ett-enl.s or iiig . 1 .
Fig . 2j (r1 ~ ht ) corJ:'!.icient~ 01' t he expnncion ol' t lte pe riodically co 11 Lluued '1i.n e and cos1'1e el e:n'lntr. oJ' fl5 . 1 ir> a
series of 1-eL'iodic Wa:sh ~unctions c01 l (i,B l ""rt ;.;nl(i , e, .
Fig . 22 :"l10"r.'!1 tho ~..1 ~l.sb-.Fou:·.:d11 t.cannfO?"D!l o!.' diJJ~ anc
cosine p,ulnr.n 4;hnt vanisl out.s!de che intcrvn- -1 ~ 0 < ,- .
One may read.i:)· se~ bo.: the or:;hogonn~ity o ..~ t.h ~r~..!)s
!ormed tunction!:" i:; rrcserved . ~ig . ~; ehO'An the coefficients 11( 0) , e cO) ;ind a 5 I i or the expo.n61<>n o;· rc1·.io<li.z
sinf' and cosine !'unctions in a se.r!es cf i crtorlic 1.1a: :i~1
func ... ions . The ba.nd .::>pecti·a of Fl g . 22 ,1;1rt:o 1'Cl 1aced t~· line
"J>ectre . The analogy to E'ottrier t.rano rorci of a puls<o anc
Four.iar seriea of t he cor rospond i nt; pnt'ioclic fWJc Li"" i s
evident .
l'ig . 211 allows the frequency power epec~1·a u~(v) 1 a~ (v)
= G1 ( v) + G1 ( -v ) for the .fj rst i"ivo sino and cos in" pulses
? . DIREC'I' 'l'll/\NSMISSIOtl
60
of l<'ig . 9 anC. tbe block ;iulses of l"i g . 3 . Tt:e 9J.'Ca unc.leL·
t lte CUL'Ves mulciplied by T represents cl-.e energy oI the
s i glla~s .
The
cui~ves
i!l t.lte 11Jhole ince .rval - co < C <co a:r·e
obtained ·u}· co11cinui ng t.nem a.s even
inco t;!le
cont i nua:..iori is of much less :t_t.erest
funct ior~s
inte=vnl v < 0 . '!'hi~
for powe r spect:·a c!::ia.c. for the Fou.r:ie r· t.~ar:u:fcrt1 G( v } or
the Wa Lsh-Fouri e= t r il:lsl'orw G( u ) , since 1;:,ey a2·e alNayA
e ven ru.nctio:1s .
2. Direct Transmission of Signals
2.1 Orthogonal Division as Generalization of Time and
Frequency Division
2. 11 Representation of Signals
Gon::1ide1· u L"e:ee;L·a_µb;v r"l,lµhtibet containine-~
·~ er
e fini. te
num-
of cha.raci;;ers . Im examp:e is che r.e; <?t~To alphabet
having
chara<;tcr::o . It i!i u.su a l
j2
se-c-s o:.. 5
r;oeffici~nt!"l
~o :-opr~sent
theo by
with V:":tJ ue +1 or- - 1 :
character 1: <-1 +1 +1 +1 +1 -1
character ? : ~1 +1 ~1 ~1 +1 - 1 etc .
[n e;:e!lernl , tne cLlll'UCt..ers lllU.j' CO!lSist Of Gets Of m coef -
a ....,,r,,\ b f,.,\c
z· ·I
~-\ '::
{
t
I
-:.
~·
~
'·
:
I,,' "•
JI
( v ·, :.\ .d
:
''\ ·,.,
:'
~
·"'t
'"'-'
to
j
0
£6
........ ", ···.:~:?"-'j.!....·r - · -- · - •
s
6.,_ 7tt3
w ;ti J3j
1.0 f[Hl)-~
t
:?"15 . 211 F.cequen.cy r10~·1c~ spcctrn nf{'V).t.a!('V ) ~ G1 ( v ) 1·G 2 (-")
of the sine and cosine puJ.seo o;' Fir; . 9 . a) J:(0 , 0) ; b)
!'('< , 6),f(2 , e) ; c) r(3 , e) , f(4 , 0) . Curved is ~he frequency
pow~r spectrua:. of t.l1e block pulsen of Fig . 3 if the;y· !:a.ve
Sive ;:;in:c!i t;he ece:r·gy of Lhe block puloe a~· Fig . 9 . The
fT'f'!q_ucncy scale i!1 He::-t:z .holds i'o= T = 150 :us .
2 . 11
~r:f•RESE:~TATION
OJ'' S I GNA.J,S
ficicut.~ >:1;&v1 r.~ ar!.litrar:r valv.es l'Lil.t@r than just the ~ta1ues +1 or -1 . Tho; follO'-•ir.g Lotut.io:. is appropri'1tc in
-;his casd:
(1)
rr.he T'C-_?!"t!:.icntnt i on of c:iaractcrB by ti:ne !'unctions .:.s
a.ootheT' import;nnc r·epresenc.at icr~ . Con!"liclcr m time 1'urlc tious t( j , 9) . Let the l' unct;.ions be muli;ipllerl by the coe£i'iCi•··11tr, "x (j) 'lnd the products b~ •Hl<loc'.l . Cnr. obtains
tJ1c rcprci;r.lnLu LioJJ o! the cbaracter x by Lile tlme J wiction Px(e) :
111~1
Fx(B )•
2: •x(J)l'(j , e)
•••
(2;
~e
eoef!'icients ax(J) na_y be !'~gained 1r,dividually
froo Fx(9), 1: •he sysi.e:n of fm:c~ions (f(J ,e J) is li::early
independen~ .
he ;>roce~s if'> particularly dn:rte l: the
funetionr :"Ir" ortnoi:;onat. Let i;nen: be ort~.ogon-.l and ::iormalized in the intcr•ral -! ;; e < ' . T~r. coafficieHL a..,(kj
i s obLai!'lcd ty mult:ipl;;-L"!> Fx(B) with f(l. , 9) and intc gra"t;ing the p1·odui.:t i the shortor c-xp!'<J:J3iOll 'co:-:i."'r.lnt ing
F.\' (e) wi~h .C(l. , 9) ' ie e;ener ull y usod f oi· ~ :,i~ p:·oceca; :
"'J Fx(a)f(k,9)<10
-112
m -1
•
i •:I
Lot
!I>
lf2
L; "x<J) J
f(J
, a lf(k ,a ),10
ax'k) (~)
·1 11
equ11l ) ; let a,.(OJ , "xCc), i;x<3} equal ~1 =d
Bx( 1), a,,. ( •1) e<;u"-'. - 1 . F x( e ) hn" thcr: tho !l:i~re sh.om: :.n
Fig . 3, i!" the !"unctions f(j ,ii ' axe the block pul10ee of
Fib · 3 · F..,(e) represents vo2.ta._i:;e or Cll.!'I"ent or t~e asli.:L.
teletype signalo s:- f·mction or tirte .
~e values of tne coef.:~icicnt!'.1 ax< J / :.rr:t..r.scai~ ted. by tt..i-:
signal Fx(B) of ?i15 . 3 =Y also be nttain<:<.1 L,Y «lll>!H1.t<le
snmpline; nt proper times . lier1ce , t;hn cc r:r1n t ran:;rninsion
by time multiplox or by time di.vision Ill'•' u •ed .
The block pulues of Fig . 3 moy ~• l!lo l>e frtai·pr·eted as
frequency functions f(j ,v ) . The cha1·11ctor x in then represented by Lhe f!'equen.cy f unction P x(v) . Ir Jlx(v) is nppliad
I
?
:o
2 . D!REC'I' 'i'l!ANSJ-JISSION
5
~uita.ble .:-.:~eq~er.cy
~he coeffic~errLs
ba...."'ldpass .:il te!"s t one lil~.v recover
by Sl!Jllpli'lB the ou:pul volL"t;;e.> of these
ri ltcrs . f:-equenc;1 tLultiplex or frcgue!l..:;; lllvision are
usual ;;arms J'ot· i;hlo; Ly~e oI trani:md :'lnion .
llecovery oJ' the 'l. nw~mitte<l cOEd' Ucionli; 1i;y Sillllplillg
in Lime or J'roqu"l ncy domuill without fUJ'1;hcr com1,.,t a t i on
i . not po:::siblc 1'01· Ol08t syscems of orthogonnl funct i ons 1 .
Recovery !:>:; ~cnns o! ~JJei.r or~tbogonAlit;y i.::- gl"·a:,·~ :pos3ible 3CCO!'Cing :o ~) . The Le~s orthogon~l civ~sio~ or
ort!:ogonal !!ult~p:e:x /il'e upprop::i!n;e in tl.i~ c:;::e . 'fi:u:
1:1dv.9.l'lt9;>e of or;;;hogonal division ic ~t.flt the r:·=1eo' of
u.r.cru..1. systiett.s of i\Ln.Ct. ion a is much lnrgct· t.hw1 for t1in.r::
or t'1"·cq11 cncy divi~l(H1 . Hence , t'"iet'e is ·no:•o fL·eedoru of
choice f or the be ti~ ~.YG~Oin for e par t icul w: upµli~at ion .
'l'hcoreti ca..l ir1vest..igstion s irequc::c!y t·etµl.\eoent cbarnc'ter~ oy vecc;o.r·:i iu ~ :;ignal npnce . Hot~· .i ... Lt.is vec~or
reprei::ei:tatio!! relEtted to the re_rresentttt..ion
~y
ortr.ogo-
nul fu.nctions·. Con:1idcr ca-dicensio::a:,
r. ::-az.E. • a!'" carLesian coordinatf·n ha·1:.n6 the, ..-.; v ·~·ec:.ors- e , . ... t1P. lcnglih.
o.f chese vector:: C'qunl. the ictegral of tt.(! .·qua.re of the
or-i:--honorr.ial :'unctiontJ f{ ti ,a) :
'"
J' r'(j ,9 )d9
=
e,e ,
(4)
• 1
..117
Ihe seal~ rro1uct o~· ~·o 'lectc:·:: e ; •1.nd e "' j # k,
v:a.nisl..es sir.ce i;hey nr,.. rer;ie.: :dicul·~r t.o et1c· othe!" . Tt. ..,,
colUlect!..on
'::>e;;•e"'~
sen:;ntion iiay
-chu~
orthocoual funct!on a.ntl vocto:- :'."eprcbe cXJ:1''.!s8etl ':.Jy che or:t-.oe:on'1li~y re-
ll,ticr :
'II
J.
C(j , 9)f(k , 8JdS • e , eK
c
l>Jk
(5)
• II ~
A
c!l~Jr;"lcter x
is roproccnted by t!.e vectcr Ft' i n s i gnal
1
1"i0re tn3.L one F<l!lrlitude sumule is .,hen needed to cozpute
r.::.·oCess is , howcvc.r , a Jnnthod t.o
ColOpute the integrRl (.?) a.nd thls in not what is generally
1.Lnd.erttood as cimc or .t"t·equency division .
~Ile cocfficier~i;!i . -Such u
qpnce :
....
"
f' ~ /~
I 0
.
& ,,.f ") e
(6)
Itl~tcad
of" orthoi;onal vector::: one mny nlso uttr.- c til:early independent vectors . '!'his :-ep:-e5entation l~ obtL'll'd, tr cne runction,; f(: ,3 )
11ne11-r:!.~t irJt.ie~,eoUt'.!n L .
Aa " µr~ct lcal example con,;i.,10r q t:olr.ty-pe charac ter
F.,(IJ) oou.:.. oswd of 5 o;ine and co:;ine clcmont., nccordine; t o
.Fig . 1:
~/Olf(0 , 0) + ax("'{2si.a4ne t
"x(2)'{2coa 1+r0 +
(7
~
tu;,:.i.,:e "' .,, .:r.R :..r tts- dura\;i0n cf a t"let;tpe
i~
1.50 ac ,
w-~ich
in 9
lH.i.C:t-u~~c ~tP-~d~rd .
Th~
charact~!·
coe.t"fic!l:!"nts
a,.(J) ar:u +1 or -1 for" balanced •;:~~""• unJ >1 or O for
an on- oft" ~yotcm . Let F,.'3) be :,.v~l1el at tic r"ceive=
simuli;nnr,-oualy to ~ multiplie.es whic:1.11lllt:i!'l:; 1''v(E J wit!:
oh~
·
val
-i
.func~ •Outl
f(v,S) to \{2 cos c•n9 . '!~.<' ou LJ'••'· vol~u.,:,es
of the !i mul t. .i.?1.ieL'S are tnr:egra.ted ct•.J"'i:"IC' ";;h~ Clu.e interti 9 <
0.
'!'lie ouvput voltngos of t..hi.: five
tors roprr1no11L l;he valuaG of t ho coc!~!'i1.;iut.Lu
LnCee_:J •a~.ix (J)
of
{7) at tho cio.c> ~ = t . Fii:; . 25 ,;how• cscillo1.rrou.r of t:-.e
output voltngc-:: of the 5 in~egraL01·s <lul'1n~ t.he i ntcrval
-i ~ 3 < i . Tb~re are 32 a::rfe!'enl ~l'aC~S :o:" egch of t:.e
5 output voltoges Cue to tt~e 32 c~sr'lcr.ern or th(! tale!;;~-pe
alp?:sbets . 16 truce<> t'each a fO:"itiva valt:o ( •1) t"<>:· 3 = !
and. 16 n nognti-:.~c- val:ic \ -1 ) . Tr.:!.n ir:~t:.c:'ltos o balanced
-Cele~yµe £iyotc.:u . In ru: on- o f±· .;:-;,·::;tf:'c., '1', ~t."nc.: c .-.::>v.lJ. ~s
SU!le the value 0 instead o.r " ne15:.it.i.ve value i.;; a - t .
'.!!he apparent 1ack or syff.mec:ry ·oe t;ween lhe Ll''..lces '?ndi.r,g
al +1 r.md those endi.ng at - 1 in cn~!:nd by ;-in t1r..;i-1or:.P.L
signal \('2 sin 2ri 0 addeu to the chaJ~"c L"~·e f.CI!' '1;11u:l t.::oui;ou~l.on . The elomontu '{2 nin 2n9 and >[2 co" l'n9 dn not arrear
in ( 7) for LIU.a l'ea<ion.
2 . DIRECT TR/C:lS!HSSIOt;
"'ig . 2;, Detection of the coefl'i cionts +·I and -1 by cross-correl ation
of 52 d i ffer ent
relet,v1~e
sit";nals
composed of sine a:iC. cosine pU:i:len .
Duration o: the tracer; T ;:
1~
ros .
A.11 three O.iscussed represem;ation~ of s i.g na l s contain
t Jle coefficients ax( j) . ~~he vec-cors e ; permi t -che raprese11tat ion of m coefficients Dy one vocto :r Fx, th1; time
:·unctio:1s I (j , a) the repres;cr:tetion o;:r ct:;;;? 1;i:oe func tion
F e ) . !>ome s i gnals ' s•.icb as the output •:ol'tae;e o r " miCl'O-
x<
a.re usuall,y uva.ilable us tirne funct i ons . 1J'heir 1·e:presentaci on by coa.fficient.s 1...1111 be discussed i n 2 . 13 .
)J~'10lle,
2.12 Examp les of Signals
I•'tg . 26.9 shot..lS ti... 0 c'.:1aractern
F0 = e 0 and F1 = - e 0 in
sign.::i.1 .:;pac~ . ~1hn ~a.rr.-c cbar act-e.i·s are shot·t n bel o w as time
:"1mctioi:.s ro::- the 'olock pul!Je f (D , S) Ol' the Wa l sh pu1se
1
sal(1 , a ) :
Fa( 9 ) = •f(0 , 0) , £" 1 (0) ~ - r ( 0 ,0 )
or
-so.1 ( 1 ,0 ) , To',(a
+sal( 1 ' e)
2 . 12 r;xAMJ J,J·.8 01· SIGNAl·S
..
:j~
,,·f·..
J
, -9:r
_
CJ. <>
-9:r
iJ1
LJ'
c
.b
.~
Cl 0
•.);:>
i=i-
2
-i:P-
l•+Oo Go 3'
d
CJ. 0
.-=..
·'I.
-9:r
' ""1::r
~
' -d""
L:T J i:P
(!
e,
·---=r,r-'o
1•
)o
60
S
0
24
1So \lo
'11,
lo
.cc:io.D_ ~ • .D._ .c:ci o"°-
rn.•~ J : : ] _ ,.r::=::t
i::P 2~ -9=r ,_D.
LL.J
..c:i.!.!!!!
1
~
rl=.1 Cb.
G...:,-2..-0.
Jo- -~
3-'=b- -9=r>-°
- , __ .=CL,Cb,
~ wal{C,0)
~sal(l,S)
4tr"°Hi.vr-o.Ti
Fig . ?.t L!hnrac ~e1·s represen;; c·d t:1 poin:..::1 ill ow an'1
dicnnniono l !'" i~nol sr-nce+-5 and by ti~rio fu.nct:lous .
Figt: tors e 0
the en r
~
..ud r silo~~ c~ar-ac- cr~ con ... t.:,•l..i.ctit:i .f1"'om vec I e ,, or fro=i: t•no :·~c'tic!'. ~ . '.1:-it.tt.in i'!ld~t~ilt
... r .... er l-ig . 26d have the .:'""o:l.,, 1111' ··o ...·1:. :
d,
l!"
hi~
Fa= e 1 + e 1
,
F, •
e0
-
e ,
F 2 =- -e 0
• 01 ,
F
or
Fo (e) . !(O , e) + :(1,a, , F, (e) = :(o ,e) - f(1 ,e J
J.', (9) ·-l'(O,O) + f(1 , e) , F,(e) =-f~0.9) - f(1 ,i! )
or
f'o (e) • wnl(O,eJ - oa.1( 1,e), F 1 (S) • wnJ.(O,e) • uoJ.(1 , 0 )
1' 1 (9)
- wa l (0 , 9) - sul(1,a ) . F,(9)= -w al(0 , 9) • au1(1 ,9 )
2 . !JD\ECT TF.AllSMISSICN
'!'ho fut:c&ioc.s i'{O , ~) , f(1 , B), wal(O,i!) und sa1(1 ,8 J =~
~hown belo·• F'ig. 26d ; t>ie charact.:-rn t" 0(e) to F , (e) coopose<!. o: Zhese func:;iocs ~re .... !,own above the:o .
..
,/
2'1"" - - \. -s>Q
'
,
\
\
\
I
/'o
3
,
b
c
,~,
1
\ 0
0
1
oO
• '•
...
I
u-rsu
~e>.D_ wal/0.6)
f~
%
-6--
-sal(l,0)
-11r6V"z .;;;--tlii
:'ig . 2'/ Gl1a!"~cters represen ted ·uy p0l nto io a two- di..vJensionnl nign~l ~pace nnd by ;;ime f11nct1.ons .
Th,. te:.·raz
pLe.i to the
lJin~y ,
tern9ry and q_uru·teruu.ry aay be apof Fig . 2»>, aince the individua~
'.lector. or .!"1G1c-:ionG arc mul::iplied Uy coe:r:.cients that
fltl!1Ul':'\C' 2 , j or 4 :'lifi"r.-rent v·alues . Flg . 27 shows that terms
like 11 bin!11"y cnnrac~er" .s.r·t ,...en.ers.1 . . . y not appl i cable , if
n c:1n1•ncter cousist:s c :.. more t;hnn o ne vector or f 1.u1ction .
F'ie; . ."'7t1 <>hows tlte ch"t'eo c hnroctartJ <Jf u ~ocal 1 ed trans01•L hogonnl nlphabet; . The c h aractor·s 1·ond in vector repreaentu~jon ai; follows :
c!l3i'ac~er.:;
~
-·12 ElW1J'I,J:S OJF0 = !'f3eo
80
te 1
,
F1
Oe 0
--
is mult f • i d by one
or-!(;, e ,
.~
6'/
SIG:Hi.LS
-
::.r
1 e ,,. F2 -='
-i•l'!e 3
..._
tt:J·· -:nrf"-. <"OC'.rf ~' e·
j:e 1
~
f {',
0
1
or - 1 . ! :"
tbe vect:or~ e 11 ruid e 1 arr:- rotated rol&.tiv<; t;O the sif".Jlal
points, r<';>!"6:"Cnt~raonn are obtained tl'st have tt.ree J.if-
<>nc cf "he '"'" coefficient
ferent coc.f.!'J.cieute for ea..c~ vecrJo.t·, o:· t.\·,·o c.:.iffei•enL coefficients fo L' 0 0 anu three fo:;, e , . Signals compor;ed. of
tha .runctiooa f(O ,e) and f( 1, o)orwal(0,9) uw.: - "al( 1 ,a)
a1•e shown below t:he veccor rnodel:
F 0 (e)
• !iv,rco,a)
+ !:'.\1,>) , F , (e} - -r(1,a),
F,(a) - -1•t'.if(O,a} • t f( 1,S
J
F0 (a) • tlf3w11l(O,e , - 1sal(1,9), F,(e) • •sul(1,9)
P 1 (9) •
-11::Swe.(O,e) -
!:ial(1,~)
Fig. 27b shows the fou:thogonnl alphab •t :
char9c~e:;,s
o:
a socAlleO
bio~
These clwracter" look very sioi_n.r• to tl:OSf' oi: r'ig . 26tl. The
similari l:y dlsappenrs, il' the chru'act d!'l< llL'e cou1,oser.:. o!'
more than two vectot•i; or· fW'.lccious .
The d11oho<l I inoG i.n 1''ig . 27 show dist9ncon betwocr. certain signRl pointG . All signal pointa of the tra.ni;;or~ho
gonal alphabet (f'ig . ?.7a) have tho <:a.mo distnnc<> J'rom e<ich
other. The vectors froo signal points 0 to 1, 1 to 2 and
2 to O are F 1 - F0 ,
F1 - F 1 =d Fo - F1 • ih<> square o!'
•heir length equals 3:
I F, - F, )' • ( -f'{::S eo
( F, - F, )' • ( o'f3e.
+ i e' )
( Fo - F, )' • ('f"5 e o)'
:;
- t
e , )'
.
1
. ;,
t.
i.
- ~.
4
~. :;
4
Ir th~ charcicters are represented by funct iona rather
i;h!Ul by vectors one must replace scal ar products by ·~he
integi•aln of tile products of the re11peci1vc fLlllctiono as
shown by (5) . It; follows :
••
2.
llJ
"'Jl - i1[3:'{C ,3 )
Pe ( e ) 12 dfl
) F1 (B) -
• 112
prru;•"'r
rRAKSJ'i!SSION
- .:i'·''1
.-. \
,. 31J'
, ~
r.:.·a
·'
i.r(1 ,ajJ' ae
3
2
-111
!'[F,(e)
F,(e'] 1 de
..
;[ tv:'r(0 ,9)
llZ
-111
• t(J
!.'2
,.
II 2
j(F 0 (9)
'" 2 (9
) j ' cl0
. 1/2
i)f(0 , 9)J' d 9
[ 1
3
· 112
F 1 (a J - £' 0 (8) i s ~Ile func.ci on that muse bi= o.tl•ied to tna
character F 0 (0) ; n order i;o obi;ain ~he chru:·ucter F 1 (0) .
1/2
J [F
. t r1
I
1 (e)
- F 0 {0 )}'as is tho cncre;y required to transform
c:1ara.cte1' }' 0 (a) !..nto char ac ~'"' F , (a) , 1-:. the integral
112
J1;i;(a )de iE; 1::-ie ez:ergyof t:he cnara.c7;eL' Fx(e ) . The squnre
- l/2
of t;!1e Ui!;taLce of a signal _point :·ron- t::ie ;)rigir represcncs the onerg;'l of tl1!:! t ch!:l!·a.ctei· .
fo'ie; . 28 ChP.ir.ac1:er:i -rcpresentod Oy poi11ts in a. th.reedimensional s i g:1a: spt-1.ce .
•
i
'- 4
0
7
J'-..
c
'-
<
F:i:; . 28 shc'.'H:I chara.ctcr!J composed of t hl~ee vectors . The
sphe.r'el:i .represont the signP.1 ;>oin-c;s . ~ 1l1e rods bevween ·them
represent cbe distances betwcet? adjacont po i nts . ]fo unit
vectors e 6 , e 1 and e z are shown . Norm ali~ atio n is dlffe!' ent from Fi ti;s . 2& and 27 . It io chosen so that ·the d i s t..nces between adjacent ei51.11;l points iJ.L Fii:;s . 28a , b and
c are ec1ual . 'J'he values of the coefficient" in vector space
depend on the oi•iencation of the Wlit vec·l;ors e I . F'o r in:;tance, the f our c l1aract ers o f the transorthogon a1 a l.pha-
2 .1 2 EXAJ·IPJ,EO 01· SIGtlA.LS
..
·:-r .:. e
+i e, t !.,r e , - .!.ir
'l' ;:;.. ' ... l
e
oe, - i\" e, - l.Jr-..":;
•z\ ~' _,
~
F,
,,,.: e - i..'-i.'' e
F, --t e • • .~ I
rz' ,,, 2
oe,
oe
F, •
+ t 1rc{; e 2
0
Fo
;
'The energy o I" nll !°Oil!' char"c Le~·s .i.; oqua l:
"
1J
'I'be di1ltnncel' between Lhe .'.'our d ionsl
po.i.nta n:-.-·
sl~o
equal :
1
F0
-
F,
'
1'
( F, - Fl •
..
c
b
r--...... oA
.Lo_.__
o.C:=::t
1
CF"
• 1
ri.-.
'L?
i=R=-20
r::-i
c::::J
LLL1 0 ~
CCJCJ '.JJ--.=
--~Cl~ z OL_JO
9=P2==-=°
--.-o-. 3
9=:cJ-3~
CJ
,...,
w w
LJ
4i:=P
0 CCJ • 9c:J="
~LJ~-
Si::==:J"
-r::Fbs=A=o
-.,..u-.-31t:P-
i:::c:P-6....,__,i='
..c===i. wol (0,81
I
I
I
• 7
cr-.. . .
r=:-iL::J ... ar(t.iJ.
I (Z.gj
0
0
Cl -col
c:::I
(~8)
Fig. 29 Charnc te1·u of Fi.,; . 28
Fig . 29n show~ e representation of Lr.enc four •·hor:i" tcr.,
b:rth.ree block pulGe:; f(O , e), .f(1 , 8) and ~(2,a) nt ;;cl.
asby threeWalshpuloe" wal(O,a), - sal(1 , e) nn11 - col(1,D ) .
In the coee of tho biortbogonal alpl\fJbeL 01· J!'l,; . 28b ,
it is i·easonuble to orient tho coordinate nyntem tio that
t'1o opposed aigrial points are located on eoch llxir, . 'rl:w
roll owing simple vector reprGsentation resultn:
2 . lJ!H:.CT l'J'.A:ISJHSSIOJ;
'Iv
r. ulses or t::r ee
The ae characte-r!' compo!::ietl cf three blo..::Y.:
Wnloh functionc nr" iiltown i n 1"i.g . 29b .
The charactern or the a l phabec of Fig . :'8c ••ay "b G wriL ~on ii_ a purcicu I nr·l:y s l mpl e fo~·t:1 , i r thr- :>:<c s of the coordinate s:,,ste:n lntnrnnct the Slll'f ~ce~ or tr.c cuoe at. t~eir
conter.s :
Fo
-
e,
' e,
F,
- e0
eo - e ,
- e,
F,
- e o ..
e, - e'
- e,
+ e l
F,
- eo -
o,
F, = eo - e I - e,
F,
eo
F,
-=-
F,
=- e -0
~
i
e
- e0 - e ,
• e,
I
o,
- 02
.F it; . 2<:.Jc s!'loo.;s tLAHe CLfil"octers conposed Of three block
;. ulr;e5 anc t!:u·et- ·.·.'•tls~ ru1 ses .
'rhc i:erspicuit::;· or :he vecto1· .r e;;renn:-lt:1tio:: iti :ost,
ir tt~c cll:s!.·act-:rs conr.l.zt of :!lCI't :;tlc...91 t.t.r~c vect.ors . 'rhe
charac~eL'S o.:.. 90??~ alfh3te:..s may -r~nC.ily be sveci;-ied !'O!'
L'ou.r· 01· more vectorc or ftui.;vion!'i . ·rni: .i.s t.ctle 1 e . g . ,
fol' Lhe cr.aractct"I of Lhe biortnogon nl WlO t he bin=y alphube Ls o f J"ir;11 . ?9b nntl c . Tr ansoro hoe;o nul £1lph'1l1ets al 1·ead;y req"llir0 conside1~a1>le computnt i on . o~·.o rna;y cor.ipose
.:t111 cl~aractors of !1 t.ra.n:;ortJiogo!lr1:. blJ llnhct !.":-ou. :n f!J.nCt.ion~ . The!;o :o+1 cJH.il~acter~ are s?eciflcd by :n{c+"1) coe:"f1cicntz ~x(j ) ; ~. o..... ~-1, x - o .. .. a . :he ~c ::owin6
ccndivior:s are available for t:iei:· co:r.:.iutatiou :
a
:'he encr.t.y or n LL u.•1
utul. ettent:
~tiol
character~
is eq·,1a.l . This
ctn m cond i t.ions .
b) 1l'he dint!l!!.cen Uetween t he mt-1
qual . Thc::-c :.u:e u.
1
cL~ractr.rs
ar e e-
(rr.- 1 '.• + (m-2) ' .... < 1 = }m(m-1 )
distRr:C('Z betwcf">r. rr.-t1 c!"la::-acters .
t. totli o:!" i:n(D-3 ) "'qu&tions '!!'<> nvailoblo ~or the detera.ina-;.ior: or t~e ::n(m+1) coef!icie::ts . ;,, consitlerab~e nW!lbor o:· coef:'icicntz co.n be chosen freely or fixed ':>y addi t.lonat cond.i t;ions .
2 . 13 AJ1PI 17'tJ!JE SANPLI'.'IG
71
2.13 Amplitude Sampling and Orthogonal Decomposition
The nWDpl i.ng Lheorex of Fot;rier P-'181 :;sis
:;ta;e~,
thn"t:
a signnl consisting of a saperpoEition or p~rioc!ic sine
a.aci conioe fllOCC"ions sin en rt anC. cos 2nft .,,lLL Irei:tuencies iu the interval C ;: f ~ t.f :.s complete:y dete1'11'.iinec
l:>y 2llf lll!l]llHud& samples per second i f t:.t h meusuL·ed in
Hertz (1 .. GJ. 'Pl1io srunµl i ng i;lteorem l1i;a be: en t(eueralizecc
by XLU 1/AN.EC ror other compl ece sys~ems of o rL ho1~o nal functions [7J. In essence , KLUVA n,,.c ' s swnplln.,; LLeo.ce:n sta Les
that II Sign&.J. conzioting o;- a m1perpoc.i t (On Of funci;ions
fc(<;>T,t/T) 8nd f,(;n;,t/~J with neque11des in t~.e incerval
O 2 cp i llcp is co~.; [P~,.1~· dete=in~<l hy 2ll:p ·i.~.pl itudn !'arr.plea per second i;· 6:P i,; o:casi:rcd in Zf'" · It "ill be "ho>.-,:
in this section, ti".e!; a:lp!.itt.:d(' ::-ampl i.r.g or n frequi?ncy
limited signul i" 1Latt:.ecr:atic..-.lly cquivnlt!r.t ;o its dccon:position into the incoi:q:i~ete ort!.ogonnl ~ysLcu r~:-l;~;~·i
'· I
j • 0 1 %1 , ~2 , .... The co.rrespor:.ding retsul t fo1· sequenc:..r
limitod nii;ncile coLtposed of Wald:. functioris will be discussed lnter on . It turns oui:; tc ·u e so ain.ple ihal; it is
evident without cal c ul nt i on .
A i'roqu ency limited signnl F(O) with no r.om:ronontc lrn ving a normn 11.zod I1•equ ency v = IT > ~ <!Ill~; b<' "Xplllilcd in
a series or orthogonal functions t.l:ut vw'i' h out~lde the
intenral -1 ~ v ~ i . Sine- cosine pulses , ·1..'ultL ;;•ulses,
Legendra polynottlals , etc . are suit&b!e f;rnct.io113 . The
following ayotea of sin~ -cosi~c rul :-:;on \o::.11 be used , vhe
ph~se flllgle tn being introduced to sicipli;'y t.l·n ::-e:;nlt :
e;(O,v)
: 1
1;
g(j,v }
~ '{2cos {criv+*n)
g(2i , v)
[
(5)
g(2i- 1 ,v ) = y2sin(2rilain)
S(J,v J • 0 for v >
j
-
0' 2i ' 2i-1 ; i
t
and v <
-!';
1' 2 '
~'he J•ourl.ar tL·ansform G( v ) of a signal F( O) io e:qianded J.n a serioo of these pulses :
'?2
..
Lr
G( 'J ) =a ( 0 )-
ll (
2i - 1)\f2sin{ ?.n i v..1.* n )·!--0. ( 2:. )'/"~·co::;( 2n:. v+~T" )]
i• I
a (C) -
'"
J G(v)dv ,
a. (2i) =
"'J G(
-1 I
\I'
)\i2co:1( 2r: i v.:.:\-n )r.1v
c
"'
o.(2i - 1) = ~- G(v)>[2sin(2niv+tn)c1 v
(9)
-~n
'J.1l1e
inverse Fourier t.:ransform yields 1•'( fl) :
1"(9)
=
"'
J
(G(v){ cos 2n ve +
sin 2n vo)a v
•r11e surn (9) i s substivutecl for G( v ) . Kee;>i::ginmind chat
G( v ) is zero out!;iOe t!:lc interval
F(S)=a(C)
-t
:!! o.; a ,
onr. obtai-!lr;; :
sinrr6 +~[ ("i- 1 )sinrr(9 - ij ( 2 i)sin n (a,+i) ] ( 1 0)
n6
,~ "- ~
n l8 - 1) +<I
n(o +i)
o: I
A :'r·v:quf!ncy li:niteC: s l B!tal F (0) n:l~l t}-_1.ie ·o~ .r·epresent~ci
by e series of tbe incompl ete or~"Of'IOHal system o.[ si~ x
fu nctions . 11; follows fro:n section 1 . 4:; thtu; ~hese function:;; are ori;hogona 1 . One '.ll!!Y pr ove i1; :lirect LY oy ev<J.lua'(;:i ng t!1e integraJ
~
J s ix.n (9·1·k)
_.,.
n( e+kJ
k, j
ol
sin'T(o~.J) d~
n(e~j;
=
6,,
(11)
±1 , :1:2 , ...
~'lie coefficicni;n a(O), a(2j - 1 ) anc a (2i) or (10) may
be obtaineO -07/ n.a11rpl ing tile amrlitud~ o.: the aignal .F{&)
nt ~hn times El ~ t/T =- 0 1 ±1 , ±2 , .. . Fo1· i o.st.ance , all
sin19n(9•\- i} ~•-';,
pn-"
stn n~il+l.)
!ll'""
9 = o
l ·un. ........ t-->on~"
. ; .a.--"-'
l
l
iJ
,,;; ,,e~o fo.,..
~
and
sl n ne . r'
- i ~
4
n +
1 . Ee:icc , ~t holcls Jl'(O)
,
.
.._
rre
i s
= a (O; .
11' follows from ( 11 ) t!iat tlce coefficient s a (O) , o. (2i)
ru:cd o. (2i-1 ) :nay also be obtainecl by ortJ1ogonal decomposi-
tion of F(B) by s~:-x functions .
~
';' F(S'/ sin rr( 9+1)d0
J
n{G+J
-~
-!
cr.(O)
a{2i)
a (2i - 1)
Q
ll'(O)
fo.r j
•
0
l"( - i)
fO!' j
~
i
F(i)
for
= -i
j
(12)
-·14 CIRCUITS }OR OH'J'IIOGOl;AJ
'.>
75
DIV lSIO:l
'!'he equlv1dcncc o~ ncpl.C ~u<lt swi:plill• a.r.d o::-tnot;onal
decompooition i:J not rrstricte.i ~o frequency li:nlL~d :;ignals . Let B finite nuober of <!i'1crcto oscillations
A•oin211v.e una 11.cos.?T"v.3 >.-ir;h "•'>I hoeddcdto ?(a) .
An idenl lol»JluSS filter witt. cnt- off freqn~ncy v s ;. would
suppress the::c ndtli t i ona: ot:cl llut..i1:it:s, :.trv.i u.bplituC.e san;plinp: would ogni_n ,vielC. Lhe C<.l~f!it.:ient.r a( 0), a.{ 2i) and
n (2i-1) . OrthOGDn'll :l.,cartpodLio11 of Lh•· 11ew c it':I•"- F(e) .,
~lso
A,, ai112nv ha ·• Bh..:o.s?n 'V 11 9
t
n (?i-1) , ainco the
yi~ld
!'u.nc~ io1J~
;yl1"ld1.•
a.~C) ,
J1 . i;iil2nv 0 6 CJ.ml
;i ,
a.(2iJ 1:1.nd
<;oi;2nvha
no contributloa:
m
-
•
l(F(G)+A,,sin21v e-1;,cosc; v,9J
v-'
jij:L'
s iz:.n (" • ~ ) . "l?F( .,;u" ( a I
nto-J
iJ " ' ~.j;
OD
(1;
It
rel\nin~
to be
s!:·:i~-=
Jhr;.t
~.:aLtl::•.:ou.
Cw1ds cf o.scil-
lations do not ;field 'lil;: <:onL=il:·.i~fo::. cl c~ r . Let: 9 :-=ct ion 0(9) be <•lcl~d Lo J.'fS ~. >;).0.,:i. c· n••,iri; no". i.Lat.i.c::
witih frequency I"' < ~ - 'l'te i'ou.r1~rtrn.no.t.'o:·co .. ~(3, uu.::t.
t henbllzcrointltc interval -t ~ v " t · Cn the 0L,,c1• Laud ,
t h e Fourior
·
t.ronntcrrn
·
1
•
·
rirt(:J+J
r: "' ( h.1) 1t;.
·
o r t.1c
:1.1.!l~::io::!:!
zero
ou~aidc thia intr.:"Va. .. • The tl-:o ?ou r- ic1· t.ru..usi'tl!~ms a.re -c>hus
ot'thogonul ~o eai.;h otllet· and t:.u<~ :Jwu<~ •1111.· 1 lio J d for- the
t ime functiono:
m
J
-oO
D(9 ~ llin" ( 9 .,J ' d9 ; 0
,,
,
( 111,
I&+
'
'
2.14 Circuits for Orthogonal Division'
?ti''" . ;r
1
•"low1:1
u
b.lock d.it:.r,:ra?n _f or t.?':c t.ro.nu:tl.::;:~ion of
5 coefficientR e.t(J) by orcho;:;o:oa~ 1iv1Gic-n. ;, !'unct: o:c
generator FG gnnerate•; ftmction• f(O , e) ... f('•,e • "~ t',>!!
~rans0tiLLe1-, which are ori;ho~;onnl in tile int.erv<i: - ; - ~
t ~ i L The rive coefi'icinm:::s a,(0) . . . 11~('1) 're rerres.,nted by volt1.1gon , which h•Ve a constnnt vol·~a d Ltring ch"'
'See [1] - ( 5] fol' a more detailed dincu:•::iion
or
ci:rcuits .
2 . DIRECl' ':"n;.:1sr·tISSIOt•
intcr~~a ...
~;v
- i'T
~ t
<
; T.
the cceffici.-ntr.
.'l:e f.mctions f(.' ,e ) are O'.!C.:.L ip~i~
ax j,
iI: the
cult.iplier-~.
M.
'11he
five products axlJ ) f(j . ~) &.re added c1;1 '.h, ~-eei>1to rs ll
!Ind the ope ra ~iona I •iniplifier T.1,. 'fi1e l'<!~ul tint'. si gn~l is
transmit: Ced and cntO'rA tLti reco ivar tJ1rOUt::ll t.llc nrr.p L:i..fier
ll A. It is then "PPlied to 5 multipli.ers ~I . Tl1e ~ i gr.al i s
mu tiplied s:.mul~flnOOllAly ~-:itf1 ench •'.'Jn(? Oft~~ r :'unctiono
f(J , e) useC ia tr.... ;runnoi!..terns- carri.or.1 !ct' the coefl'icient:: . F'..mct:.on gc-ner41t.ors PG in the tru.c. :t.!.t=c:- anC. :.·eCl'iver a:us: l:e s;rr.chroni~~<l . Tte '.: procuct:-of tte ::-eceived
~1gr.al
•11it.l.t !:l::e fu11ction:J !(j,€)
a.re :.ntegruted in the
T during tlic i t'i:!i·vo.l -t · < 1. < ~ 'r . ':iie •101tr.gon a t &He iut;egr!ltor outputs r eprcnr-nt. Llte coc:ffici e11i;s
intr·cr~i:-or.r;i
!',(0) t o ~,,. (L) ut ~h<" l,i.ine t
= ~T .
Another set or Jive coe ffi cien t s drnotcl! by a , (O) t o
ux;'1) i~ transmitt"d d. trir,g;;tte ~!'.1Ce1·vnl t r~ t; < }T. The
Cunct.ione f (O , e)
to
£(11-- , S
of the ;_"w1ction g('ner&to1.. PG
in thf' t !"m:soittcr wid rece:.~,-er are rect~ircJ. cc_'1in . Hence ,
-~#0!"0 !"1.tnct..ionE Rrc periodic ·oiith reriotl T . T!Jf' ~.:-ol-tages
-~T
ao;
t < i T io
th(\ :rw1s1:i it tcr .nro chci 1 l·"ttd SL<.dder:.ly at
T • t~ »n d 1'epre~c>nt che coeffici"n~o a x ( j) <iu!'ing the intei:vnl t '£ ~ ;; < f'!' . rhc ir.te!!;!'<eCOrS in l.h(' receiver a.re
R~ctlYtr
ng. 7 v .5lock dio ·.:-IUl rcr ~ignal -rans i~"iOfi by oi·tbogonol runctions !°(j ,a) . ;,x; func-cion generator, t! cu.!.tiplier,
I iLt:egrator, T.ii trnr.:-mit.te:• a.mylifior, RA r~cciver a.:aplifiel' .
2 . 14 CIRCllITS 5'0R CRT!iOGC',.U ll!'.'lSIO:I
i r· 6.:'ld start i.nte~·rat1n~ the ~:o:!.ta~o:; de!_ivei·ed !rom LLe muJ tirliers Jt.riaf t o interval ~ r ~ ~ < fr .
E'or practicol ui;e the block cli~ir,t'W:t of Fig . 30 has to
be augmented by o ~:yncltroni:rntioa <!ii'cui i; . F11r~lJCH'llOre ,
1>odems ai·e r.;q~rcd to trruisforn. tLe coerfici cnt!l "x< j J
delivered ~o t!1e t:--r.n.;:ni~t..er in;o t!':.,.. requi!'et1 Core a;:..d
also to t:-ansi"ortt ttc coef~icients otta..:...:ie<.i ac ~he receivn1· at tne time ! T into the clesir"d fo~'Cl .
reset at -:. =
•-
w.,l(q£11
o•Jt1.e>
~---;...---
- ttH1.&I
'==-h---1 t r - - - ,..u.11
H-tfr-
- u1111E)
1-t-li+i---~.,113.m
J1'ig. 31
-
coltJ,!ti
;--=='-'--+t+t-1 1-
..al {,,91
Gener<l'~Ol'
l'or pe i'icdic Walsh fu.'1ctioJ>s . B bin"-
ry counter,
x oultipller ball' adde:- , " i:i1 ut ror trit::ger pulses, __. in::'ut fur ::.-eeec pulses .
1
:l!.}t11 !' 'fl)
Fig . ;1 shows u circui·t for LllO grneration of roriodicully t•epeaT.cd Wnlsh func~iono wn 1 ( j , 9 .l :>r c. I~ i , e) und
Dal(i,8) . This circuit is ba~ed rm tlte n.u1r11 l i ·a~io:;
thaoroc of tLe fWlCtions wal ( ~, 9) as p:ive:i ':>y ( 1 . 29) . ''ino.ry counter!! E1 to P,A :;::-oduc;, the func~~c::" wnl(1 . ~) =
snl(1,8), wal(3,6 ) • eal(2,8) 1 wa.J.(7 . 8) = •rnl(tl,'l) :;nd
wal.(15 , 9) = sal(S , 9) . The multipliers shoM1 in Fii,; . 31
Prod uoo from tho so Rademacher functions ·tho complo~e system or Halal· functions se.1(1,A), cu1(1 , 8), .. , nnl(l:l,3 ) .
1he fU!lction wnl(O,e) is a constnnt ~·osi;;ivc voltut:;e . rte
2.
u1HF.1''i' TF~~!fSErssrou
:tulc1i.;lie=s: a=c gates hnving e '!.rut:~ tal.:!.c n= sbo~11 i~
TriblE' ;:. , since ',iglsU functions aSSWL~ t:.;:i1:a. values ..... -; ()!"' - 1
only . ':on:pa=:.sor: of thin trui:ll Labl" -.:!tl.i tl!~t 01' t :::e half
addel' shows tl:at the 0111 I i:ipl iers in l"ig . ;;1 -nay be nal.f
adders , ii' an output 0 si:ands fol' a po altiYn voltage •V
1..\l'ld a.:.J output. 1 l'or a negative vol vage -'v' . Tho gener alizar.1on of the circuit i·or functions with l.L.ighor values. of
i poses no d.i:ficultic~ .
a
b
+1
-1
+1
•1
-1
0
-1
-1
+1
1
0
0
1
-able 2 . '!rut!::
1
1
tin~ier
t~bles
for a llUl-
fo:..· t\\·o ·~,'nl ::!.J fu::::-.;ior..s
(a) anc: :'o:r a 1,~1! enae:- (b) .
0
i.:onsider a \Valsh J'unci:ion genor:itor h&'Jlng 20 uinary
count:C'rs rather ~httn 11 as !ihOt·; n in lo'ig . 31 . A total of
2
20
1<1
•
1 C40 :>76 d:._f'reren!. '"A L!lh fu.:ic:;io!".!l C3J! be obr-a.inei.l. .
hal.:' "c!ders ai·e required to proJ.uce nny one of c~.e
t'
;>0:-:!lible functions . The accuracy of their nc!}uency 'Ali!.l
dept'nd on the •-ii<e;e1· rul !'le 1·e::erator dri,irig the binary
.:oW1tors . T!:ero are no dri.ft or ngini;; pr•oblcm,-; . It is
«orthwhile to comp1u'e tt.o Gi mplic i ty 01· su~h n generator
to ~!lat of " .fr·equ0ncy syntt1osi2.ei· deliv1):rinv, a mllll on
1tiacrotf' sioe i·uncti ors . On the other hnnrt, !'•)Jli'e~enta.tive
::wi tchl.!lg times ol' the fast.est digi-;>.U ci.t·cuic~ are pre,;en".:y betweer; 100 pa nnd 10 '1S . This restricts i; e highest
tS~quqncy o~ Walsh fu..Ylctions f=oo 10 1 zps • "i VJ :".izps to
10
10
~J.tS =- 1U Gz1•s at -:!le p::-esent ti!TI(!: . Sit.e waves t;ith
fi·equcncies of 100 Mliz to 10 Gliz were produccu dccadeo
ago .
.Fig . 32 shows a fwwLion generaLol:· for· ;·:enoru t ior1 of
phuee !ltable sine nni! cosine oscillatiou" J'o~ the pulses
or Fig . 1 and 9 . The binui·y COU.'lters B1 wid l:!2 rroduce Ra<le11ucher :!':mctiona, froo' whlch <:he filLe1·s extt'c.ct the
fundaoental sinuso.i.d:il !unci;ior:s. The first hn!'llonic has
~hrce ci11es the frequency of ~he ;'u.nda.nental oscillation.
ln practical 1<pplicationo ii: is betrer to leave out ehe
2 .1q c:hCUl'l'!J l'CI', o:'lTJIOGO}IA.1 LIHS:o:1
Fig . :32 Ger.nrntol' Soi· pilaSe lockeU siuo t.t.Ud cor:inc funcliions . B bina-r:t cowJter . F s _:te.r , .z. lHJ ,Jt. !01~ trigge:pulsos . x and y are cott_;.lerte:Jt.ary out~~Jt:!: of the coWJters .
f ilters and to prod'..lce a bei.\.er arproxiriac1on of the .::;:ice
.functions by a superpos:!::ior: of E:adennc::.~r .:.'1.1.nctiou.s .
~
m------,..-~,_
,.1 (1,e1
t-:::-----~«I U91
P.J-i'-+--'<!tr
"1(1,9)
r-=-1- ... Ml
-~ ... •.Gl
_,.i..ai
coll),91
--~<1111.111
f-j.~Q:t--<•lllJI)
".ig . 33 Mult.lplioro ~or "Cite ctul~.:. plicat_on of a.u n"'·bitrary
~un
) ctJ.o n by Watah funct i ons . a) eir.gln a;ultiplicl'ltion ,
mulul.ple rr.ul ~ipl lcat ion (e . g . filt.-r bnnk) .
Thex·e e.re t ltreo basic types or multi)>lio.ro . rho fi1·i; ~
lllUltipllee two voltages tha t
Clill asewoe ~wo vu.lucH only.
2 . _, wc:c·r
'/l:l
say 1.1 V and - 1 'v' . Th ::.s
t~·;rpe
f'R~J:s:;:ss1m1
of mtL.tiplicr· i::1 i:uple:tented
·uy lor,ic circui ts . The sec o!ld Lyp~ cr.c.l~i;:dies a voltage
V, J1a11ir:g zirbitr·u.:..·;y values •..:iLt a ··lOltetKB- V2 that can assume o few values only . Fig . 33a .:;towG a.n
~>:aJ?:fl 3
of t!lis
type . Voltage '1 2 assumes the val.uen +1 o r - 1 onl;y . ':'he
output vole.age cqua.1!.'l c j ther+V 1 or - V, , wl1ere './ 1 Jia;y t:tl:v-e
any v-a1 ue wit hin the vol 1;age range of the ope1\:;.c;:ionul run-
plii'icr· J.. . '!11"1e
~i:rcnit ~:orl~s
as follc\-.'S: \'f1he no:J.- i.nve.r·t.ing
input terminal ( -r) of t:he aDilllifier is 15t•otmt.led > if the
fie1d effect transistor FET i s fuJ.ly conduc;;i q; . V, ll!us1;
equal -v~ to bring_ the in-veri;i?)g inpu~ 'termin~l ( - ) also
to e;r-otL'1d potentia_ . Le~ F'E'l' be non- conCucttng . rrhe nouinverting terminul is teen at V 1 and the i n-.re:rtinl'.t: terminttl must tilso be at v, . T!:is rcquirc:.:J 't'J to eque.J 1.'1 • A
variation of this multiplier is s.::ioNn i!l Fig. 3)b .
T.'le ~!li£-d baSi<.! t;>~pe o.: mu:tiplier mu_tiplio:; ~~·iQ <.1.rbiL.'.rar y iroltages. In p.r•inciple , ·t.l:is type ca11 be i:nple mem;ec by Hall effect -nu:~i.plie1·s , fiel ti ~cnissio:.i L!'Wls iolrors nnd logax·it-r.rn1c 0-!.en:ont~1 . l'h.ese C..evi.ce::i GJ'.'C usually .uJsatisfactory ~or pract i cal appl ication5 rlue to low
irupedance , te:nper atu:·e d.rii'~ , price: , c'tc . Ji'e.il.·Jy ~ui table
is- the C.iode quad :i:ultipl i er . F:.~ . 3l• show:J n .:·.zr,r esentalitve ci.ecuiL . Jt; deviates fr.•oJJ tbe u:::~n.1 one by J1ot using
tro:1nsformers .
E'ig . j4
The voltage
'.'luJ. ti plier using diode quad .
v,
in Fig . 5)a assumes the values +1 or -1
2 . 14 CIRGUlTS !'OH OR::HOGOJ>h.L ;,:vrs101;
o.nl.Y and :i.ay be coni;idt1red to op re;>rcnented by on" ·uina-
ry digit . Four binary digit,; =-~~:-n. e::t 9 v:il;;ege i;h~t ca;:.
asSUlllO 16 values . A cort'espo10d.ir-6 oultiplicr requires .!'ou.r
f ield emiseion tranci1;.;to=-s ratr.c:- than the one in 1''ig . ;3a
and a D1ore coJDplica\;cd res1si:;or nei:;~·ork . Such b. multiplie.r·
is due ;;o P . SCl!M!D . r, yield<' excellerit 1·esults , but the
ono voltage wust be a •rail ab1.e ir dip;Hul ro1·u .
li'ig . ,;, shown an integrator . 'fl:e capacitive J'et>dbo.ck of
tlle operatlona amplif i er :1ieldo an ou~ ;ut vol t«1 ;e tha~
is proportional to iche integral o!' ;;ho i nput vol tage with
great accu1'ucy . 1'he swi;;cr '" resets i;ne im•et;":t'UtOL' b ,y uischarging
~he
capacitor . r.l'hP. J)!'t;Ct i
this switch ill uoually O"J
~
R
"?Ui•· ~ [
fi~Jd
I
Y.
~!il
1 n11 1 e·uent.u Lio11 of
effect trandntor .
:r :!'
Multiplier, lr;.tcgrn.tor :1.nd :· mcT.inn 1~r,n1 .r·,~o:· r:uCfice
in principle l'or' the detecLior. of ·my L'un~L.Lo::i . Surnrior·
circuits are avail(,bl e fot · specfo: .f•.Jn= ~ iano . L"ie; . 'it. "haws
a detecto1• l'or n1 ne and cosj.n" pul ;;e, ~coJoJ ·uilJµ; Lo f"lfl; . 1 .
This circui L mu.Jt;:os u:-...c o f the fa.c't , r.l1t1t ni r.r u nc co~ir1e
pul ses sin 2nH/T m:d co~ 2 ni T/l' "re · •ip:orr•;nct i on~ or
the i'ollowing di!!erect.i a:. egua:_on:
1
1
Y" + 4n 1 1 T·'y • O,
T4e output volte.ge vJ
Yl - H)=y0', y ' ( -t T)·~: · ( ~ 'I)
t)
of an:r·! _i"icr h 1 or Pig . ;t
(15)
i.~
v,(t ) • -(R,c, )"' j.
The output voltage v,(t) = - v 1 (c) of A, "qunlc :
V.(t ) • - CR, C1 )"
J v 2 (b)dc
( 17)
~
<n,c, H,c,r' JJv,Ct · Jd;;dt ' q:t,c ,R,c,r' fJ-,,cc · )<JLdt.'
r
-v, ( t )
2 . OJJ i{.f:C'I' 1'JlAt;Si1LSSIOH
80
Diffe~'em;iai;ing
~wice
v{(t) ~ (H 2 c1 1,,c, )
1
.,,(I)
In
and reordering
~he
terms ,;tielcis :
v 2 (t) = - (R 1 C1 H, c , )" 1 v 1 (t)
8 r:~
•
~
(18)
,.
~ ~§
11.,~~"1
I
,
Az
11
QR R
~
.Fig . 36 Dctec-i:or fo!"' ~ine and cosil1e ;>ulses sin 2nit/:r and
cos 2ni·~/'!' according to l'ig . 1 . R2 C, = 1'/21'i, R, C 2 = T/2ni,
H1
= niR 2 ; s 1 and s 2 are closed
2
Choosing R2G1 .R4 C2 = (?/2ni)
a~ t
ma..tte:J
= ±T/2.
the
I eJ'\f t.and side
of (15) and (18) idem;ical . The .in.l;omoge.noouo tera v 1 (t)
is equal to V, cos 2nh/T or "I ~ s i ti 2 nkt/T Sor -F ~ t " ! ·r .
the stape of v 1 (t:) outsid~ of l;}il i= inte.'!-' .r-al is of no in1
Le.t'et;t ,
civ.c~
tbe swi"tches s 1
t~..nd
s 2 a re closed
a-c t
=
'1<~T'. v 3 (T/2) and v, {1'/2) a.re "ero for i of k and R1 = niRl;
v , ('L·/2) =
and v , (·r/2) =
1.
Fig . ;;7 shows o!lcillograms of v 3 (t) rutd v, (~)for i = k ~
'l; sin 2nit/T yiel<ls
= ' .
o
c-1Yv
Fig . ;58 i;hows oscillogramo of v 3 (t) for i = 128 and
in:!)UC \rol tages Vk COS 2 nkt/'I' WiVh
1 30 ; &his uerllli:l that tho c i rcu;it
1:;
equal to 128 , 129 and
ir-. tu.n ed 1'or de1;ec.tion
of a cosine pulse of 128 cycles a..11d tJ1.::1t cosi.ne pul ses
dith 125, 129 OL' ~1 30 cycl0s arc !'ad to i1;o i!-1pu1; .
1
'l'he loS:ses o.f the circu.i t o.:' Pig . }F. arc comparable to
t!"lo~a of mechanicu: L'esonul;o.rs . Q-£uctors of several tbou.sa"ld at ..:i I::equenc:y of '1 00 3z- a.re z'ecdll y obl>ailled without uso of regeneration . The fi·eqi>ency i·an1:5e 1'or its applicat. l on lles between fractions of 1 Hz and about 100 kHz .
'l'he lo><e1• limit is determi.ued by leakage , the upper by
the i'requency response of operational Amplii':i..ers (6) .
2 . -15 SUJI·, Al.) Gl.S!J!r; J-·UJ,5ES
81
"
Fig . 37 (le!'~) Tnicd volla,:·e,; or Ll.•.· clrclll~ 'JI° .• i J!. . :·o .
A: i::put voltnt;o ·.r 1 \t 1 - 'i/ ::: in :)ttr/T ; B ru~.J. 1 ~: l'~S:.tlt;_,ng
vo ltage~ v 3 (t)
=tl v1, \ l ; ;
D: inl-•.ll V~ l tni;:c v 1 (tJ ~ V coe 2nt;/'r ; J· '1:1d I•: rec.ult iq; •101 LnC"'' '' 3 ( L) w..J v ,, ( L) .
Horizontal scole : 1 5 1:is/div .
Fig. 38 (t·it;JiL) f;v~ lc ,,l ·:oH:;ee<. OJ' Lil" ·1 ···.lito'· ;"i ,, . .;'F .
Circu it is tuned for t!;c dctccl.ion of !:iur h.l.1,: CoQiJJ.t- IJUl..
aes witJ1 i • 128 c;i/cle::: . O·..tt;.uL volt.nr-·~tJ .· 3 ~ "'+ 8'.0"··n -:iro
caused by i.Ur·ut volt;Jf_;c: v, (t) wi-:;J, 1 ,, ..c rc_e.,.; ~A . . , 1?l·
cycle~ (ll) "nd "1~0 c:1cl~< (C; . tu1'~ticm o·· :1.0 t:rocr:: i"
T :' 78 •·~ · (Cou:-~e<:; P . S:::H!·'.!l• , !i._•t1d.::' I .,,,~ ::J ••• ihlJ.:,: of
,.._ - -~1·ucllP.: -.n- . '
2.15 Transmission of Digital Signals by Sine and Cosine Pulses'
ca I er u~...:ou:l
and tert:: th.rough an ide::i.li:.ec. f.:·eq_:.i ·r.cv lo\\· pas:: filt.eL~ .
Thia ia the lin.it ror cte-;ecc1on o:· th" ~loc. pul;;es ~:'!
lll!Jplii:uae swnplin& wittou<: correcrio11 of lnLe1·,~:rm·ool influence . 1.L'he awui.: t.;1•a..nsinl~si on rntc l1nl,la J'or r,,,, 11 :.~·_1i.red
0
-i
bloc:·
f i.tlQt= o.t fit . 7
C:l.!"I..
Lt ... 1 ...01.. .:i.l
Ge~ [1) - [11) for more r,x=ples of t1·ru10mi.ioior
l.lSl ng ortnogo11•1l !'unctions .
1
"Y"L""'~
82
cosine µulses 11 in tirne domain . Some of tLeti a.r·e shC \-.'!l in
Pig . 39 . 'l'hose pulses are not: ort:-1ogonal but ll11ear:;y i:ide;>endent . Thej~ may be detec~ed bJ' ampl::tude ::-;ampliI1~ . De-r;ectiiou b,y· c.r osscorrelution t-:ith sru:nple funci;ions rcquir~s c ircuit.s L.o CO!'l."'ect the intersy:n·o ol in.f _ucr:ce .
.
~.
~
r
1r
,_
F i g . )9 Haised cosine puls.;s i n tioe do:nai u : 1 + cos 2 r-t/T ,
1 + cos 2 n(t - T)/T ar,d 1 '
cos 2n(t-2·r)/'.P .
Tt~e pulser; sinn (~ !j·)) pex'IT.i t or:e to tr·nntool·t 2 pulses
per second and nertz . Howc.,.rer , t hese p tLl::es c~ot be used
i n pract i ce .
.t_1"oiti r a.~·ily
lnrge ampl l tudel:i c a.:_
occtu~ ,
i
-r
sequence of' such pulses is tr&...Smi LbeC ; ~-:,· de·viation
.frorn ideal .::;:yncl-'..i·oni :i\.i.L l on i.:ta,y lead to arbitra.ril:t· large
;i
cross Li.Uk bet:wee.u the .lJUl:;es . There Qoes not :;eem "to be
any w:.:.;.,y t.o i.;.i·ans1ait. Ja.ste.r• Chan at nal.!.' ~he ~yquirrr. ra:l:ie
if a11plitude sam)lling is u.sed 1 at le;Jst not '.\'i th out paying a P0'1!6.t' J)enal '!;~; •
Sine and cosine pul::us of Fie; . 1 o.r· 'SI pe r·1:1.:.L L.: -ansmis!Jion ra'tjcs r_ighe!" cnan one pulzc per :;eco::d :u1l lle-rl:iz.
[12,13 : . 'I'he !·lyqt.::ist limi t of tc>O pulses per seco nd and
Hertz c an be app.:·oached ai·biti'tu•il.y ~lose by usi.ng more
LJ.!lti moi·e comµ : ex egtti~ment . 2lil s -x.u.;r be e.ee~~ i'1~om Pig . 40
\,•ti ch sl:ows r;;.;ree syster:1s o.f function:_:; . 1l'ho t'irs-t coJ:tsists
oi' A hl oC':k pu L:;e Of dUT'ation r only . Itn f re<luenc:,y poweI.'
.:;pectru.m is :;bo•;-m on the rig}"_;; . The .frequenc~r l>Hn<l requi-
red f'or tr.an;;mir;:::ion r;hal~ be de.fined - somewC.s:t arbitrary
- ao 0 ti J' " r g = 1/'r . 0::.e block pulse C.9ll t!:O!l be l;raC!.Smivt:ed _per seco11d and H~l"tz. .
It is recsso:1a·o1 e to it!ent-ii',y tlie block. ptLlse as ruoct:..o;;, o·;a!( o , e ) oi .J:c'ig . 1 a."ld to ;:;ransmit a block })µlse , a
s:..:::lc pul se ru:d a c-osine 1.:ulse of duration 3'L' instead of
3 block Jlulse~ o! duration T each . The power spec•ra oJ:
2 .15 SINE All:• GOfH>r•. tr..,SES
.
~e pulser- u.1·e zt.o·,.;n
+..,
~
8j
fiJ:: . -£1 .
_!Je ·uundwidtl.. reii;::-ed:
for tran:i:nissior... is .reJ.u::ed t;o 0 1! f : !"; /?J • -1;; I . ller.ce ,
" pulses brc tz.·ru.Js::.!.ttetl per .Jecon:t n...rtc: Her;:- .
1 . ....
0
T
0
a
D
JT
b~
·~
•-!===============:::;;i,
o
..........-----...........sr
b ..............
___..
c
~=======----=~~
=---.
d
...c:::.=-.......__,,...,
Pig . 40 l!ompa..t~isou of
aystems ot .functions .
0
l1's
th.E bru:.dwidth rc1uirt'!o
';
':I:; ·.~n:-1ou:
Consider a rurtlier· Sliep . Instead. o!"' "Cr·nn:;mit·l;ing a. !>Cries of 5 block pulses of ,1u raL ion T e11cl 1, ou<> nw:; Li·atlsmi• simuHMoously one blo clr pulse , 2 i;J.ne 1 ~l:ie :' •Lau 2
cosine pulses of duration 5T . The l'equi.red r.-eque::c:y bend
is .reduced to O " f ;; 3 fg /5 = 5hT •Cco1·dint'" 1.o Fii;. 4.V .
This means that 1 . 67 pu-ses ere transnitied r.er second
and Hertz .
The required .frequeccybacd for tt:r ~iis••ltancous .,rar.~
m.i!lsion of one block pulse , :_ t>ine :u:id _ co~ine pulzc£ oi
duration (21+1 )T equals O ~ f ;; (i+'I J/(21 11 )? . The trar.s~i ssion rate oqu<lls (2i+1 )/(i+1 ) µulcle• p~r ~econd n..~d
liert?.'. Th1.e rate appr oaches 2 for hu·i;a V"1Ut!O! of i .
'See [14) ror a detailed diSCUSSiOJ1 or the .l: J'nc tion o l'
~errr1 outaide this band . This paper ul:io di.!lcun.1013 'l;h.,
a. P 1 J.cation or KRETZSCHJ1l':R ' s principle 01· po..rLinl re~pon
se to 'lignals consisting of sw;i" of sine an~ cosi11a pulses .
9
••
2.
f_\1.1
urn.Ee~·
'i'lllJ,Sr: rss1mr
•ra1·le ~ ~ho1·1s values i'or t~. e nwnb<>r ?.i+1 c>f <lifferent
pulse .:hapes nnd .ro:· the nwn·oer ( 2i.+1 )/(i+1) o.t pulses
r.;1~t1usr:i i tted
secoud
per
anU
lit!l'tz . One ir.a:r !;CC! t!:1at l.;.he
.:1tu1:oer of dtf:·erent pul~e shape:: a:1tl thu~;; tt~e complcxitY"
o.r U~e equir:r.eni; ir:croaaea :·apidly ;is (2i+1 }/{ i+1 ) approaches 2 .
Tnble 3 - Nu1nber 2 .i +'1 of dif.fcreut _pU.:se sh3.:pen and rr-ui:1 ber (2i •1 )/(i+1) of p1: l <;cs tnurnmiHed µeL' ,;eco11d ei:d
::-1e1·tz. 1:or a trru1~J1:.ssio?: ~y;:;Lea: using !;:in!? and cosine
pulses .
2i +
2i + ·J
1
1
1
"
1
1
3c
1 -5
'l
• <)
1 . 75
..,1
1 .83
-1. 07
1 .8
'Pa.Ole 4 . Utilizatlor. of u 12v I:z wid~ ·~f~ J e-:;ype cLan.nel .
T1..ansmissio:-i T'fl.to i~; 6 . L;'? cl1ar·ac ... t:l.'$ ;:.el'.' secorrd ;. duration
of n ct:aract.er· is 1:.>:'.."!- ms . l•'ir:;:: cclt:.rn.n l i:.ot.:; t!1e tulse,
:;ccond vhe fI·eqnency of Lhe f1mc~i.or: from \';l·.ic:.. l t i!:i gaLed ~ Ltird th.e ~ul'~ch~nnt?l ( 5u . ) and dic;i t (di . ) for which
t:-ie pulnc is ut.'ed . c:.!.!.'.L' . st.ands £or
ca:eJ::•icr· :.:·tnch:-oniza-
tiou , !.IJ11C . ror c:1arn.c-trer s;;•11ehJ."011i..:.at i on. .
.r[Hz] Stl . di. 1
f.U~SP.
f[HzJ
µulse
r•
1
8
1
9
u
.••
wal(O , a )
C'a1'1" .
si11
0
CCJG 1 8TiO
::;l11
CQ;'.$
.;,j
2 n0
2 ne
n u_ne
coc- 4n9
!:lill
6nq
COf' · -irta
si11 8ne
cos sna
;;:in 11;119
cos 10n€1
sin "'I 2 n'3
cot· 1?ne
!;i.Jt 14ne
coo 1Lne
.SiH 1~nfl
cos 16n0
G . G7
6 .&7
13 . 3;.
1~ . 33
20
2<)
26 .;,7
26 .f7
s.v11c .
1
1
1
1
1
2
1
2
3
"15
33. 33
c
2
JjQ
2
2
3
)) . 33
40
46 . 67
4b .6?
~? . 33
53- 33
~
...-.
j
)
;;
3
"5
1
2
,,
5
si11 .?Ona
coe 20r.a
sin 22113
co~ 22-na
sin 24 n B
L'.Of:i
2';na
ci11 ?6na
co~ 26n9
SiJJ 28ne
c os 2s n0
sin 30n6
cos 30n9
:-3ln ;>2 n0
cos 32na
sin 34n9
cos )'I ;r9
$U .
3
~(J
1.
0G . <:.>7
L
1,
L
L
f6 .•6'7
/3 . ;.3
7;: . 33
q(,
Or1
'-''··
8G . i://
86 . <:i7
')3 . 33
~3 . :;;.
100
'100
100 . (.:.7
"'iOb . 6'7
1'13 -33
113 . 33
di.
5
1
2
3
1+
5
5
1
2
5
11
6
6
6
G
6
1
2
5
~
5
3
5
3
I~
5
2
_1 5 SINE ;,ND CClSllb. I l1LS3S
Table u li!"C::
~i:le
n.::d c::sinc p4l!"o.
1
of lihC po1-iodic "'-'Ave!" fr:Jc ·...·h.:.:;n tl.ey 8re
use in a i:tul t lcnru..:tc _ t-,-. - et)Tc :-.y. t.en. .
no&: •star:--nt.op' but
ving nsyncht"<H.O\.ir-l y
-,,bich chey 01·~ fe<I
~~.'nc:..ro:--.01 :~ .
m·Jrrt be
!"ed
.-i, ...
.rr~qU.eIJcie:
:.sto..U , wid t.hei:_·
rw.sai..;siou is
l'e let:;ve si !it.1.lo ti.l"ri-
tt:rQ'lf:h a
·ourrH' 11-o:n
trltnsc:itter o::
Fig. ;o. sc~rt u.uc. : ~ 01 rul se~ 1:1u.:.:t Le utld1 .I to t.hti .:irn3l.e
at the reco.Lver , !10 l:i.~1t they m.'~ Y be l'r.1d 1:1to t.he USL..ol
,,~1chror:3u<ly
to
~he
telel:ypc oqu.Lptuent• • rt.e d u:·«-bz. of L ·" ~11lflt.;ype clrnz·ac -
ca.rG is
~oouued ~ o
lie
-.;,o
rr.i; . A ce:·loJic ru ncL toll ni n2r:t/1'
T • 150 :o.:-s a.rn;. 1 ::: = O.•::it1 H~ l.o.<:J ~ "u:.•o ..;,1·tJds~.:ii·5
: 1 Cl'f a!: ~cgj r:-:iing r1:-.: rr. I or I he rnrFJcter~
a.nd is ueed :.s . y1.chror.1:.ai;io" :;i;;nal .
A telet~'pe s:,•. tr·n. ~tcco'!"d.:.::g -;.o l't:.l" lo ..,. or. OJ ('":rAZc sc::ic
100 to 200 telet;r,+- ::ha=<' ls in -, I•'. ·phony • ""nel , d<·_pending on the quoli~y 01~ :.!le te.!ep!.0:1:,· i;;ha!U1'1l . ;.:ith .'='.uc::
a large number the que:::.t:on of oo-.-=-:· .!oc<l.LUt; b"'cO:nes b portBn t . Tel!~s have sho'.<::. 1 t:.;,~ er·~-or· r·nt.cu cf 1C
ru'd
less cnn be obtllir.ed 1<ii;hout ~xc,,,,t.l ln
L.1e I'" I"' in:-i t I -;
with
wit;h
n~gnti.,Je
po~er
londinb . This fi;~u.!'e ILOl"s f er· ,r·ccn i::i. -oio n betweer.
n mucL. mo:-e sever? i:•.r•n .• 1r1()11 !"'." 'lll t1··~n.5mi.Geion between two ~olcphone ~xch ani;:es . r,o cooing- or
othe r 1~r1·or- ·octuciL.v I.lie ~Lot::: h•ere u :;c1l . .FuJ · .... rn1 ..,r:.oac ,
the ',•: idoly ua~d 'J'!::Lt>: s;vs ten: o.c.;::ot1oanLe!;l 2l1 ljcli:;L:r' J e c:.annelr: in ono telephony cl...aru...e: , ':Jul .:xc•:e•Jc ti:o _p'l1..:r.i!".sibl c power loading oy auouL a r act o:- ~ - J:;xcor-1Hng the ;>ower -Oading is quit.e t.:.~ual :'or .r..i.gh ~~~ei1 lb-tJ rr~n.-ll1~1Jiou aynteas . Iherc is ~t l~!lst 0!10 ..;:;r:~c:J, -s.n:t. :-c1u _::-e:;
~he baedwidth Of one t(•leph.on. Ci.a~.ue! ~~~ tho pCAl'l' loaC.
O! e .. ght channel:; to L!'ansai ~ 2-00 bits/ . I~ ~.. o~l~ le
PDinted out that synct:ronou~ t::r~?:!;r.Li ~'!aiou t
ve:·:: een.~ i tive to ph11150 jumpn •,.b.lch occ·~r in ,;wi tche<: L"l<.1 t. ••.e uel wo '
ove1· lonf,' dic.tanci=s .
Error ri:itf n ll'l'1s increase 1.0
'10
nnd moro , depend ins on how fust.. lo.;t; ~:,~r •• ~~.1·oni£.;.;tio1J
COL b0 reao~obliahed .
t~o ®bnc1·iber~ ,
:r-
On~
.t.'OHAOll 1
whynj n~&.nd.cosine pule.its
;vJ.n l d ver•::t reli-
abl+, trllno:niooion , is that t;e:Cephony clle:rnel:i a.re design<><)
2 . DIREc:
86
'.I'RP.1·1s:nssrmr
foi" disi;ortion i'ree tra.nsmission of poriodic s i n.: and cosine 1't.~.ncti ons . Sine and cosine pulses contai ning 11ery1.1a.n:y cycles come close to the periodic ['unctions and suf-
fer litt;le delay er attenuation di storti.ons.
J~1o t he r
rea-
son is "t!1at err o1\s in telephone cilru-inels a.re oa :i nly c.aui;ed. by pi.:.lse-l;ype i!ller.::erence rathei' thru: t hermal no i se .
It will be silowr\ in ~ll;;.pi;er 5 ,;hat cr.erma.l noise a f fects
all or l hogonal pulse shapes equa:ly . ?ulse- l ype interference , ~'owever , ei.ffects block pu-ses more l;hw~ otter s ,
particulsi•ly i f run;.>litude se.m;ilin;; is used I or· dei;ection.
2.2 Characterization of Communication Channels
2.2r Frequency Response of Attenuation and Phase Shift of a
Communication Channel
Co1;ununicatiol.'I chei.nnels ai~e u.sual:y specified by bhe
attG-nuati.on and _phct:::e shifi; of harmon ic o5cil l atiorLS as
run ctio.n or vhc i r l'r·equency . ;\ voltage 'J cos. wt i s appl ied
co the input arLd t!lc stoad::t· state vo-tagc at chc o utput
V,( u. ) cos w[t-tc( JJ)] i s rns~1 sured . 'Jhc quancitic;; lg V/Vc(w)
~
a c ( u, ) and .ll'tr. ( w) ~ b , (w) a::e a t ten-;atior: and :;mase shut
the t r eq'..lerLcy w . Ttc Pf.l=1.tmcter c may be
omitted , i f attenuatio~ and phase z.'lift of V cos 1.1t and
V s.Lrt wt C:U~e equ~~ l . Sir1cc :._t i s weli kntH·n1 that: periodic
.:;ine a nd cc~inr. .:'tL'1ctio::i.e: l:.E:ausll!il.. illi'orm.:.u ;ion a t rate zero
otly , i t i!'! L'tteres<ting; t o invesi;igat e why -chose functions
tire used i'or c.:1aracverizat1.on of communl.car.ion cna.nnels .
Let the commun i cation cl:.ennel be divided into t;!le t.ransi.l>iSBio1L l i n e a...:d t he c i rcuitr y at it-a euds . Tte line is
descr·ibetl by a part i al tli~feren b ial equat i on or e p9-rt:io.l
Cifference- Ci .ffe l.'e n tial equ.a:V i o11 . Tte ter:ninal c;i rcuitry
i~ del:IC_f·i ·ueC. by 01'd.iiJaL ;l differ:ent..ial Or dif' f'~renCe-dli
J' erenr;ia J equati.or1s ) ii' its dimenaion~ aro not too large .
in :>a:roi.c11l ,,r , it will b\'l doscribed by a diffe:rontial
equat.:orL with COJ)f:it.~t; coefficients , if the circ1.ti·t component B .s1~e suci: t j me in~...-nz·iant items. as coils , c-a.pacins .C LWL:tio.r1 o.f
1
87
2.21 FREQUENCY RESFO:JSE
tors 8Jld resistors . A ,;ir.u~oida~ volL:.,;e n111 li"d to the
input of :iuch o circt..i;; appear:: in tLe :itcudy st"te as an
attenuated und ~hase sbi.f1;ea vo~ ;;,,ge at the outrut; ;;he
sinusoidal ~hare a::d the frequenc:; a ro prcuerved . llet:ce ,
the circuit oa~r be cha.rac-:;erizea by tte f:·equenc:,- res_ponse of attenuation !ln<i phaEe s!:ift . A chnr-r1cte1·~~ation by
other funCtionS - for iLscance ".iRl:;r functlOU3 - j S fe r £ecl;ly possible , ouc 1Doi·e complicoterl cince the Sltape of'
these func1;ioHs i.r. chrutgec .
Consider a tr·ansiussi oll l ine descri heel Lty Lh<' L"1legr~.
pber ' s equatioJl [ 1 j - [3: .
~~~ - LC~ - (LA
+
RC)~'.'
- RAw = 0
( 19)
L, C, R and A a.re the ~Cucrivity 1 cqracit.:,-, :·esist.ivity
and conductivity ~er u."li;; lengtr. . -~e liuti is clntor~io~
free, i1" LA is e<;.uu.l RC . J:t:s gene:-?-!. eol;J-...10r:. iz :..:i ~h:. s
case ae .follows :
( 20)
a = (LA
1
RC)/2LC ,
c
1/ 1/T]',
LA - flC • O.
J:(x-ct ) m1d l:)(x- ct ) a1·e iu·b H o•o..r;y Cu nctio". .: de Lel'mined
by the initial and boundary conditio n•'- 'l'he o.uls change
suffered by th"'OO !'unctions C.urir:g tra"51uaalv.u i. an a<:tenuation and a a clay . 'l'Jois :';;ature al•<.> t,olU.: l°or :-:ot
distortion- free liues , ii' t1ie;t :oi·e • tl,,ci;'iCiu I y ~ho rt ' .
According to K. ~i . ·..:AG!IER [ 1 J a l.2_e of leJJ6iU x I a c. ect:!'ically short, if ~he .followin~ condition holdn fo= x :2~ :
x <
ii/F • ¥,
z
=
VF
,21)
Ar. o:n exo:nple consider sn open ~. t::-e lirct: . The two conductors sre copper wires
3
diru:ic tcr llt: a tli ctn."lce
or
=
of 18 cw . The following typic"l value!l npply :
L • 2 . 01x1o·l Ht:tnry/l<m ,
ll • 4 .95 Otun/km,
A
c = 5 . 9 x1 0· • l"ornrl/l<tn
*0
One• obtulna ~ • 540 Ohm and 2Z/R • 225 ~.m . '!'his J ine is
88
:i. ci~i;ort10.:i-!1·ee !.ir.e
:·or !Jist:..it.ces :::r.Q.l . . er t!:an
225 lt.!U . In~e:-;in5 i·eoer.r-=a.tivc 3...'l] li;-ie.1.· .. ut c:Jort;er dist.;n--:ccs, o:Je :ns.,y t1•tu1t:oit :.!i;nals d!..st.o.z.·t:iou-frcc o,.rer au;y
!.ike
distance .
As a ftu•vilei· o.x.WJ~j,le , consider u ~clnJ>hone C(:lbl e betwee!l cxcha:ig:e. and ~u·~scribc:... . •rhe cor:.du;-•;o!'t1 arc p.apcrinsu!atcC. COfl.;E:IJ.' wi1·0:.i of C.8 mm c.ll,w~tc?t' . 1he fo::.:JwinB
t;Tical values :.iprl;; :
L
=
7 •10-' :ienry/IUI., C • } . 5x1:i·• l'ar:.id/b:i,
P. • 70 Ch!!:/kll, Z • 11.~ Oi;;r. , .:ZIF
Thls l .:.!le •·:il_ bo o1
~ct.1·:.cally
= ''
k.:11
.snort "''Ol"' SOL1t:
~uoscri'::>e.rs
\mi:; not for· al_ .
'rhe uBual co.sixi.a I cub l ~::; i.u1v,.,. a WHVO in.;.eUwic.e Z between :,o and. 100 Oh:n . !,et t;be L'esi,;tlvity bo 10 OLm/k11:.
2Z/R will theu bo bnh·ocn 10 li!ld 20 kD: . L't,i.s i.:; clle :n·d.e.
of !Dngnitude of the usun! rlis-cances l;~l.t-:t!>f'n amr-i.:'ie.r.•s.
Ho·..,tvt:!'." , one ou~t kettp ; .... :nind. -hat t.he ~e!.cgrapher's
e<;,uatior. .:l~c:.: !lo:. a I lO'n' for !:.he st.:n ,11!"feC1,; . >~ i.r:vc!ltigation of tte eiii.LJ i:?l.foct fo1• f·.wctionr nt-~el' tLwi s!.:le and
cosi!"le seer.ls l.o 'Le lH<"-lri ng 1 •
Desrite --C:!'tese .L·e::ulti:i, 3ine aud \:O.~i:-..n funct i ons do
play n d i sti!"<gutehad rolo in t r..i thtior·y of' trnr:•mission
line~ . One iopo1•·t;r:1rjt t·eason is l::!.BHNOUtf
' o: L'letltod for tho
~olutio11_ of partiut dlf.:'ez·entiP.l e111nt<1on w~t.!i t.ime invn:-ia.n;; coeffic.:.cnt:'I. t.ssll.'!le :>h<lt '" x, r
in (19) oay be
r~;-i!"esente:i at: t!lc prod'.lct of a
n tine varial.:le "(:;) :
w(x , t)
=
u\x v(t)
:;..?nee ,,aria-Ole u(.x) and
(22)
1 A ~ractical di~tortion- fL·ee line ucing ~ci:ticonductor.s
wns rtcr.cribeG. "J.Y K:EkCH~ [3] . Supc:-cond'.Jc:.ivc en bl es at~t.
al:r.:.osc C.istortion-l'rcc and transmit r:·it1i tching cransientfi
in the nano,.econd region [ t;, 5: . Such uupe1·conrtuc,;ive ca-
bles ~oul.:!ba•1e 1-;:.·ett':. ;!·act.icttl potential, if organic cocii:ounds can be developed ~hat axe sttpe:-conducting at 1'0olll
tea.perature , a.s ::ome phyDicist.s believe t.o be possible .
1J(XJY(~) i:: to ~1:) yi.i;ltic
Sutstii;1Ji;ion ur
d~f!ercntial
~~
+
2
LC
(1. - !lA)n -
d V
dfY"
-
(L'
0
two
01~;.i:i.n~.r:-1
eq11at1on;c :
1
('2:;)
i.;
c.o
R")-ct·
.v
I
·;
The il' e1f;oiu'w1ctlon<> <>1·e coc V
\ i.-P..~ )>. ,
1 , where y iu tlt!fined a.!l Lollo·.-.1 s :
1
0
'i.t
_
LA+IW , [ - ( ,A+RC)t- 1 ]'"
Y - - -ZW • •
LG
lZ
V i,-l!A ) x
n.nd
~·
1,_..-:11
'
I
JlEHJfOULL l ' ,. >iet.;od _s o:' g1·oat .:.er 01·tw•c~ !'01· .'.'ind.i:g
solutionw. or the :c!er,-:-··_.,1.11e-.:_· ' ... i.-4r_._tftlo1 .. -:?.d cf ot.JJer J:.~L~
tial di.!"!erential equa:;ior.s tr. at s~t;i; :"':1 cert:-.u1 :llit.iti..!.
and bo=d:i..ry cona1-b.:is . Eo-.;-=~~!' , i~ i. tile aet110J or .>o-
lut.ion tb.=i.t :f.'avo.rs .:;1ne a..'!c: cotl!.!le fl.U\C"t:.iO!"'•.::: . :.:.n.e ~::;;.
.riable lineo t:ou ..._.. of i..:.uLL:·sr: uot rer..11it !J s:e, ··...:.·ae-ion or'
the :JOlution lnto n t i:ci.e 011u a Sf.~1cc 1er-enu.t.1.ut; .!.' actor .
Tne propngnt:ion ol' d-=ctr·~r.i~i;::c.ct Le '""Jlv ·.-1:w~~ i;; de scribed by t;b.o ~'t'UV(1 equat;ion . Lt i!': o~ t a1 11 eo :'or oue -
dilncne:lonUl propui;:;i; ton a" a :or cdn l
pher ' o equrttion with B
a:
'"'~" o.r t ho :;o l ct;i ·:t-
A -=- 0 . :1..s i;e11crH
r-olut. i o..:J is :
( 2'··)
This solutiou bu& L..e .;ar.ie form no (" ! , 1>xco;•l "h91:
the 'ltter.uation zer:r. ,~-• t :..!: :r.i!'."t:.:_"ig . toe::~e , ~ i·ao:o _ir_!.:
l:e!:nY')B liko a
di=T.o:--::ior:.-::·1·ee t:..rie . R• dio \'.'&vc.- do no!:
have to be sine wa\·es o.r be Ucoc!·~t r:·l c:f· nir." runct;ion!:; .
D1.1J.'ercnt tt•ansaitte!'~ d~ .:.ot ll'-'~'" to Ol'G!'llto ir. c.ii'i'erenr; l'requonoy bo.nds; they rr.ay in~teatl ope:'et0 -:.n Cl.:---.:-e rent sequency bnnd,, . The:-e ariJ cxc.-,llr.nL prrn.: Lical L't!Ctsone i'or alloc:.:t ing radio channe ls nccor·olnc; to frequ .. u.cy ,
but tneao t'Gaaons ru:e m'1iill,y cue simJi1 1c1 ty of fop l cmcntill 1.- oceivoro uncl transm:ltterG rat~.0·1· Ll:,.n lrtwu of n:ii'Ur~. I t will be aho\-in later on Lhu~ mobi' e x·u.uio couw;un:lcat ion is indeed theoret ically po!lrdbl o •,iith Wralch ;iuvcr; .
90
2 . Dl '1£CT l'TI AhS~llSSION
lcr; u:a co!l::.i 1er cl::a:-~cte=izatioL of coru:iunicatior_ cbon:-:els .f.::or.i ano;,,hr r ~l,. . It i r: =~ea.scr:nh e to co:-c.ribc the
r'eaturo. of a chrcme l 1Jy r'unct..Lonn t lw l• u1'e di st l.ng;li.shect
b~·
t ht!
trnns.ci.iLt~d 2i..,;r.£;l~ .
. ... ap.~:: ?ro_i:-:::i:._ed &ntl ti.:>cd
I'C.crt: t:.a.ve lit>enr:o'.11.tu\y;n,;.lse
:o::
c.ig!ta~
t:T'&.Jl~:r.:.:;a:'\.ou
rhat .i.t
would ho hu.rd t.o claim a pari:icul= one as tll<1 only ure.rul o ur l'O.t' c'..nract.erl7i..Lf
ri
cl.wme? l . ,..,ll is dOE':-
~ot
hold
tor tt. e .hon:.· • · .,..n.al~ . ..-t. .:.:. bt'l.c!"a:l J-~1·nctice t.o r-et;n.t•d
~e}:.:~Ur-ny sienr: ls ut: n suµerj,0'3 itio:; or ~ino W!d co sine
ftu:ction:i . :!en~e, ~hcne :u.nc~io"" ltpp~nr ;;»eemir:cn".i !'or
-he c:_::.r,cteri:":.At::oL of r,e:e;u.on.1
CJ:.annel~ .
'l't.e tl.i::... icu1-
T:; i:; ti.at ttnl"A is no over~-;h rl 1:1i.!lg r 1JHt1Jn •,.,ihy l.<:-epllony
uigr:ult: r houltl t)0 regarried u1:1 ::n1l.)e.::·r o!! I tion~ oJ'
e~e
w1d
cor: ine l'unctionc . Con~i,le!" Yo ice s.:.t;r: 0 1~ rcr-resel!"tcd by
tl.c O...\ .. •u& volt..n~o of u :rn.c1"0tlJor.e . ;... lont!, 5untl::'i.!Jed vo'.'l cl will ~roduco a voHnt:e co:;:.;i,;ting wi t:h t:;ood ><p;n·o1"1!!1!lc- i ou ! ~ su.o or: a f~w t-:r.:-iusoidDl C.6c-l la-ciouc . ,..·he ..-yst.e=i rroaucing t. e VC\\v- i:; r-i e -ir:varit11l:. iinJ _s activut.etl b:'l 1,r..~ ··.:oct•l cordE '''ith .Cl :.~inc· .!'unc t.ion . S uch a sy!"ten ir. describod b:: n r'u·Lln- "4il'Ser~nt..lnl C((~ut.ion wi.th
rif!.-e ... r.~e]'.:i:t:Iicnt co-ef!.1Cien-c~ 8.JIC a .5)'"· 120:.Cal excitation
;'ur.ctior. . 711is i!l not. 110 !'o::" voj ¢€oles.:; corusonan t.s , par~ i
cula.,...l,v e:i unt::.s li1-re p , t o.:· k . 'l1he !:::.-rect p:.·ot111cing tho
.:s:ound.~ !.s des<..:.ri:·,cd L ·.· n parli9~ 1.:.!"J"c-,..e:::.\..iaJ equat"\.on
,·.il!:. .&.~n.":' vnri HL:P- coeftlcientl ar:o tJ1c1·0 is .:.to sinusoif1 :tl excl.c:: aLio11 f'LL!1cti :'In .
Thel"i.! ls no ;.. n!-'t;J culn.r· .r.·easo11
~!:yon~
. !:ou.ld
:-uc=.. zour._dl:i tc
.con~ide1·
cons i~ '!;
o.,. a sup&t'-
i:o::;it..:.on c;"' siun a..nc. cori ue f~mct.i.onn nnU not of _..wlctiun.::: of ~01:11'! otl1l'J' com1lt: I,\ . r:i;op·oL..:r.i of or·l.l1oi;onal functions .
E;qeL'io,,!H,lll worK 1y k.U-.IJi, !:!Oli'=OSWl:-r'.!Sll , 7ASTO , ::.ltKE,
t:A:::..: "c. at!:er. ~;F.!.!i ;.t.own i;r1:1 .. voice r-iignal.:. ma~· indeed
·oe conci <.lerco oO b" a s upoTpcsicion of' Wal sh 1'1mctiona ·
[,01:.E ~l"ld "ll!.TI.,r; kove ':;:lu ... t. a .... elr:-phony !fll.~lt i µlcx syste~
to!"'illt; .!1l:i!.!.'Z" .. hat rer:r..it; •..;al~h functions :ir t a sequancy of 11 ouo .:.p. to 1.1auu th1·out!;h ra-r;llr:r than oine- cosino
!'ttnctic;uJ ll!J ~o u !'r-equency o r 4000 H7. . ·r11ere .io no dir.1
91
acrniblo difi'~-r-enl~I;!. of re:-fc rmAnct" . .BOESS'.·.'E?.r:·=:R -.lttts ·:..uil t
an a.rwlyzer U!ld n ";'mLhesii.er for o vocoder usi:ii:; 1•~ fil tero that fiHel· e.ccordin5 Lo ~llfl r,pq•.iency o.f \.lnluh fWlc tion• rather than 9<"cordini::; Lo t.l•<J freque::cy oi" nine fWlc tiOnL · JU,EI.:i hO!: S)lO;;r. for '3 few "XaJ:l~les , ;;hnt 'IOice C.t composcd by "A'Al!':!:. !'unc;:1or.~ contalus ··~ec;uency rorm:..nts"
Just e.s voic*= dccocii:csec bys i:ie-cosine .!"u.nctionr- co::.talc.s
ri•equc-ncy formnnttl-; tbe~e invc!lt.i1:~ut.:ions arc conLi:.rJirt.'£_ .
A theoretical ar(!WU<mt: cxplnin ini; ~llese resu I er lt1 g iver.
in
~ectio!I
S .11 . SJ.;JJlY
11~;; ~sed
W:dsb
fw1ction~
ill Liieo-
retical •,;oi-1-: on t!re~cll a.raly-~i!l .:1u c:a:.~:y a5 1 C:-•;;:> (1 ) .
2.22 Characterization of a Communication Channel by
O oostalk Parameters
Uuving shown t:1et cocw:.unici;Lion chru:L:lels do not have
bO lJt> characteri~.ed by si,;e- co::in" .fWlc cions r9i.r.os the
question , how c.-.se l..l:.e;y maybe
c!'-.nt"Llcl..~.r'i.;.ed .
A consititer:t
theory of co:r.:mUii.icat:ion Oasea or.. ort.lioi.:onal i·urict.iout! re-
method o.:: cha!'acte-!".!.zst.1on -=.hat wi:l arr.1y to
all or a;; least 11>any :-;ysteir.s of 01·tl:ogonal fw1ccionr-. !.s
f'j, eide et'fect th-i.r. mnre e;ene:_·al chE•ract;cri::..u.t.iou wi t.l aimplify the discusslo" of chamie' CElJH>cihy in :;eci:ion 6 . 1 .
Consider a cow1let<:> aystr>n: of orthogonal fwlclions
[£( J, 9, I . Let them be di viced into "ven fi.::octiori" f c< l , & ) ,
odd functions r,(1,e) and t!:l" couutunt 1'(,,~J . :'he vol tage Vfc (i, &) in:-;tc-ad o:... Ve cos iJJt.. i:: ai::p~icd t.O the inr\~;
or u channel . l•'or t.he tine bein..-; , l~t; the chru!.t:el l;e suc:i
Lhut the steady stute vol ta;;e 'I, ( t )f, [i , 8- oc< i; J I. 01:t&illod at t he ouL put . This will ho l<l for a la.i·i:;" cl~~" of
3 Y'3Vomg of fw1ctiOllA { f j , B)) in tho cnse of a dist. ··t ionfreo transnis,;ion line . Vc(iJ/V • K,(i) or 1 1- V/'.'c{i ) =
• ac(1J is the genere.li::ed atten\!"ltiioc. 0£ t:.:e CO:rlllt:.nicn.tion chanr1el.
e,(i) = b , (i.) is the ['.eneo·"1ize I !Ir.lay ,
nlnce the ter1n phc;~e Blliit in r,n l icoble to sica rwd coeinr, tunci;ions on t:y .
quirds
a
AJ1 input vol~llt;e l'f• (i , B) p.roduco" the outrut voltnge
92
i J] . Ae~cnt-at.iou .ind deluy a:·~ deL:nec1 1,y
V,(i )/V • K, (iJ , -15 V/Vs (i) • n,l i ) and 9 1 (iJ = b,(J ) .
':be· coc~t=t \If( O, J J yields 'i(O) r[ o, o-n (OJ; 1 , 11( ;;i=f;( o),
15 V/w u - a(-1 .r.J •(O) = b(O) .
Let; t1e ~i;nctl..ons of t!lc ~ystes:i { !(C , t:) ,i'c \ .. 1 9 ) , ~ 5 \i , O 1
be i:;trotc!.Jed 1:y tho r.;ubstitut ioJJ l - l./g ~ i.. . 1·et l an~
5 illcron:ie be.Vo'1d 11 l I bounds . 'l'IW fl,'/ lit mn ( .Cc ( µ , J ) , f s ( 1.- , 9 ) 1
i2obtoin°c.sccordin·Lo•ectio'11 . 22 . Kc(i.', K,<i) , ac(i),
a.:·i 1 , b,{i) and o,(l) beco:te KcC.1) •.• 0,1.1) . Jn ~ic·tic:i
la;.· 1 one> obt'1i.n:c !'O- t ..e qeci1.l :11nct10:;~ ='c ' ", ~, V 5 ( i )J: s [ i , ;J - 6 5
(
t2 cos ?.'r.u9 • '[2 cos wt' aud fs ( µ., i;) c: [ 2 sin 21µ.J =\t; s i oUJt
the f.t·eq~oncy .rur..ct1nn& K, (w) ... b,,( w) . ·rhc in<ltces c a11d
..
1
s muy be 01::iit ted, J...t' nine a.nC.. canine .funcl.i-=>nr. of t!i~ name
rrequen<.:.:t n.:-e eq>.J.'~l ly '!.'J...._ten'..lu\.oc<l and de:n.v~i.l; tt".e funccions KCi.J , ,,(..,) w::.d :>(.11J a.;.•e ohtnined . !Jene., , i;!:.e drnL·act~ ri:-at.:. or:. oJ' com.c·..uUca'tioi:. C;J'1Jlr.Cl!1 ·':>~" tl1e frcquenc:;
i•espon•c o f "tt cnuatior1 a:-.d p he.sc shi.'.'o i:· i Hc J uded a~
spec.Lal cR.!>e .
rn 15enernl , t.be i'unctiOOS V::,(i,9)
~tI'e
nnt 0l'2:J at;tc-
nuated w..d d<!laye<.:. 1o.1ut: f.:istoi~t..!d . -he ce·11 outJ ·.iv f·.mction
Vgc{i , e) is obtained ins:;cad o! Vc\i)f, -i , 9-0,{i)) . Let
g 0 (i , e) 1;e e :-.'])m1ded i JJ to a serie .. o f <: lie :;yt<L<>m [ f[0 , 9-
3 c ( i) , !', [k , e -9c (i) J, .r, [ k , e - a c(i)) ]. 'rbe ""-- ~" o:· LLe ci.e lay
9c( i ) l<i 11 be de "ined :ntC!' 011 . ·r!:<' '"ari ·o~l!: i~ no" k 1
~L: :e 1 i~ a constnn~ ( k = 1, ~, ... i l .. . ) :
K(ci,O)r[0,9 -o c{i)J ~
..
L;
"'
(K (ci , c k)f0 [k ,9-ac< i )J•
..
K(ci , CkJ • """i;;,\i , &)l'c 'k , r1 -~,(i )Jda
K(ci,sk)
_,..:"gc \ i,e )r,c~ . • -ec(i )]do
(26)
'H
c:onside:r tl1e inL &p:ra! K{ci , ck) fo~· i • J<. It a vulue
depends Oll 9c (i) . ~~t e,(i) t>r· C!JOse:; ~O th~t K(d,ci)
assut1es it abso:".Jt.:.· cu.xi.cu.;.::: . Th~ _-;e:.el'!:il c-.eC de~n:; 9c(i)
• bcCi) end e,(i)
t,(i; is then deffotJ ,-,o , c;hnL it a~
proacne s the value 1'01· the dintortion- 1'1\ee I i ne wiLh <lecreasinl< distoi't ion,· .
The co~fficie~~,. K<ci,C·l, t:(ri , ck) wid l'.(ci , ak) ar·e
generalizations 01 tl.L atrennntion Ec(i ) 'ora d -'. .i:.01·tin;;
commu.nicu~ion c!..rumnl. Kc(j_J Lon the one v;;riable i
and
may be represem;ed t:.·
ll.
vector .
K(ci,O), K(ci,cll) ror.d
K(ci, sk) have !;he two "~·i 11>ler i "'"' k , and o";; be represented by a T.at~ix K(ci1 :
K~c1 ,O) K(c1 , c1) K(c1,.,1; ?.( c1, c2~ K ( C1 1 s,>~ . ·
~(c~ 1 a1) i<(c~e , o'°' K(c2 , n? ..
K c} , O) K(c3 , c 1 ) K(c,. , ::01, Z(c;:,c, K(c3,s2 ..
K c2, 0) K{c2 , c1)
K(ci )
1
(27)
!!:he output volt ai:rc~ Vg,(i,0)
instead ot Uc (i , 3) i~ np;>~ied
K(si, O), K(si , ck) nnd K(si ,~ k)
(26 ) . The matrix K(si) h".s tho
but ci io replaced by ci .
"re ob t ained , i f Vl',(i ,& )
i!l~~t. . Coc:-.ficieu;;G
n:·e o·oGain"d in ru.aloc;y !;o
Io x·u. of th<' mnldx (2?) ,
to t:..c
Transoission 01' Vf( 1,3) 7.!.elds '.'g(ll,~J und t:.c co!'fficients K(O , O) , K(O , ck) and K(O , sk) whic:: 1ta;1 ce writle:..
as line matrix :
K(O)
a
(K(O , O) K(O , c1) K{C , 51 ) K(O , c2) i<{O , u2) ... )
The tlu·ee ""'"rices K( ci), K( ai
bined. into one :
X!O,
K
a1 O)
,O
K
~ s2:0~
K~O , s1)
K( O, c1)
K s1 , s·1 ~ K( s1 , c1 ~
K C1 0 K
c"i, n1 t: c' , c1
;;
K s2, s1
K c2, 0) K(c2 , n1, ;:( c2, c1
K~s2 , c1
:u:d K(
K~O ,s?)
lLrJY
te
co:r:-
K(O ,c2 ), ..
K s1, s2) K~c1 1 c2~ -·
K c1 , s2 ' c"i , c.') ..
K(s2 , ~2/ :• ~2 , c2 ..
K(c2 , ~2)
K c2 , c2) ..
(28)
The t"rma outi>iuo of the ctoin diagona l oJ: K vnnil!l11 , iJ'
the fucctions f(j,6) 111'6 not distoroed , he tcr:i:z iJ:i tle
94
·:>ccoa.e t!"tc onc-di11e:1~io!".al
tion coafficinnts X(O) = K(O , OJ , Kc1.J)
a:'Ji-
di~go:i!tl
~c"'
of ~tteni.10K(Cl,c.:.j nnd
K; (i) • K(ni,si) .
Ttc delC:&:,· • i~i.?s 9 c, i) o! ~2'__, nnC. the corr·e!1rOLd .nr,
del:.iy ti.r.1e~ 9 s ( i) nnC. 9 ( 0) ::or the> tr~n:1mi.s.sioll of tte
r·uncLions VJ' 5 (i , e) ~r.l Vf(0 , 3) ma~· <il<"O ·"e "'!"' CLen aR
ll~tri 'X:
e<o J
e
0
()
0
...
ec ( ~)
u
c
()
t)
~,('!,
0
0
'I
0
(29)
~ c (2 J .
1'he t.wo u.nt:rice-....:- K an::i 9 cLa.r:icteri;:n c!ie co:rununicacior. e :.uxc fol-· ... '~ sy~~._m .:J! !"·.L'l'\ct:o:J:- {r(~,9 l ·
Dl ~~ tortior~t.1 ir: u r.hc-0...ucl cau!-5'' croei;;t~i k in n"t:.ltiplex
trnn ..aui.::.;:;:.on . One
'f -ic-.:t ior: of the ru: t.r~x K i: for ._ 1e
I a: f, :111-:;ri.x
corrt"Ction of thin Cl'OS!;t 11-t. Hp•ice , c··o,
tin a.rp1•op1·in l;~ ter•11
l'o .t· K, i·;Lil o 9 m.:.:.;v ~'~' cal led tltf! d~
_;._y CJ'ltrix .
2.3 Sequency Filters Based on Walsh Functions
2.31 Sequency Lowpass Filters
... t
h.as
h('len
s:10t·.1:1
c~ o~ c!::nr 'c -ri·:.t... '1 -=Y
ir.
~
sect ion
S=/fi- C'Jl
1 . 5c..
!low e
.:"ilter
-=-1.i.!)
L i~nc (
r, ( t... e)'
of
t
f, lu , a )j "-""1.0t<d a·· •in<> a•1J co<ii no f unrl lons [1) - [ '/) •
t·:qu~tic:1 ("i . 11~ .} hr11.:. bef:r. olt::i.ir 1)J.t thfll .:·eprc '-h?nts the
_.i,,,al F 0 (~) llt thr filt"r ·1u~f,., i:· tnc .'it:;!lul ?(9)
sp~·liod
t. o tr.a i.J1f,~Jt . SubP>tit. ution of t~H~ nyf.:tem oi" Wolrh
funcclm:s (c-.(µ , e',:;l .i,d 1into'1 . 1·1"') yiel<ls :
..,
•
..·{:.tr.~ .J )}:c{µ 1 cn1:.J,tt - a{ (i..J] ~
(50)
0
~
u,(u lK 5
.,).~<U.'1
'.I'bf fo!_lol-:i!•~ relation.
derive l'ilL!l>"" l'rou; (30J:
,:-e,(.,
])di..;
of' St!t.:.tion 1 . ;'4
ai~e
r'!oeded to
•
•
•
.:111 (µ, a)
csl (il, 9'
nnl(u . 3J
-t
ti
e
<
i
i
1,
=
""
"
.'
~
~j gc.a~ G(O) b r. div.i<lci.! i nto tim~ o~ct i::ln:"
L et s
9 < ; ,
~ 1.,, 81
-
•_(i,~
~.
( .51 )
0 ;; u < 1
< i.+i
!. - 1 <
~ i
wnl(O:J)
cnl(i , 9)
f , ..... f(B}
2 B <
-t :
=J
ati•iction.:1 un
<lc:o·~~
< : . S.lch u Ui- - ""ion :.loe!" !";Ol J.lacc
I
he n1 £:..io: G-(0
j
-!
~
Lhe i:ectiouL"1~11' in-
ri!"l:,· .l'e-
~ut. ~ c;.r:-.c~1.L'Oni:.ation oi-
t
KDa.l is rcq ulretl l'ror.i i'Hilc;-i 1ihn bt-!r.i;i nri n11; WJC
l end nJ' t. Le
intervals can be de-ive<l . l'hc- co~ff ir.iente
3.:\. ..1 /
tL"'ld Ni.( ~ )
of (30) mny be ec:qu:cd for F(3, •it; tl.e hel:;:. of (~1) :
.,,
o, (u )
u(O)
5
=
j°l"( 8 ) Wtl( 0 , ~ )do
0
~
()2)
u < 1
-1/l
I/)
o,(i,
lie ( '..! )
= _F<a><:s1(i ,· )dil
i
- fl l
•• <1o.1 ) =
!).
.< i)
=
J-'F(e)n1l( i , ~1do :.- 1
"! "
<
i-4 "
<
::
.l
µ
- Ill
l'he specific
cf'
l°'"A•·ure~
"':l•C
~,1~ 1 SiJ
!'unctio112 .:11a.;;c
;:oss..ible to tran!;for:n t!J.e .: :eprt-.a<.·nts-:ion n1 c
ted l'.Wlction 1·'( 3 ) by Hn in;o5rnl i uto ~
"
it
ti:r.~ L1~i
t ..{·r· t· ":':;;e nr. nciou
·oy
91.ur. :
F(n =
1 (:ic<u)caJ.(u 1 ~)
•
~(O)wal(O, e)
1
"'
2::
id
-
a,(..ih~l(LC , 9)]>1>1
[a ,(:t)c~ J (i , O )
1
i.,( i ;on l(i , 01 )
1'"ne attenuati on co~ffieient~ K,(u)
(30 .. drtr-J.:ai1u:: r. ·1e fi tor . The;y m11y b e cboset. f re el,v wit l1in \;h~ lii::1its OJ' !· ·~:t'
OS
the del11,yo 8c{uJ <JJld li s ( Ll
!I
Ol~al renli ~r~tion . The f oll O'-"in1
ct:r>:_co is
i:i::i..Ji:-
to be able to reprencnt ;;he outr ut S~f7!"-.:il b.:
tnru; au int;egr al :
r:, (µ ) K(O) , ae(u ) • B( 0)
0 ;;u < 1
K, (u) c K,(i) , 9, ( u ) •
µ <
+1
1:$ ' '-i ) ; K,(i) ,
a , (~ )
•
-1 < "
.J
ir orr.rr·
S'_tn rat-ier
~
F,c~,-q(\) K{O)\<C•.L'.l,S-•l 0 1~r1~,.11K,li)c&l,i,o-6,~i,] ..
ht
~(O)
• 1, 9(0) • 1 : Kc(i• • K, i) • 0
F0 L9) follow"
fro~
\?0) to L35
J~c(l-,~-1 ;Jµ
F 0 (9; -
(}<)
• '<( ;wal(u, d-'lj
(37)
0
fl.e forll of t.l.e int~l l'!il $'..l'! ,-.; ts call-nr Ll...!~ ~ low _pao- !."'llt~r Ol' - JJ~r,.., 1•1'1.Ci::.el7y - .., Cf,..q_...lcncy .. 0;·1:i.1 ··s::; filLoJ· jn OL·Ue J'
l:JWfir·:~!"
!l.hd o
l:o dist.lHf:llj
fc::-
fil'-P-r .
t~oe
·~·lock
!'.~!..
i L .!' :-r,11, Llt ueunl frequency
~Lc-.-111
1:bi-;ru ir
in
Fig . 410 1
Jil!ern.!rl. in f'i ..· . t11:> . /J.1 ir.tegrj.to!"' I de-:eI·.J..i.r.ca
FIOloFl</l) ~ ,__ :,':...._~ ~~(01
~,.~
"
1
~!!>
GIB)
____ .............
/
~~
•
i;1J'
-----It.== :rF
·-
l'Jc; . '-1 Seque;i·oy lowpn"" r'i I ~er . n) I tock .linr;rarR, b) time
-1i ·1 l'{·UU , CJ r1· 1 1ctical cit".! llli; .
.... ll1l.eb~btO..L' , H holding
circ•tit. , A ouc-r"lt.!.onul
~'"''r-i!'le_c· .
!l(O) according to d2J . n(O) e::ul c;, r-a:nplea at =ne em.:. of
the inCei~rnl _, ;; a < t ot t:::.c ir.Lt.'f'rt<I 01' OU tput by awi tcil
8 I • The integruLor in th"n r eseL 1.:,y 3Witc h s 1. u(O) fo 1•
tllo aection of C( ~) in the inL e!:"val ; z> 9 < t i=> o'L Lui:•"d
b.)' integrating G(e; d11rir:g tLut ~i:1.e .nterval , ecc . a(O)
aust be 11ultipiieJ by wal ( O, 3 - ~) '1CC:>:-a1r:g to (::;7\
-..al (O, 9) is" cor..otnnL ..-i;h 'rn!u" 1 . !.euce, a(OJ\"•I (0, ~-1)
is the voltage e(O) obtsined. in Lh., inter•·al -~ ;l: q < i ,
sampled at 8=+~ nnd s Lored rturini;; Ll.e incerva l lt £1 n :l 1.
A holding circuit 11 ls r>llown it. f.!.e; . 111a , and " p1·acUc.ul
version of this sequ~ncy lowpns:o .fllt,er is ;;nown in F~p; . "-1c .
For numerical valGcc consider !1 rr~quc::cy :a·,..,i.:u . :i1 ter with L ~:;:= cut-off !".:-e-1u.enc:.· - :... ~igr.al :it t~e: outp·Jt
o! tru& <'ilter has; 8( J() indepclrld~r.t =rl: t.tdes :per ::econtl
naoording to the "lllt;>ling i:heorcm o.r l!'ourier '1nnl:;~is .
Ttie output signal ol' tlle "'"quency to>ipnss i"ilte!' oC Fig . 4 1
will J.lave tl1e same itllor:r.ai;io n rat.e , if :it a_ea h:u~ 8GGO
indepe.oden1; a.:op!i tudes per secoutl. . Hence , t!.o:: Stit:;>r. of
Go(9) in Fig . '11":> aunt be ~ = 1/b
e 'lc5 J.S :ong; ;he
cut-on· sequency oqunls ; j = 1/2'!'
4
1 q;:; = '• k7:pr . l,;se
is made here Of tl'le 8rutpl in(o; t!.l'Ot"C:n of \hl~tl-~'Ourier
analysis , which heppeu~ to b e triviolly sirr.:ple i'or soq~ten
cy l owpass filte r s [ti) - [1 0 ] .
2.32 Sequency Bandpass Filters
us derive 1>imp~~ seqaency bo11dp1"s filterr . fhe
mult pl:ca;;ion ;;hcorcms of',:nl>il I mctions d.erived in ::.oction 1.14 a~e needed :
r t
col (i , a)cal(k , ~) • cal(i <:!k, B)
(38)
Bal(i,9)cal(k,B) • aoll [ke{ i - 1 ))+1 , •1)
sal{i,9)sal(k,ll). cal:(i- 1)e{%-1),9]
[C'1 l(0 , 8) = t:al(0 , 9))
'l'hr: multiplication cheorects (381 are 'lery sictilnr co LLrrne
OJ' ainc and cosine !unct;ions, except Lhllt one • e1·1:1 only
Qtnndo on the rii;:ht hnnd side in"to·•d or cwo tor!tll for the
)
kt1"
th.
T~tSS.01' ol tr.torftlehOl'I
'.)8
r;.u:n and th~ diZ'ferDnce of" the ..:·reC!_:iencies . .A consequence
is tnat: tr.c modulavion uf a 1.,'al~n ca!':·ie r ·'Jy u !3igna1
yiel ds
A singlP. (seque:ic;';) s iJeband modulation . ':'his mak.ee
ic posnible to imµleme!lt. oeque111,;y bandy,..·;ss fi I cers b;i1 a
principle well k:io\·1n ':JuL l i ttle .i:;ed for rr-cqucc.cy bnud-
pass filte.t·s .
Let the s i gnal F(e) of
(33) h" " sequenc;y ol1i.Jted " by
muHiplication witll cul ( >: , s) . U,;ing (38) one obtains :
i'( e)Ctil(k 1 9) = ll{0,1Cfil (f; > e; +
.,.,f (
i!c (i
'"
• a s ( i )i;al[[k9(i - 1)J.,.1 ,6J )
1
)Cel( i9k , a}
I
(
39)
a,(k)wa1(0 , e) +
l:'az!;ir.g thj s sigrl.:.i.l ~1:.z.•o u;.i;h ~ lo•·•pati:.-> f~lt.er desci:·:lbed
by 06) yields in "nalogy to (57) tLe o u ~put sign., ;l :F 01 (0) :
F 0 ,(a) =a,( k )wal(0 , ~ -1)=
c al (lt , 13 - 1 ) ~ c al(k 1 t1 )
..,
J a , ( µ )cal( u, 6)c'Jl(;i , 5- 1)du
(40 )
'
Mu~ tlpli<.;ation o.r F ., 1 { e ) by en L( k. 1 e - 1 ) ~hif t5 the fi1-i:;er·c:d oignf.11 to i cs original yc.1:..:;i t;~ou i n lihE" sequenc;t doaain ;
I>!· ~
F 0 ( 9 )=F 01 (9 ) cul(k , &- 1 )=n.( k ) CF• l(k , 8 - 1 )1
j' a,(µ ;cal( µ , a )dµ
'
wal(O , u- 1, CalC:<,9 - 'i ) =cul (k ,S - 1 ) , cd ( k , S- 1 )a1
(t11)
-:.'.'l1e last intebral sugF7es·1;s 1;he name seq_uency bar.dpa.ss
.:il tier. J''oJ' its p:-ac\;ical i!1:plementation OJ1 f' must put a
:r.ult::iplier i n front of r.lle sequency lowpass filter of
:Fi5. 'l1 ;;o T erf'orm t l1e multiplication (;<)) . A seconci multiplier after -che f:;equency lo~·l))ass f i l t:.·:::r pe.rl'o:rn:s the mult.i plic:it i o 11 (lJ1 ) . 2i g . l~ ? sho~·.·::; !;Uch a bruJdµass £ill;er .
1'he san1e .runccion ca.l(k ,9 ) is fed to bot h multipliers ,
sinc12 eal(k , '3 ) has ti:le i:eriod 1 a.nd is ·t;nus identi cal with
oal{~ , '3 -1 ) . Suitable multiplie1~sare.shown inii'ig . 33 . Note
. 2 SEQUENCY BAJ<Ji'ASS ;'IJ,'l'l:JiS
2 3
Fi!; . 42 Scquei.ey bti.!Jd) :1:\n fi l tcr . j·l
11ult# .:.plier :or ·,,•at~!l :·ur..ctoiOc..li, LP
se~uency low~ass filter .
coat multi plication hy " Wa::sL funct' on ~.cnn~ mul Li;>li ca~ion by +1 or -1 only ; !IlulttpUc~tion l!y 11 leuves
a
signal unchru:1ged , mul bi plic::ation by - 1 rcve1·,;o" i Ls ampli t11de .
Fig.43 ahow!.l 1.<ttenuat i on lil'U C.ela,y '11' f\ll1cLiOJ 1 of BBquenc;y l"ox· a cequency lo•.-:paos fil~e1' wir.l1 K(O) = 1 , and
aevera.l bandpua& fit tcrs . The coe:"ficfonco Ke (i) Md K, (i
axe zero, l)XCept Jor tl:c val ..e,; of i for wtich Lhe:y are
shown to be 1 . Tue hlltched areas at i;hc r:ind limits IJ. • i
indicate thot tl:.e run""" ion cal( i, s) or snl ( i, 9 ) pa:rnc~
~ou gh
the !ilter i
they do not pas~ .
0
0
cro~e -!:.al;cJ·1eC
µ-
I
r-
iudtca;c,
t!:.~t
l
~8000s'"'
7· 12Sp1
~?"~4le
0
.:.:XO
'·'llp~
1-
1:1:-i) ~ lX t '
0
"1.i)):'
f•'-31n
~:-:Of"
1-
F1g . '~3 .1\tt.enuation and delay of r;err1nncy i'll~e1·cr.
Ti.,. normali;;od bnndwidth v 7 - µ 1 ~i, - i 1 •Aµ equols 1 ror
nu tilt era or b'ig .J.>3. Let u s denote Lhc quo Lien L 1.1an<l1tjdU11/(lowu bnnd limiL ) = C>µ/µ 1 aa t'<'lntive brmd>J.i.dth .
The followios relations may be seen to hold for A\l/µ 1
,.
100
2 . JHP.ECT TRAliSNISS I O!i
from Fig . 1+3 :
4µ/µ '
1
6u/u 1 = 1/2
.!'or K, ( 1 )
:·01•
1 or K 0 ( 1 ) " 1
K0 l2) = 1 o r K,(?.) = 1
6u/µ 1 = 1/k for K, (.<) = 1 or K5 (k) = 1
The fu.ncLions cal(k , 0 ) oz· snl(k , 0) fed .i.nto the seqttency
bartdpas!$ .:'iltor of Fig .. 1.~2 determine the ~"e 91;j,,re ban.d-
1nidtt. . Relutive b.::wdwidths sntallcr or equa I. 1 onl.:y can be
nc~tievecl wi t!:l tl:liB ci!·cu.i t .
The nOr!llulized sequenc~t µ. ns well i;is the r...nn- normalize-d
sequenc,y qo = µ/T .!'or T = 125 ps a.r e plotted in E"ig . 1~.3 .
rl1e values of G' show l;he channels. l;bnt on{' -i,·:ould use in
1
:ti\ll~ipJ
exing telephon:)'
si~na.J.s ,
if
e:iiC!J
eig.nal oay assume
BOO<: ind epena.em; amJili1;uuee per second.
~~e
sec:_uency response of a tr;cnuat ion an<l. delay shown,
CH'e idealize<'! . In p:·acr.ical filters the root
mear... ::1quai.·e deviat i on o.r ;:;he .:"iljer output vo I t;age- lies
.i.n Fig . 43
between 0 . 01
Md 0 . 00'1 at
the present . rhe a:ean squal'I~
deviation is thus beLNeen 10·4 and 10 -6 1 \..·hich means the·
attenuation i.n t he stop- bands of Fig . 43 is bet•...1 een - J.1Q
and - 60 cU3 .
The infir.itely ~ti Rep fi lto1., cdge!l ni"io•1m i n Ei'i ,g . 43 also
hold for practical fil tcr-s . 'l~h t s resul ~ i s startling lio
an ent<;lneer use<l l;o think ir: t~rms oi' frequency filters .
T10•1,1 eve.r·, seqttency .fi~ 1..ers use a·.·1itc!1es t!:tn t- i..11t.t·oducc a
ti.me quantizacion of t:l:e ~ie;cal . Keeplr:.e; tftis iil mind,
the <Hscom;:Lnu.ous chru1t,e of "~ ~enua~ioJJ is iio ~ SU!'prising.
l'hc di.sconi;inui;;y would disappear, if i;lle Walsh functions
of t!.le -w hole in-erva l -w -~ .a < +QO were used rather tban
vhc s0cti.0~1s in ~he il1ter~1a.J -~ ~ e < .; .
At-ccnuation and de!a~rof r; 1;;qucnc;y· filters a.t'e constant
2n the -pa;:;s -band accorc.ing -:;o F'ig . ll3 . Hc11ce , tl1er e are
in.liez'ently l!O :iLLeuuaLion or del:iy distortions . Deley ill
olie sLoµ-b~.nds is not de£inec1 .for ideal fil.ters , since no
.!?Oet"'gy i::: :passed . Rcfll i'ilterr.i. pass energy in the sto_p ...
bru1us . 'l'his delay is s]lown by <lashed lines in Pig . 1f3 .
1 01
Fig. 44 (left) Approxi:r.ation of •ir.tW<Hdt<l 1 unctiou~ by
Walsh functions . A: sinusoid~ - wnve , i'rcqucnc:.Y 2~C' f.7.. ;
ll: a(O)wal(O , o> ; C : a.(1)sal~1 ,i< ) ;
: a,( 1 )cu:.(1, 9 , ;
E: a,(2)3111(2 , a ) ; F is t!le "um ~:· B ond " ; c; is tl.e £Wll or
B, t: and D; Ii is t.J1e ~u;r. of 3~ C, J -i:__j E. '11.ne ·~ar.e :=: •
~, os; ho1·izootol scale o . ~ :n"/div .
Pig . 45 (rit)lt) w,,1,,t- ?our:.er t~·"-.'lSfo:rm: of S.lll~oO-<i~l >18ve.: A: ainusoid3l t:nves , 1'reqt=enc:: 1 k!I:-. , vnrio•J. .... i.;ha:s~s;
-horizol!tal ecnle O. "i ms/div . "E and C eho\, 11:t1 .. s - J!t1urir-r
;,rllllsl"orms n,(<;fl') and a,(<11I) o!: _<. ; Liae b '"" ! • 1 . . ~ "i
'-' 0 r:t.zontal scu.l.e 625 zp=>/div . (lio- h o,;cil l ot;t"O:t.; courle~y
C. BOESSWETn:R a nd W. KLEm of Techni:Jct.e llocbochu.le Da..-m91;adL) •
.Fig . i11+ ahowo a sine wave (A) ac ~ he wpu~ uf "everal
•equency .filtero nnd the resulting ou1;put. ~J.gnalH . ll t8
th~ ou~puc of a sequency lo1·ipass fili:er wl.tlt Kc(O) = 1 .
102
2 . DIREC'r ·rRANSMISS!ON
C, :> 3£d E :u·e the outputs of baucipass f i ltoers '.'iiLh K, {1)
- 1 , K, (1) = 1 and K, (2) = 1 . F , G anc H are tl:e o utputs
obCaj ncd _from several bandpass i'ilte.L·s c onnected in parall o I: Kc(O) = K,(1 ) = 1 (!') ; Ke(O) = K,( 1 ) = Ke ( 1 ) ='i ( G) ;
Kc( C) = K, ( 1 ) = K c (1) = K 5 ( 2) = 1 (II) .
f!'ig . '+~ ~how~ ~equ enCJ' 8.J;lplitude $>pectr.a o f si:-i;,isoidal_
funct;io ns (A) of equal frequency olld. o.mp l i tude but various
phAse!i . The aarr>litude opectra a.Cu)= a<(q>T) ar.eshownby
ii.<
1:1 , the nn:p li;;ud;: npcctra a, ( ~ ) =
:p'r) are zho;m by c .
Thr. oscillogr;ims E a nd C were obt;;dnon by samp l ing the
o~t:put voltages of a b r,ink of i6 s.cqucncy fi l ters . Squru•ing
ai1d addlng the traces B and C y i elds t f'.e sr::-(lur.nc.::y _power
sp-ec ~ra .
. ,6
.6
b6,.
6
c
f0(0)~
0
S
10
O•t/T-
IS
l•'ig . 41:,:; Sequenc;y wide bru~dpass .fllter . 1,p sequoncy low_pnss
.fil tf!r , H holdine; cirr.:LL.t .
103
3ondpass filtc-re accor<l:ur: to Fil'" . -< fe;•mit rcl<ot;v(;
baJ'.1WidthD "4J/U1 a ~ , 112, 1 ' , • • • Only . fLf . 4· . •O\ff o
bandP ss 1'iltur fo1· :·e:at.i·.·~ bar.:l.\•i<!th LJl-.J.,. -- , ~ , : , ..
This circuit Utit!"$ n laxpn!:e r:.:.:e:· L?1 :.i.ccordin :.o i"ie ~ ~... ,
which iatcgrnLes the i:irut ric,nnl.:i over ti1t:1 int.;rvoJ!= of
durat ion T . A !"u.rtl1e ..... _0~1:;ias!: _:'il:;cr :_p: .inL'~fJ.'utes •Jver
ticle intcrvn In 01 du:-r:"";ion T 12 , D!.' T/3 , or T1:.1 t • • • 'l't-.e
output voltogot: or r'·io t wo lowpas~ fllt.1·1·n · 1·1.••• hnwn -:.n
Fi g . 4bt• w.1d b j tlrn integr~tion period of .,J.·1 !llld LP2 in
T/2 . 'the dif;'eren~ rieluy kl.mil~ QJ' I i'1 '.l.ni;I
1,.P2 ar~ cornnc1;ua t e<l by tlce holdin l:' cl~·cuit Sl . The difference oJ tho vol I ai;oo 0 1 Fi i:; •.1+6LJ "n'l t· ,y iol".s :>hu ou tpui;
signal p (9) or ti·"' 'didc bru:dpnse .!'ilt· r- .
chosen oqual
~o
Tht;re ia a.oot11e:- irt.porta.nt d i fft-=.·t::lCt: bor:·,.;crr tte bL~C
;nss !liters of Fig . 46 ~d '•2 be£iJ0;.-s tliP. t11frnr-ent. :_--e~a
:ive b1Wdwicth . T~.e func~io::.s snl(i.;,3 e:-woll si:;cqJ(u , g)
may pa:;:; the fille.r o: ?ig . 4&in !:!'".<" rae-s-tnr.u 1 while only s&l(u , a) or cul(i.l ,5 ) a:,.y Fa~:: :; l'ilt1n· 1ccor<1i"•5 to
F'i>; . 4l.
f 0 (0)
~
l•J.g . ~7
Ser;uHHl!.'/
lli 1 and
:-f"r,ur·Hc:r·
,,,~r:;1tJL":' !'ilce1·
Lu
J•• r-;-,f
~·::.ter
(b ' . r,1 ;;~~-.1f· 11 cY to\\'11a:i ::t .::i1:.e.r·,
ET ::;equenc~; tq· :IJ 'J.- .. !-l.t •.;':. ,
I:·
d elay circt:i ~ .
A grea.c vru.•iety of filters may ce ae:-.i.·:ud !.:."Ua. t.Lt.' ba.-
flic !:ypee discusaed. Fi,e; . .(·7 shows a ... equcnc:r L..it:"h;iuss ar.d
" sequency bandatop filter derived from lo~. pa~" and band -
pass filters . Parallel conne ction of novorn l hnndpn:;r 1ilte:r~ accordiJJe; to Fig . 112 yields fi1Le1·s ~l 1u~ le~ pas s
sal (u ,9 ) os well os cal(µ , 9 ) and have rr•l&tl.ve bandwidtils
Au/u 1 • 1, 1/2 , 1/}, ... ; on t he otl1eL· h:.u1d , one may ol-
ta_!! 4<icle bandi:tts: i"'.il:;.1:-:-s
cal u,9 ' or. :; a.:;.
1
:;.Lat
-!:t
pn!"'r-
5s,'(t.o. , B)
or
t ")\\"n Ly tz...e osc!.l:og:-ncn cf Fig . t-1.;_f- H.
2. 33 Digital Sequency Filters
t1 o~d..
One of -C.ht:!
pr•nmi?.i:jg a pplicat.ior.n o f' .sequtncy fil -
t-ors based ou '.·uJ .. rh t'unctio.ris. nre nig1r~1l :"ilLers . Th'l
rf"O!iOD ts thliL t.l.i..Cll"C·i.C&1 \..'nlsh.-F:iu:-iP I' trw1sfo.:·m.:atioo of
eigna.ls requires a.dd it !O!'t~ ar.:.d
:::ubt ?"~ct i. O!'lS Q[_ly i while
!J:.Ietr.rical ?'ourie1· t:-M:"':"'o : :i:.fitic::_
:-~1uJr,... s n:.ult~i:;:ica-;ions .
~~;!@]-§14~~~91
p
'"ig . 48 Block dingt'nm of" ilie:i tal nriJU'-':ic:1 !'ll~cr . l·P nerp..tf.:!lCS l ot\'ffl:l:l l'i 1 r.et' u! F.ig. 41 ; JUI a.r~r. lo£;/ l i@:il.ul con"l~rt:e!' ~ ST ciit l t~~ :,:,t.oi·ngr:· , AIJ ~r·it..:w-.,..l.:.c Jn i L ~~r~ornin.g
\.il.lU.L.t.:.onz e:nd .o·..41.:t.rnct"!.ons , JA Ci£:iZ::ll/ar.~:.J;:- conver.'t..er .
?oi.· ~ ci--i lanollou c:· a dii;..:t~l ... .-.qur-r.cy f.!.lt..er concii.-·r vbe ·~:ocr~ dingr:_,11 of Fig . --:S . A c.it;;!u.il F(e. ispa.s ·ed
t.11rot;gf a sequcnc;v
· lor: F , ( 6 )
~t 11...~
}<)~'! 4:;::-:
ti_ t,.r
ou::r,i.r. ltr:t'./e
l
.
.... ~·I. ":nC' r;':".cp .;"unc-
E:.P;•.:1 ("If Gll!'0.1ior.
:;16 .
rne
x1·11 l lt.udc~ !"!~· the ~tCJ.·~ r,tre c9UV'C'rtr.d hy the t.utul og/dlgi-
l;ill
.::onve:-~cr
l~~l ~v=.t:e~
Afl i11t.o I innry Ci f:.:tul c:hnJ·•·1r•t:p1·t1 , whicl. are
nt tt..
2·•1Lt..~
or
1 1~. ~';.-1ar:.1ct>;;!rl:i
P"r
Lim~-.
unit ':' to
t?1E <lit-~ital ~';;,..t'L,i.~·.·· S'I'" . .... er s set v!' 11J i;;onr:ecu~ive chsr6.""tll;"r$ l:e 1.-·!'lotc-0 ty A., l-, •. _ ~ ~ , ••• 1 .f. . 'l'l-e -~9;Si; f:als1".-
1'0l4rie.r
OC ·eCt.iou 1 . ?- ;r,ny tf": 1:~-ec. 1=0 obtain
lro~ -:r;o~~ 1 1 c .... r-ractJ<":"'rf th~ 1 L• t.:o~f.fJ.clc:uls a('J) , a 5 {1) ,
1-1,(-:) , .•• , a._(01 . Ar.oi 11on~ b:llC. su::>t..i·uut.:""ou:: o!:ily have to
t.:'3.lt::~·onr.
l>e J.•e!·fortr.n:i
l1~·
t.i.""
r
1•il:.r.i.-·tic unit AU Lo uLLai;_ one , se-
vei"lul ci· eilJ c.t' t li O$f• <.:OC'-l'J'icienr.;s . J°lSS1.Hno 1,,he coefficient
,(le(!;.) ul o!ll": in ~on:;.1•.~f. Hd c.::; shown in 1"1g . 1~8 . 'l'he 1 5 coefficients -ac(~) 1 +nc (~J , -rac(5) . -Dc(5J, .. with the oigns
.;ar:-~!ipon.Jiul". to th~ ai gr.s or cal(~ , 6) in t·ig . ? are trana-
Jerr'><I in1':> ti.~ digH 11 s~or:;ge 5'1'2 . P.enuin;:; &nese 16 co •:ffic:.c;::::;" o"c cnr1&lly t!lroup 3 dir,itel/i;malog convert.-·r D.h yi~ld:- ti,e nr~~lot) ou~put signal ~" 0 (9) . The connec-
105
tion bet;wenn inp:.i~ ,;igntl F( e) and o"t»ut cii:;na l F 0 ( a )
foll ows from (;2) ""d \~', :
rn
'F(S }c~l'5.~ )de =
'
• 1/l
(it?}
!• ac\~Jc•l(u,O)da
s
Let f' 0(G) 1.>e obtaine<lroy:f,.eain,. cql('.,a) int;o Lite se1
q-uency .rill.:er of Fig . l.!2 . Let 11 0 (9~· df""vin~·.p f1·o:ri lta nocij.no.1 val u~ accorJiu~~ l,oaG~ussian Ci rtrib11L io11wiLl1 .u.ea.:J
aqual'P. dcviacion 10 · • i'efeJ.'red t o n u:iH vuH;;i,i-te . 'l'rie
crosstttlk tto nu'1tioL of ohe coeffi cirnt!I "c ( l J, l f. :> ,
lll!d n5(i) iP. n.on _,·,o dE . A wucll 11ir;hrt' cr-os~tulk "+.temiation cnn 1.>e o bt;:.in.-d 1::; a c i £(it~ l :'i lL er . l·t! t F 1 (ii l in
Pig . US have ~hn cenn ~qaare Ce,:in~ion "1L ~• 1~f~!':>~d to 3
uni• voltage . If 2n sa:i.yles of F 1 (3) ..re uee~ to "<' ·~~te
ac<5J. and i t the :;<.o:o;<;/d.ig2 ta. =or:•:er-.ion introduc~z "
negligibl~ et'ro1·, ou~ oO."tainz :::. oe:1n SQU9:t"e ~Q\.. :.a!:ic:? o~~
ae(5 ) o! 10" 6 ;2n 1·eferrcd to ~ "'11it voH· i;c . 'I"H:> c=·osstalk nt:ten11ation ir: d:I is L!.:.11~ 10 log 10 ';~·" = - : uO '
-'- 10n log 2) . l•'o r ;:in -= ,,,_ as ~!'ted in Fig . 110 one o'tit~ i n:::
• cro sstnlk nttenuat .i.or. ur - (bO - 40 !<>c; :' J : - 72 <11' .
The u~uul rurnlog l'll~ers are caJ>nbl<: L1> !"il ter f·J nctions or one v11__riable s ucl1 as th~ out;puL volt.:u('.:l;c or a llicro_phone, which in a j''i..~ction of t.itl;fJ . 'l'ho g r ny:1c~s of
8 black-nnd-•..JhitO picture i.s :l ~WlC~iOlA Of tWO :O!'O.-.:e V8.riablcs . Colow· pict.ui~e:; or vel cvi~ion pi.:t.lL.!·ea a.!'9 fur:rtion s ot two space Vhl..ja0.l"f!~ ~"";j n t"?"-Jird vu.l~i~:ole "&taj
standn !or the -;h.ree ba.Eic colo~=~ or
tho ;in:,~ . Digi"":n.l
tilt rs can be proe-rBl:iDed to .fil~er ~ttch fttnc";io~u o.:· two
or t:irce variables even t!.touf!;t. the ~nnction~ qr.-• l'ej in-:;o
tile !ilter iserinl ly like !unction" o:· or:e v:u·.'-:il.le . '!'he
C:O!r!J'luting effort bccorr.cs cnoroou!l nJ1d tl:.e lrWe1·enL :--elativ<) simpl icic;y of digito.1 sequency f illct'S com":ired ;;i tr.
frequency !ilter~ becomes an ir&por~a.nL t\uvori l op·e . PllA'l"l',
KAl{L W1(1 ANDREWS hove proi;rrunmecl n digi t nl f il t01· ;·or
functions of two variables and used it for Lite reduction or
in.fo1•mntion of still pictures , as was point I'd out in 1 . ?;, .
3. Carrier Transmission of Signals
3. 1 Amplitude Modulation (AM)
3.11 Modulation and Synchronous Demodulation
TLe t.r aL~oission of cons~ant~ ax l ,; ) by o .:.y!;te:o of ort!'logor.al f w1ctlou:, I I'( ..1, :]) I t as b een <li8CJdee d i n the
previ ou s chap Le1· . Tl •O t r1m~mission of Ll31t' [ unc tion s F(9 )
or F x( e ) by mea n:• oJ' <• :;ys t e:n or ti me r unco.io us 11•(k , e) J
~·:ill h e d i.SC1lSStH.i. HOW .
Thi"! !IC furi.Ction~
1~ (k
,a ) wil l be Cal-
len carriers . F(9) denote ~ a ny ciao~ fu11ctio n, e . g . the
ouLput voltage or a a.icropl:c::e . l't.e nOLb.tion F.( ~ ) ~ s usP.d
-o f:-JJIJ!;.as-i ze ~i.JJ.o a •net.ions t!lat con:ain n J'.inite number
o f coefficieut.:-;; tt.--(~ ) , .1 uch .e.!:i t..eletyr.o :r:.gn:tl:t :
m-1
F,(oJ
=
.L:a/,i.1f(.1 , aJ
i:O
rh ~
~(" , &J are ;n·ed.omir.n:>tly ~ i ue and cosine
a - -:..11~ ;:. t"('(if'?: r . r h e =e ir , hawove r' ne it her n
n:.at.:_et!U3t l ceii. no;· u ptiy:::ica..... t·eason il.'t-,y ot:her functions
carr ierc
fll..!lct lon ~
1
Ct1UlJ :iot; be used . TCi.:: hc·lds !'01·
t.rrulfitli~~ion
via '"ire
llnr.= , ~f!J.·.re £"Ji:j.ES: , rnd.!.o links , etc . I~rio.1tc r;rai ns o f
?t.:.l!Se~ arr uti. d t.o soa:e ext.l:!nt. tsS cu:·:·icrs in c a bles .
:•\1nc t. io11:J t:-1at ffJl'ID n f!ro up •...• 1~h reap1.::ct;. to !Du l tipl;:.-
block
cntion ai·e va r ticuJ '1r• .Y we-1~ s u:.4>cd an cArl'id !'S from the
mn Lf:-. e m.:jtical p oint: o f' vi e w. Ampl :Ltuoo modul a:cion oi' s uch
l\.uu.:. Lion::: yield:-; .i!lhl'r~n't l.Y a s l1tgl e t-i ld r- 1 And ood ul !it i on .
r·1u:t.;l plcx r:ysteai- ucing riu ch carr~ierti do not need Sinble
. tJe1and filtPJ'. • fhe te~Cl sir.g!e sideLn:ic.l ILOdulati on i S
uGc.i h.a1·e 'l'fith n 111ore gene-rat r::.eaning then usual . Tne
exuct cear:ine: of t':l1. ana o't!!er :.er::r.z t;.:;ed in a genernli~Fd 5{'DSC' 1.r; be~t
' xpln..i!:ed by
OD OXW:1 J
le .
1 0?
_; . 11 SYllCilRO:JOUS DEMOJW..JcTIOtJ
Consider ltJll }i tuc" 2JOdcil at ion O!" 6 co:;ino !'uc.ction cy
i:;ignal .i'(li) . uet !'~ 'l / be eXpUlde<l illl:O A !'°ourier .;erie'°'
in tne inter·:!ll _, ~ o < ~ :
El
-
r (e) • n(O) + 1{2 /. [acCi) cos 2irr& • '.l,(i) cin 2i-ril]
(1)
i:.I
Let F(9) rusr. ~tu·out;h u lowpas" l'ilter tb1L s uppre3,;es
all terms 0£ t!-.e sum with index i > k . Suc!l l'ilteJ:S can
be implemcnLcd voi·y ltlttch lilr<> t he seq u e11c,y l'iltern i n c;oc-
tian ? . 31, bu~ 1 mp leweutai;ion iG oJ' no .i.ropo1·Lt1JlC~ be.r:e ,
'J'lle rilroreu signal F 7 ( e ) has i • I< ::-nth er L!Hui i • ~ ;;s
upp"r limi~ or the ,;um ( 1 ) . Ar.1~'ituJ<' mo<.lu-~cioll or 10::ie
carri• r 1{2 cos 11 0 1 b;1 F · ta l yie-do :
F• (9)(2co,,.0 1
•
9=L:[a«i1cos(:1, -?.,i;~-a,(i)dn(.1
i•I
+a(J)1{2 coe 0 0 9-
(2)
0 -21i)3]
• [o.«i}co~((l0 4?..,i)~+u.(i)cin(n,+--, i :s ;
2:
,,,
The first llUJ:I rcr resent" t;:,., :owr,,• d,lebuutl . [t ~ol
lows tho term wi.th tne f.:·equenc;; i.1 0 ot' tile cw:rier i ·w hich
is p!'oduocd by the DC compouent of F ' (~) . Tlie second mm
reprecente tbe upper sideba.nct Lot 1•'(6) be exptt..tldecl i !.lto a \falv!1 . "l'ie. :
F(e ) • a(O) +
..
2; [a,(i)cac.(i,» 1
..
•
~,(l)o,.1(i , 91J
(5)
"I
• n(O) +
L, [acCi)w"1(2i , 3)
..
+ ns(i),..:iJ.(21 -1 , ~)]
Let F(e) pass Lh.t·out:;h a sequcc,cy L-•r•:u.•S .filter that
suppr.. s.ses all terms witl1 index i > /. . !'Lt l'iHeroct sisnai E"(o) ha~ i = It as upper limit of tl.o ou:n (~) . Tile
serJo s ~x:pa.nsiou o.r F 1 (9) an(l F 11 (SJ hovo thus Lile sa1:1e
nll!nboi· ol' t;ormn . Amplitude modulation of o W0lno carrier
wal (j , a) by Ftt(a) yields :
108
f •
a(o) .,.\·a1( ;; , ")
r:
+
£.. t
~<
r,) +
, i) •1· ' \ J''>•-·
\ '--irt. t ~
r:1
;Jomnari son of (4) rued (2) shows Lh«L L'•<" DC con.ponent
u(U) i ;; ~ranswi1..tcJ in boLL c~6es
by
t..hu u.nchan~e,.:._ car-
c·ier. rhere is , !10•11f'vrr, o!le su:n only iu {'~ ~ . liependh •
on t.=.e value of J, this ~wr. ts&y de!lcr1be ~;upper ', ' ...o1
o:- 'part!.s U!lper, partl:; loner nide°:Jb.!l<!' . Cor..s:._der,
!"O!) ex!l!!r>la , h. u\1.'.tbc-r 2k l:avict: 11 C.igi~. l:; binary r:ctatio1J . :.et j be ~8l',i5<"'r :;nan 2k a.r..d .... P.t ~i )1!ive- Z~I":>S at the
"' lowe!:it bi:i~ry pluc1.1s . J•he _.. ollo•.-.•ing f'('1 nt.icns hold :
w~1· '
e
21
j
= j
{ 2i - 1)
i
;o
e
+ 2, J ,
j
j
I
11 1
j
• • • •
1, J +
-i
2 1t > j
3, .... .
I<~ -
(5)
1
>
J
1 , 2 ' . . . . 2r. < J
All !JJU.ices 2i9j u..nc! (~'i-1 :e~ J.· t.t,c s~a 1 1- .~·e larger ;:hn:: "t":Je iu<!ex j o.:- t:1e c:'!rri .., •. •,;1Jl( .~ , ; J for tr c
Cl!oice o:' j . Thls corr!'~ponUs -!;n .'!.!I u;•;, r ,"'.-idebnnd a:odul~t.lor:. .
As u f urt her exUJ:1plc , :.et j !111vF
binary pl uccz .
2i
I
e J
2,. -
~ j
0Ju.~ oh~nir:s
- 2,
1) 6 j
, -
4
1
- J - 1, j
• • • •
- j ,
~
-
~nnl' 1>1;
?k < .1
. •. .
Lile 'l 101·1est
ca~ri :
in -clti:::
j
(!:.)
- 2l( < j
. .;o-..· i;t.e icdice3 219j cwC {?:. - 1)1:tj in t.=.a
su=:i (4) a.re
all smaller than tLe ind<>;: of tl:e c:u·:·icr wn (j , 3) . This
corr-espo1:ds
~o
a
.owt-!~
Hi::ic:band moci.ult..it..ion .
'file nun:b~rn 21El,; u.r~d \?i- 1 )aj wil: be Co!~ CIC3.cte:in vaLucc of i l tu·i:;c1· c'oo n j !l.nrl ::or other va J ue~ :imaller than
has nei tlir.-r ;1,01'0.s only nor on or. only on -che fl lo\..'r;;::st t: i r.a.ry plucet: . Thin corr· e~pond.::i to n :partly upper,
psrtly lower sidel>iud <1odut,,•ior..
W'_y does =1·lit:ide l!lodulation or ~ico nnd co~ine carriers y!.e:d two sic!eba."1tls , but a:uplitude rnortulation o~
1
1/nl:;b carriers ouly one? sideband? For the rul£','/E!' consider
,j 1 if j
~ . 11 SYNCJIROliOUS DEJ-JODU:.AT IOii
&he i:iul¥1J'll1Catiou t.!:.eore:os Of sine
ttr.1 cosine :
.
-cor. 1. i-r. }3 T .:oe(iT.<.)3
(7)
2 cos 19 co:i ite
•Sin(
i
ltj9
T
~rn(
i~k);
..:e
=
2 sin 18 cos
cin(i•t:H
2 COP e sin ;. = - sin~i-i-!)9
;.
s
•COS(i-%)~
co,-;(iTk)9
=
2 sin i8 sin
There is a tiWL of two sir.P or cosL"le !'u.ncttor.s on <::he
r i ght hand siues of these e qunc i om; . Le L Cu~ ke 01 sill M
be cuJ.·.t'iO.L'a and cof:) ie or ein i9 Fourl er compoJ1ents of a
ai snal t hE1L :u·e amplitude modulated o nLo ~J1ose carriers .
.
An upper wul n l o wer side-oscilloLioH l " p1·o<luced . He nce ,
the double nidebancl modu-ation of si11e Wld cosir.e ca:-riers
1 s a consoqueuce of tt:c mul ".:iplicati•>n th~or<'tw (7) .
!.et us consider once I:lO:'.'e tne t1t!- ti f ! J.Cntion t.Leo1:eos
of \lalsh J'unctlons :
cal(i, 9 )cal (i< , 9) e cal"r ,9 )
sal(i, 8 )cul(k ,e) • 5:-;l(r,a)
cal(i, 9)sal(;. ,9 )
sal(r , 9)
eal(i, e)sal(;. , 9)
cal(:- ,• )
:.'
r
r
r
i
e
c-o,\
It
- 1) J ~ 1
~i6 (k - 1:) • 1
" l i - 1) e (~ - 1 )
(ke ( i
There is only one Walsh fuu ctlon on tht: rif;hL r.and
sides ol (8) . Lot cnl(k , 9) or sal (k , 8) 'or. or.i.n·iers anc
e al(i,e ) or ~al(i , 9) Walsl" con:poncnts of (1 signal chaL
are ruopli tude module1ted onto ~ha cnr1•i.el'tJ . r·t~a ro i s no L
one upper and one:- lowc-r 1 side-fw1ctiou • l.:·ut one fll!Jct.lon
only. This ia the reason ·1;l!y .ampl ituUe :r10llJlr.1tion of ··~'o.lst
f unctioi::i ;yields a •~gle sideband rr.od.1.•>tiou .
A circuit for a:nplitude eod\:la":tion of 11 Wo.: .. b carr=.ar
is &hown in Pig ."9a.
T>ie co nine cnrrier '{2 co" r.,e l!:odula Le.I t::• f '( e) iL (2'
=>ay be demodulated by multi;;licntion wHI. 'f2 cvo r:0 e :
(F
1
Ce ) '{2 cos o 0 aJ 'f2 cos 0 0 e
=
P 1 (e)(1 +coo
2r. 0 ~)
(9)
The Ure~ term on Lhe right nand side 1·op:t•N1cnt:' tr.o dclltO<;l1Llat ed signa l. 'I'he second t;erm mua l, be aurµr·esscd b;y
~ . .fil ter . Halt the power is lost on tho nvornge by ~liis
l l.li;eJ:'ing . Thio poweJ:> loss is unimportant, if \.lie produce
; • <.:APJlIER 'l'llAr:s:nssrmr
·110
:··(u )1{2 cos n 0 a can be a.:nplifi<'.I before mu: ':.iplication
with '{2 cos (l0 3 .
I.el a Big,.-"', Dt ( e ) be -::ran:n:ai tt~~l l;y a Ch.rl~ie= \r2 CO!! a, 3 .
Syru.:bi·o:Jou~
demodulation by -,f-2 cos 0 0 8 v.:.~:d!'· :
[!>'(a )•[2 cos n , a :•r2 cos 0 0 e=JJ '(e )[ ooaC n 0 -11 1 )a• co"( n0 +o, )eJ
(10)
LC1t t he oJgnals 1" ' (9) aud D'(e) cont~in oscillatio::os
wlLh 1·,·cq uoncicn in ti1e in:Le1·vu.1 0 ~ v "' v, only and le1;
~lw
<lemodulated signal r, pass
~tu·ouf:!i
a Jrornency !'il te:c
'.<lLh cut-off frequency "• . F '( D) wL l be rl'.>co' ved l<ithoLLt
i11t.111•rere nc.:e l'rom n+(e ~ 1 if t:.ie f..L·e11u~r1cies c:- tr.c carriers
'{2 co" n 0 e nr:d l{2 ccz 111 e saLi>.r.v oh< !'o I ?win;;; condicionG :
l<n, - n,;/2rrl -v ;,; v, , C t!vliv 0 •
L<-t the d;;na~ 1' 1 (e )f2 coa n0e be fir,ci;
(11)
rtultiplied b;;
~ nu.xi~i:iry
carrier f -2 cos Gl'la !'l.JlC lc-t: tt.e ;..rc:l..ict.
lio J>odu~ ated b;.• au:t=.p~icat1011 wi t..n 2 cot! (n 0 -n,, )e :
t...,.~n
(12)
= F'(a)L1~ CO!'l
2(n. - n ,)6
I
coc ?n. a+
CO!'l
?(106)
~Le
c!csireC term li' t(a) i::; obtaino<1 . 'l lio t:~11~ee oot desired
~ e.:.·1ni:J on the right hand side 10u?:Jt bl"' J'i. Lt cl"t:-d .
Let n !ligna l D 1 ( e }l[~ cos ( n 0 -?n. )9 1-~ l"Cccivcd . Direct
r-yncl.1'0noua n:cdu.Cation 'lCCOl'dint; Lo (9) yield>::
(D'(e~{2cos
(n 0 - 2r..J&]1{2cos n 0 e •
=
(13)
D '(•)[cos 2n.a • coz 2(0 0 - 0.)e]
The ~igmtl (15) =;; be filtered , i f tho :rccpeney bandwil!th or D'"(e) i.-; Eu1~~=.cient1y s:&til!. . Henc!" , tbe.?'"e is no
int~rforenco Uetwee11 F-( ~) and this int"l "'f' sig::.a: . 'Phi s is
uot !lo , i:' •he sit-'.n"1 D' (e )'{2 cos (0 0 -20 . )S i,; first n ulti plio<l by an auxil i <i0'Y e~Tier '{2 cos n. S and l '1en delllodulriled by :nultiplication with 2 coo (n 0 -n. )0:
( [ D1(A )1[2 co~ ( 0 0 - 20.)0]1[2 c os 0.0 12 con
(n 0 - n. )a
(11t)
• JJ' (e ){cos 2r.,e +cos 2(0 0 - 2n 1,)a .1+co"2(00 - 0.)aJ
3 . 11 SYNCllllOHOuS Dlli':ODULJ1T IO:l
11 1
The tcr!ll !J 1 (S) <>Fpear,; o:i tlio !'!giJL hnnd slde of ( 11;\
irt ( e ) i .. rr ctcJ b;,· LILe i :oace sig!.'1! D•(e) . Oae :11ay see
rroo (7) am! ( 1.:.) ~hnt t!:e ~·ecepL io:.. o!· ic:ai:;e c i .,;nG.l::: is
!l consequ~uce or thr. mul :. i pl..:c:ilio:>:... theorcu. .... of sit... e nr..d
cos ir..e . 'rhc-!·e 1,oul<l b~ nc· :.?:..tc rfer·cn~c hy .i;nne;c· 8 i &nllls ,
LC c-here w<!'t:t! or.o- tcr::n rntt-.e r thn..:: two on ~he .r ic;ht hwid
side of (7) ·
F1aiJ-:-:-l F"!~;r~-11''(0lt•l(j,al
-a--~~
a
cal(j,Ol
r" tm,.1~~:e1
b
l
<al(j.9)
Fig . "o .:Jnplitude cioculo.:;or "') a~.d .1~~.octu.nlc1 ('u) :·or
'.Ja.ls!l carrieru . LP sequeney lo·n;•ass f1l ccr ; f1 a.ultirlie=
: or Walt:h fur1ction" .
Let us 0011~.i.;Je.:.· Lhe 1::1 !:11ll.e ;.. .cvct: .JSf:r 1 i 1 \\'hl:Jlt t;\J..r'.r ie!.'S
uoed inttlieacl or sine- co s ine cerJ"·i r.r:i . _.t~~ th.,... tJ.J.t;uul
Fff(e )«nl(1 , e) oJ' ( 4 ) be mul,;i pl ic(l b;y wi il(j , O) :
si~e
[P•( e)wnl(j , 9) ]wnl(j , 9 J - F'' ( BJwa1:u , a) • & • ,~,
J
(ll
( 1 5;
J • 0
?he:rc is no hish .-eque-r:cy tc-rn =a 3,.~ i"i l t~!· ..... d , coi ~r91..y
to s ynch1-onous dcmoculatior: 0 1· si:_c-co~i!l~ ca.!·riers . ~oi-;
•ver , thin diffet·ence ust.ally o.~!ll.S ve:·y litl.lr. , oinc<"
r~l:;ering in required axrp:ay in 11ul~icha11nct ::y. t.r.nE i::
Order to Separate sigr.al~ froct di.;'ferel\1; Ch;nnelr . fo ;;i'.!o"
thar;, let: a signal L·"( e ) be rra~!:ciittr.- .1 b:r t~ :-:econd. ce:.rl'ier ""1(1 , e ) :
u( e ) +
2k
2::• ( ilt ( i )co.l ( i , e ) +
"'
~~ c(r)wa J ( :r, a )
~
b,(i)nal (!. , fll ]
( 1 6)
:5 . CAh!ilI:.ll 1'P.Al'lc:1ISStotr
112
S;rr.c:.roncus ccnocul ai:ion of
yielito :
0 " le J>tal\ 1 , R )
by
we.: ( j, 9)
(17)
" c(r)~·al(-Eijar , a)
, ;?::
"'
i.~"'r ~"· tt{e) and n~r-{ e )
coutaiJ1 i''ulSJ1 i'unct: i o ns \·•a!(0 , 9 ) ,
cotlt.l . ~) "''d sal(i , e) 1-1itl i 'I< only , oi• '.·l ah;Ll func tions
1·w:(·· 1e) wi th r ~ 2lc only . I.et funller Lhe dea:odulated
.Sifl"mtl>! b" f ll te r·ed by a i:;<>qucncy OWJ)S99 f:'..l t~r wit h cut ol'I' ~equency u•k . i;o «al >Jh fw;.ceion waJ(Jlll.;er ,s Jo f (17)
d ! l ra"~· -:;hro'"t;:h tu.~ .filter-, l.f t!IP i'ol Lowing condition
1.r. sn:i~:·:.ec :
~ ;::: ,
l tZI j if r
r
= O, ' , .... 2·.
(18)
Or.ly ~~o of tL~ ccar.y ~o~s~l:lc ~·"'1'5 ~o ca;;is:'y (~8) will
be di.::cun5e:-d . Lei; ~!Je nuaUe.:.· .2k huvd' 'l t:._nnry digit~ ~ Let
,1 !lnd l t" lart;er ~har. Ck or.i.il let. t.Lem h~ve =r:ros only at
t.!1e 11 lo·,;e~t Lincir;f placer . One obL.:.aiJld :
l a;
l'.
l rr .:i '!.1
1 , l + 1, 1
,..
= i
E;
j , i
i
2,
l
I
: ,J + 1 , . . . .
;u.
( 19)
l a- ,i , 2k
condi~ion
'f'he
l ai .I -. "?.
ttll. t
·1;e
S&.t;L;.!'~eil , ir, order fo:- ( 18)
(20)
co ~.old . Aci.C.i.n15 j
mod1Jlo 2 o:o bctt. sides o:' (20J yi.,ld.; :
The l~::t ~rc...a~£o.::astic~ u~f!S tl.Ju !·elttlion j a; j = 0 and
t!.(1 Jnct: tl.f.r. 2; ':ac ~ Unary place~ cnl:1, wl:il e j
has
;.:,,l·o: nt iT,:: , lo•.,·e~t binary i l 1co~ .
AC:dir.g l '1mC:ulo ? in (20) :yicld.o " :>econd possibi~it1
LIJ a:•Ll •. f;v \18\ :
J "'
;'>k S' l = :'.'k + 1
(22)
con<it rio 1rn I 1ke (21) qr.rl (?2) divi-lc the seq uency spec-
11 5
~
ini;o 5equency c1'a=el6 just a!! the frcqufncy :-; ecie dividud into f1•equency c!:amnels by •he req'1ire trull'
ment of certair: fL'equcncy 11m.lw1dtr_s f or the chwin< .c . A
l!lO:te general incthou f or nlloci.tint; sequency c"ruinel c for
\la.leb ce rricl's baseci on f!it'O llJ theory wil l be given I nter .
It isposDible :;o multiply,; signal F " (B) wal(,j , 9) firs;,
with an nu.:xiliary carrir::r· wul (t , e n.r:d l.hPr: de11c.•ll1tlti-,,. i t
i>Y mult:iplication wich n ouJ:rirr :ml< j E!l, B) :
((F"(8)wal(j,9))wlL.(h,5 hal(jet,SJ
=
(25)
'l'herc is no inter.f crel'lco b:r lmnge signnl t lt \,.'q_!;.t--..
cal'rit.>rs are useu . 'l'o ohow tlti~ , let B 1•cc:o1ve<l >.<is;ua l
D"(e )wal.f l, 0) be multipli"1tl l'iruL b;y wa-(h , 0) and Lhou by
wal (jilh, 8) :
(2l•)
" c(r )wal ( l6JS'"
L;
..,
e,
D•(e )wol(l ej, 0) contail1r 110 •oOn>ponont Lhat: could peas
throui;h n aequency 1 owµns;i fil tcr wi tb cu L-oJ'f ::>cquency
µ • k, ea loni; as the condition (18) is sotisi'iec. . rhe
absence or i:nage signals ca.n be traced to t.?...e occurr~nc~
Of only one \./a::.r.!: function on tLe righ:; hnnd side or the
mul tiplication theore~e (8) and (1 . 29) .
J;'ig. 49b shows a block lliae;r·am :or the synchl.'onou" demo dulntion of Walsh carriero .
Amplitude modulation of i'1mco ions of o Lh~r co:op:ete ,
Ol'thogonul oyste= tlluy be <liGcu scoci in ve"'Y '"""h tile same
'<ay . Ko other systems hav& nhown pL·act i cel 9dvnntui;er- ,;o
tar, l>ut this may well bo du" to .u:: in,-.,,fi'icio!I~ er:-~rt .
'1oat Of the better known fUnctiorrn have !OUlt;i~ lic~tion
theorems that produce no~ ooe or two ter.na a" in t!:.r cuse
Of '.laleh or sine - cosine functions , but an in.finite series
9
Hn'"l'llf'», Tu1ntm•it.f1an oJ Worm-at10t1
3 . CAR.ill £R :i'RM!SIH.Ssroi'f
ol' tcr:.m!;i . Cal.'ri crs of pex·i odic block pulses !'oI·m an orthogonitl sys:cem but not; a compJ et.e one . l'h.ei1.. amp.i-; tutle rnoduJ.ntion dif_f err; str•ongl:J~ il:-OD• that of si11e- <:ot:>i:1C or \·I alah
ru.nct1ons .
3.12 Mu ltiplex Systems
One of the most iwpor~ant nppl ication!l of' ampl itude
modul.ation is in multi1:.lexing . Cons idr.r a fr equency mul t;iplex sys.t ern for telepho ny . m tel ephony signa J s are pas-
sed
tl)~ough
frequency lowpass fil te rs a:>c! e nplitude mo -
dula-ced onto m s ine or c.o s i ne CaL·r iers . In ?r:.:iciple 1 the
;nodulated ca.rrier.s a.:re ed.ded and may Lhen be t ransmitted
comm.~n
l ink . Single sideband filter,; are inserted
after tlle modulators to suppress one aideband .
'!.'here a.1.. e seve.cal method.fl to separave frequenc:y multiplex signals et i;be receiver . S;:irnchror:ous d emodulat ion is
one such method i:Ul C it ca11 b-e applied to non-sinusoidal
ca..c1·ie~s as well . '!:h e received m11l t.lplex signal is multipl ied in u 1nodulators b:.'t' t he saJne m carr•iers t!lat were
u sect for r.iultiplexinr:; at the t ransmitter. The carriers in
vi.a a
the :receiver must ·ue synch.ronizeC to thor;e in the t rrulsm.it te r . '!'hi s me;,ins -che frequency :nust be rig.ht and the
phase d l .:'fer ence ·v ery s.:nci.11 . TLe tleruodulatea sigru.tl s pass
thrcu~h
m l o-.·rpar.is fil terh which
Bup~ress
the contrl.butions
I.t·orn .signal s of wrong ctannels . A practical I"L·eq11ency mul-
t;iplex system differs al' co11r:>e from this principle , since
specific fea t ures of sine and c o sine functions-a r c utilized
in practical syotems . Here the emphasis is on those feitture::., i.·.ihir~ll s ine and cosi ne func t i on s share with other
corupl e !;e s ,y sLem" of orthogonal functions .
'f'he
t;•., 0 me"t"bodB of quadr"ature r:Iodulation and single
1
sicleband modulation ere k.uown in frequency l!lul tiplexiJ;l.g.
Two carrier s of equal frequency but 90 • phase di!'fer enM
a r--e are.pl itude modulated by two independent signals 01e4uel frequency band width i n !;he c ase of quadrat ure mod\Jlat ion . Two signals are produced ) each o f which bas twiC.t!
'!> . 12
MULTIPLEX SH»1El1S
11 5
the bandwidth or tile original signal;; . t:o aore bar.~widtt:
,ban in the baseband is occupied per nignal, since bot!:.
signal s occupy a colllllon frequency ba.~d . On'y one carrier
of 8 certain rroquer.cy is =plit·>de !!lodu loced by a signa!.
an<L one of the generated sidebar.<!~ i:< :>U?J're<':;cd in single
sideband modulation . :he:e are several method:; available
for this supprosuioo .
Corresponding atodulaLion methods ex i .st: ('01• Aeque ncy
mull;iplexing (11 ) . Lacking better termn, ono may denote;
them by quedrnture ana sint;le sideband modulation too .
'Ihere are cwo Wal~ll fwlCtions cal(i ,S ) and t.nl(i,B) fo::each sequency i . Quadrature :nodulation monnn thnt col (i , 3)
as well as Gol(i,B) are amplitude modulated by two independent signo.le . Sir:gle sideband modul~tion means that
either cel(i,e) or cal(i ,9 ) on_y nre :nodulaLed . One sequency sideband i<> gene::-aeed ir: eitb•r case , but tt.e carrier sequcncie:i have to be s-paced twice us \11.itlti uparz. for
quaclraturo moduluti ou ns fo= single sideband modulatio~ .
For explanation of tnc prir-eiple of sec:urncy ttult.iplexing by moe.naof Walsh ca::-riers r efer to Pig . SC . TL.: o utput voltages of two mic1•ophones are applied ~o pointn a
and a,'. They are p«ssi ng through two soquo11cy towpaGo fi lterij LP . Step voltagcn appear at theii' o.1t1men b an d b '.
These are fed to the multipliers M ond arr.p l ieudr> modulate
two periodic Walsh carriers applied to pointr. c •nd c '.
The c odulated carriers d and d ' are Bdd~d in S und i;h<'
output voltogn e i s obtai.ne~ . ~hi~ voleage is a~ltirlied
a.t the receiver in two multip!iers M by th'-= na:n~ ~als!:
fll!lctione used at ti.e t<'ansmi t ter . T'ne two voitagry:o appearing at the outputs g anci g • o~ the l!lul t 1.plier are i'eci
th1'0ugh eequency lowpass .filters LP , that arc coual ea
those used in the transmitter . The sti>p volt~gNi -at che
OUt"pu~a b and b. ' a r e "'lual to those aL b o.nd b '. :hey may
be l'ea diroctly into a ~elephone beadse~ . ~·no 101~rar.s rubera of Lho ti'ansmitter produce a d e ! r:>y of 1 2~ u s =d
those; oJ: the receiver produce another 125 u o delay . The
••
3.
11 fi
CPJ!RI£R 'l'RA)lSt-;ISS!ON
dnchc<l ~t-ci;ior.s of che time din;:;rn.:n of Fig . ;c indicate
the::-" !lclAys .
:'ig . 51 st:o·6s a 1mli;ir;lex Sj'Stem '1iL!! 1024 <:ha:-.nels fol'
trr:n~:rJisniar. in ore C~ec~ic-L . f~ulti?lcxing of ar.a.:.og signAlB will bt> <li.;cussed . 'l here i .c. no problem in codifying
inpue and outpu~ ci:-cuits for other -y;ies of si,;nals . For
J.u:.:.~ru1cP., r.!1e '.'oltaF;es +V 01· -V only :"JT'n applied to the
iopuL::: oJ' t.he ch..nnels for CJ·nnsmiaoi on of biuary digital.
$ignn Ln . Se\ren channel~ a.re requi1·ca fo1~ the &ransmisslon
of 11 •11:\lln I l'Cf-1 teleplloh:; e.i5nlll . Suci-1 rleca i l.s ar·e on:ibted
frori the J'ttt'l..t:ier· discussion , .oince che:y a.r~ no !llo:re importru1t ··0r tl:e princil'le of S"fJUNIC:f '!Hll ti.~1le.xing i;han
for thnt oJ' time or frequency mul t~;>l P,xing .
Tbe tn·o-wire l i.ne coming !'-om e s:.ibsc!"ib<"r is spliz b;y
a hyb:·id e~rcU.:.i; im;o a tr:umni :ting and a receiving
·~:-1.lllch . A sig.nal on the trana:Utc,ng !>rl<JlcL ~s app_ied to
OL.f• of the 1C24 input!': of t!1e t.:nne!titter . It passes a
aequeucy lowpass filter 1.1' .,,, th cut-o!"f rnquency of 4 kzps .
':he $\'. itcllen of' cl1e low-pnss fil t.eJ.'S nre driven l)y pulses
1
oJ' :he liULini:; goneratoi· SG . ~ho input Si£"lllll F(e) is i;ransfcm:iM tni:;o a step funct i on F " (O) ; P(e) and F"( a ) are
ubo«n 1 n Fig . ;i? but wi.Lhou~ tlrn d<' I ~j' of 1 25 ..,s between
'
I"( 9 ) 11nd I• tt ( 8 l .
After .fllte1·ln5 , t .h e si15nnl is nr.ipl-Ltudf' codulnted on+,;o O:lC ~ •· 32 1.v'alsh Car.ciei·s I1 to 'r;2 i.1.1 one of the [tUl ti j liers :·1 . 'rite first: four ca.r:·i~H'S wal(::!, 9) to wal(3,e}
BT'C ::ho'1n i.r. Fi,:; . 52 . Juratiou r b.DI! position of their orChogo""- ny ir.t<:rval coinc1dcn with t:h<i steps of the sigGli! F " (a) .
32. mo<'lulai:ed carriers arc
~ocbie"d by t:he ad!Oers S1 to
S52 in~o oee gr·oup . A• a :et>ul t, 32 groups with 32 chanr.t'lr r•ach are ol>cained ss showr. iii Fig . 51 . The :t:igure ;12
is cl10!1en as exacrple only ; r·i·incii:- l es for judiciouslJ'
chooni ng ai ze of grou pi; and 11upo·rt 1~oups will be discussed
lo tor .
Tr.e ou Lput voltages oI' th<> neldors cu:e arupli tude 111odu-
117
'
--
-- --
Fig . 50 Principle of u seoueI1cy mu:~i;.>lex .71s•~m . LF sequenoy lowposn filtrr , rt :riu.:.tip l :_Qr, S (11!\"!1 r .
Ft~2 51 Sequoncy multiplex
system with 102'> telephony
~""""eln tor tl'OJlSJtLi.ai;ion in one diroction . ~· J se9uency
..::;.,"Paos
tilter; 11 multiplier; S addar; 'l'G , 1''G, fiC trie;ger ,
0
t1on and timing generator .
-
~.
118
CMR [El\ TRAJIS!USS!O!;
T:i"ole 5. Gener:ltion o~· cnrrie::-s 1'1 to T3<' oy t:!.e multJ.('1ic1>;ioi: wal(;:,~)wa~(l,6) • wnl(,j,6) ttt;:lOJ t:.•' Ca.!-:-iere
T 55 to 'f-6.!. b;,· the rtulti~licaHOU w:;l (k' 9'"~1( l 'e )wal ( 3~ '9)
• wnl{~.~) .
<.u· I
1i e 1 I
,
TI
T'
0
., '
'
•'
•'
'
'
••'
,.
1' •
''
T
T I
T i
'I' II
'
I
,,..,,.
•
"
'''
''
''
""
u
"' "" """
T"
••
,. ""
,rn " """
,"' •• "
T:l
'~
,,.
""' """
,,. "" I "
""
,.,
"
"
"
"
""
" ""
"'
""'
rn
""" """
T"
111
T I!
Ito
I
•
••
'
•"
'
'
"'
"'
c:ar.1
r 11r 1
0
l 'H
'JlU
,,,
'I ~!)
flt
·1 :111
'• :tt•
'l'tll
'J 1 1
·r 12
'Ill
TH
•' " '
' "'
'"
•' "'
n:lf'(
)
I
".."
• I '
" •'
"'" "
"
,.,
'"' \lll(i "''"
'" '"' '
'l-011
"'"
'"
""n3:: '"r.11 '"'1:,.
"' '"
'"
'" ""
'", '"
,_,"
'" ...
•;• '""
1;r
'!I.Ii
!Ut
..
1:::1
lb.
!l.'1!
..... ..
.. .. .."" ,,.... ... .... ......
.. . ,...".• ..."' ......'""' .....'"'"""'"... ....,.
'"
"'.•
' .. ...
•' "'
...
'• "' " '
T"
Tll
T"
ll
T"'
rn
TW
H•
41'!
1w::i
uo
!-..~
T:.f
t•;J
U1:t
n
TU
111:1
'"'
"'
•'• '"
" "'
H•:I
l'l•fl
11 0:.11
1111!
"' '"
111~·
!ll:S
1111!1
111;.l
8'•:
....
u.
4
l'I
"
lai:ed oni:o 1;he Wal.sh carric::-n 'l'5;i t:o 'L'6" in t he multiplie::-n M. Adder $~5 aclds the r-csulting j2 voltP.ges . A seep
voli:age in obi;ained at c"ic oucput of s33 . The widch of
the ntep~ i:; equal co (52)- ~125 '"' ~ '-22 ns . The amplitude of tl:is o'"tput signal !losw:aes 8000><(32)' = 8 192 000
indej.iendent runp:!.itude" pe1 second . ':he "ignal occupies
the sequer:cy bane O :: q; :; '' . 096 117.ps . This n:ul tiplex sign• l rnay be t!'lills:Ut~ed directly to the receiver or it
moy bo used to noc ulate " sine 01· Wa.lsll carrier .
. d =· "2
At the receiver the Bi131wl ia rirnt multiplie
,.
mul LiJJliel'S M wit;h the carri era 'l'3j to TGll- and then in
j2 muli:ipl iern with tl1e carriet·s T1 t;o ~· 32 . The demodulated ~ignnls pass ·t hen Lh1·ough the sequ cncy lowpass filters LP , which ere equal to Lllose in t:he t:ransait·~er.
1
3· 1 2
119
11tfLTlPLEY. SYSTEI·tS
Fllll-- -
f'1Gl---===L=:_---;-===
(ll~hrO
I
I
• il!0,81 I
l
I
-----
1i.2" a
~
*'"•8
"-~
0
,__
m
lSG
- .
,.
!(I)
Fi;;. 52 Time diagram for the :nult1rlex
I'sble
or
tem
&. The 2x;2
c~=iers
Fig . 51 .
r;y~tcm
of the nequency aul t 11 lo3 aye,'wnt 1I11"
l1.11\cl 1on
c:olll,tl •lllli 01
.
'
T l
Tl
1 '
1
1 •
T O
.•
1 '
•
• •
••
""
"'
ru
tu
ru
...
......
......'"
........'",
r"
ru
Tll'
ru
Toi
T10
Tb
•
•
•
•
•
•
•
•
•
.
'
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.•
•
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4 et. I b1nA1y
.,,,.
..
'·.
-,.
I
I
I
rtJollll
to.lllll.•
""'"
.,,.,
•• ...,.,
•t .....
.,,
'•
• .....
'
llltlllll
11011
.
Mll•
001111
.....
•
.."
" .....
.....
•'
IS
11
m1no
"
"It
"
II
..""
111100
111 IC(l
1\11111
llllfll
111110
111110
1111 11
ti n u
""""
ettot
e1•1~
. . . .1
111111)
D14lll
(l;l ll
IHtlt
etlli)
0111•
11
l Iii
Tto
Ql)llll
.."
··....• ..... ...."'"
"
•111
I
Tn
QlllllU
WO ii
•'
1 ll l
OOlllll
:::
""
"
""e!I
:0\
••
II
nn
r ..
T"
T"
Tn
TU
T~
·-·-·
,.T••..
eun
UlllO
Tti
T"
r"
1otn
1· ~=
HUCO
T~!I
10101
10110
1111 n
T:.1
T6'
llto»
1·r.;
11001
1't>ll
1'!\li
111110
no
I IO U
11100
T°'
111(11
'l'd~
11 llD
111 11
r"
1'o!I
T ..
'
•'
'
''
•
.-'
<
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'<
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61
I
'
b ll'IA rj
df.t.
tlec.
bHhHf
...."
'"" ..
"'
.."'.
."''".. ,,.._
.*I.. . . . ,.,""
... .....
...,'..
.
'" ........... .
"'.. ,_.,,
.
"
""""' I •. """ ' I
......
'"
' I "
I
'' """ "' •' "
"
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'
'
''
'
"' • ""'
I
I
woli 1
colt+, I), "" 111. II!
;
0
,.,
"' •'• "
"
~
,.,,111r,,n1
j
dee.I bln•r1
of Yig . :;11 .
•ll)lll• •~·-··
(•
1 1111_111 \1•11~•
llV<•.v.•WllU
ll(•'(llilOOtlll
_
_
Iii:.
· ~•111(,o)ljQll
!o,;iJllllollli,1!0
·~·011•~11111
tlt•)l\00000
ll(llf•:u•lll
1:-~
•l!fJf.. 11111!1(1
lll•!•ll0\111\j
\11111\A>lll>O
111111111111111
h'!
WlOIOWOO
Wth.00000
MHltlllllOO
lll'J••'llflll
~n 1tp1ioo
OtJ"'""i..111
.1!•
n•
OU1l~
~11-.(,y
ott-.CO
"'
~..,
•111-..
•1111 ....
IJtlt..:q
l< • .fll . . ,,
!'t:
......••
...
~!'4
J~b"~···
,,, tl•r>'ott
)C>ll#I•"'
1l lll l •>'l(>J
li..>1100000
...."'~,.,
..... ..
.,,.....
OIM>I_.
ll)ll l lo:.h!
111!111111(•~·
llt•tll{o'Vl
HllllKIClh(l
1111110(1(•)
.
Miii-..»
"'
"'
u ...... .:v
6111
IOIW!\1(-..)
Ill::
lll l lllf•XOO
~Ill
lll ll (l(o(o O'\()
l<fl
11•1 ~.o\."9
tillllUlOCO>
'!\,II
111111((""-lO
lllfl
'"
!111111 \o'U'lll}
1111100::(:(1)
I Iii I HICtOOO
lllU(ll!\)00
"'
111111\IOOO
1111110(...(1
,.,
H lll illil~I
'"
11111
llb!\
11::11
I 111111).00))
l l lll(l(y.(O
' ' 1(1100»>
1111000000
111110()»)
.3 . CARRIER TRANSIHSS!ON
120
'!'he
~otock
dia151·a..:r1 of Fig . 51 holds fot'
qu~orature
modu ..
lation as well as for sin~le sd.eband modu.I at ion . The t wo
methods d.i.ffe1• otly in Ghe ca:rrierz usec .
:t:c Walnh ft;nction::o t,•e l{0 1 6) to 11.1al(31 1 e; are best used
for the carriers 'P1 i;o ·r52 . Thell· generaGior. by means ot
<.he mul tipl1c"tion i;heorem ( 1 . 29) from the Rademacher
functions -wa l. l 1 , a) , wal( ; , 9) , wal('/ ,a ,l, ... wal(2°-1, a) ..
~~ shoNn by 'l\a"ole 5 . Hade1:iacher funci;ions can be generabFd by bina'!'~/ count1;c!"!i ~r. !lhot·.'TI i.n f?ig . :;1 .
Tee :::nro·iers T?>;) ~., Tt4 musi; l:le ci1osen !'lO t!:et no crosstalk is prorluoerl tu1d no r.;equenc:y ·0R:1u;-;idth iE> wHsted . Table G sllO\\'$ a possil)le C1\oi.ce or ~hc~c C~.trriers . WEilah
!unci;ions are shown in th:.s i;able in the !1otntion Nal{j iS)
as '''el• a.c calli ,9 ) , E-al(i,9 }. One ctay ""e LLat the last
ri-.:e d i gi-c.:J c> .:· the normalized sequ.ency- written as a bina-
ry t:.W!lbcr is al•t1ays zero . The 32 ua.rrie::·s T1 to T.32 may
be :·:_t1;c:d betNean an~· two of the Ci1r•ri-ers rr33 VO 1'6l~ . ~he
av(lilable sequenc.y band is cor.rpletely used , there are no
lo at sequency bands be'~Neen a.djL-1.cent channe.Ls .
G~ncepts
oi gr·.:>U}) t;heory may be used -co ·p rogress beyond t he pu~'ely empirical wayo~ ·hosing the carriers. !!!he
11lalsh !'unctions wal(O,O) t::o wal( 1 023 , 9J formagroup wii;h
?' 0 c:lcmcn.ts . The functionn wal(O , e ) to 11~al(;)1, 9) ar{! e
s·~bgr:oup «ioh 2 5 eleaenLs . There are 2 '' /2 5 = 32 coset;s
of ti.is SLtbgroup . This is just che nur:ibe:.- oi: ca1'riers T37
to TEA . ·r he Wal.sh functions generated by the modulai;ion
of the carriers T55 i;o T64 by vhe J'imcti ons «al(O ,a ) ....
. . .ial(;i1 ,e ) are i;he elements o~ the 32 <:osets . All possible .'."unctions wal{j , e ) usable as carriers •r33 to T64 a~·e.
obtoi:iec by ruul ti1)lyi11g eact or1e of the functions wal(O~s) ,
wal(;-.2,e ) , ... , wal (992 ,e ) of Table 6 with ru:iy one .runc:;ion wal{ O, a) , wal(1 , 9 ) , ... , wal.(31,e ) . Sucll,a mult_i.pli cnti.on mean~ onl;y a reorctering of the elements of each
coseL . One c"n mllltiply wa.l(O , a) with one o.r the .?2 fu,dCtions •.·ial(O,e), wal( 1 ,9) , ... , wal.(31 ,9 ) . Onemayfurtbel
multiply wal.(32 , 9) wii;h these 32 func tions , then wal(64,e) ,
~ .12
MULTIPLE"A
SYS_S.~S
'i21
etC· There o.re atol;a.!_ of 3-2 12 .., ~ 1 •• Sllch ;,:1"0duct;s, "·hich
~eant1 there are 2 110 possible choices o! carrier!! r3;: to
T64 , none of which wou ti wust" sequency bb.!l<lwirtth or pro-
auce c!'osstalk .
ll'ig. 52 s hows sine and cou.i.ne carrier s beside~ Che Welsh
cai-riern wal(O , S ) Lo ws!(},>) . Cue may u"e Lher~ sn carl'iere T1 to T.32 in ?ig . 51 . 'I'be mult:::plierc would :.ave to
be of a more co:i:plicated LYJ•e in this cn!le . :he ;)2 ir.odulated carriers could be ndded wi.thout r.avinR to pnss "
!tingle sideband ~ilter . ~hie type of quedrntu..:·e :nortulntion
shows the close connec~ion lJetween frequer.cy e.r•. t cequency multiplexing.
One may readily seo from Fit: . 52 that any ayctcm of
functions that is ortllogonol in a finite int("rvnl (!filj be
uaed l'or the carrir;r!l '!'1 to ~32 of Fig - 51 i!' the iuput
signals F(e) are f:iltcr1'd by sequency lo'Opnss filters .
'l'bis does not hold for th" cn~·riers 1'35 to T61L They n:u"t
have c:iul tiplication !.;heoremr si:r.i lar to :.h0.$4! ol nine coeine or \lalGb functio:.s .
shows an e>xtra ~ynchronizat ion line> beLwecn
tronsmitter and receiver'. Actunlly , O"<:! o= rr."r" of ;h("
102'1 channels can be u~eeJ f'or t..L~an~mission ol & ~ynchI·o
nization signal. A Walsl· !'unction w.01(2° - 1 , q), wl ich is
a Rademacher fU!Jction , iu ti"ansTiitted if a conr·tant voltage is appried eo tho ct.o.nnel 2" . :be or;hogonsl it;; of
tne Rede111ncher function" is iovarim:t ~o ;;c.i.l"t$ . Tracking
!ilters that lock on1'o lhel\ can be buil; w.LLh relative
ease. The modulated Waleh functions iorn: n utatiGtico.l
background and ca.n be suppressed by long nvi;nq:;ir.g tim<'s
Of tho track:Wg fil t e :ra .
Fig.!>1
llequirements for synchronizat.i.on and ri"c 1oio.es may be
info r1·cd fi-om Fig . 5;>. Let the signal v.(t) re,·1·1;aent the
ou1oput voltage of n telephony multiplex sy~<:nm with 102""
Channels . The width of the steps is 122 ns . The ir..for1A tion o! the signal is compietely contained in its niL?l~
tudoa . If the signal v . (t) is translritted, it sufi'iccs
1?2
- in Lb.e ~b,.e11ce of noleo - to $O!ople the nrr.pl.it uden 0£
thtJ E'teps , iri order to ohtain ull the in.fa ·mstion . The
sMplO.ng 1rny be none at1:f"'nere :.n t~.e 122 n:- I ong intervu.~e , ::ind. t.l:is i ~ r,.be Lo-f\rlillce i .n .t."rval for t..lle eynchronizuoion . Con:: id er tile ri:ie Lillie . Let tbe r.i se L.lme be so
slow tnat; it. t1f~e~ 122 nr- to ct-ange I.roa: -...-e{T.l to v~(2T) ,
f~·oa: v.(~1) to v.(:;'IJ, etc . "" ::i.1own by v~(;; 1 i:: Fig . 53 ,
Ihc origit.al nt p vo. toi;:e mny b~ 1·egained by :>wnpling
Va (t.) exact..Lynt clie poJnt=s o, T , 2'1' , 3T , ... rue- maximum
n~c time l.; tl111" 122 n~, H ther-J ls no :;yncl1ronization
~i·z·or. Ir. ~eocral, rise cl:ne pli..;.:..: synehro:":.izution error
11u.--;t to less t;.,,r:n '22
u~1 .
Arr.plitude rnmriliut; .i.!'I ~ i;oor dr.itection met.hod in the
p1·eoeuce of noir,o . Uowevcr, one rr.ny "'f?Sdil:r ::;oe l;hat Va (t)
car: be recon•·t!"rt~C to v,(i;' >.;,:; intc~!"'&~io;;., ~1.1Ce the iatogral over v.(t) ta:.Ce.u !"!"O~ : ~ - 6 · ,
~o ?T is proportion"1 to ., • ( .,T \ - v.(r) .
'L'he c.i.rcui~ show:i in (i'ig . ::;3 will tr-nnsfon1 v.(i;) into
v.(t) , :;heor~tically wiLhou~ ar.y t•lflging . rL isa classics! proble:n OJ' :'r-9qUC:lCY t~eo_r:;- tO U??=OXillttt.e 6 .filter
·.-1J11..C•J c~ do t.Hi:: . -.ine ... ir.:t-.licity wi"";h •...•hich this problem
cen be l:iolved with i n -cha ~\· irtcr Ii-rur.rwnrk ef uoq 11 orLCY tbeor',"I l<';! dt...e to th~ usf' of n ci;r,µ vt1!'inh e el~a:irnt - tbc
fio!-.1 e:nission :;rn~ci:-1t.o: F - tr tt.e :"~lte1· .
Co?:!1itle.::' Ji'ig . ... ,. !'or a di::::cu~r ior:. of nl n1-t,l~ sideband
uiociulnt ioJJ oJ' W11i"11 rune Lions . Tne origina rignaJ. F(0)
onJ th£'l si~:nttl !•' n( 9 · !'i l :e1·ed by a ~eqt.;.ency ... owpass fll-.er orr- slio . .;1. on r.ov; r:he ticae ::::C-.1r't tetwrf"n F(e) snd
i-" C5 J ~" omittel . 'Pile ',-hl!!ll .;arri.. i·~ wal(O , e), wal(2,0) ,
1.\·~i\4,13) , W~~l\•~ 1 9) , ••••• •• f'JJ'e ;3.0C.•Wrl . rhei.r ~i'D.e base l8
"'.>() u•, <;i':ich is tidce ~h., duration o.r the ntopu of F" (a) .
·:M 'ic'ter-ed. uig:i'.11 i"•(e) fr r-crresented by the following
expression i"· t. .e interval 0 ~ t < 2r.10 µs :
(25)
r•(G) • c(OJwallO,eJ + u(1)wal(1,e)
e •
(t- t
0
)/T;
t
=
125
~s,
Ta 250 µs; -!
~ ~ <
i
12}
i=:=== ======
.. 1~ei
~•11,61 ===i=il:===:f==t:==.Jr=
"
· •1 14,91
=!:=F:t--:-f= l==F=!:=F
...t.110,01
r==-=== ====:--=
it - -..
.,,,,g -=,,,...==-=::;.-_.,.,=,,._
•
: -
Fig . 53 (left) FinHe rise ti.me ol' & ctep f uncoion nnd fil t er for t;be convei·sion o .r v. (I:) into v,(1;) . rho filter
also reconverto v 8 (L) into v,(t) .
Fir; . 5'~ (right) Time diai;ra.:n of" rl.ngle sideband ~cquency
multiplex sysvem . Time base for tho lo:.1pass filters is
µs; time bao~ for t!:e carriet's is 'I' - 2~ JS .
T • 125
Amplit:ude modulation o f one or L~e carrier:; wul (2J ,e
o! Fig.54 by F~(9) yields :
F"(e )wal(2j , 9) • c(O)wal(2j , 9) + c(1 )wal(2j;1 , q)
)
(20)
HodUl.ation produces just those lfalsh carriers m.tll2j•1,e)
that are left out in Fig . 54. !!ence , the aodul&~ed 4a:.sh
functions occupy the whole :ivailuble zcqt.:ency tnnd .
Pig . 51f <llso chows sine functions wi~h a t;irto buse of
250 µa . Their amplHude modulation by F" (e) ctoc1;1 not yi<>lci
a (frequency) singlCl aideb:ind modulation . The COl'l"r>npoudenne between llalsl; and sine .functions in th" eac.(' of
quadrature modulation is based on the face, t!:at wal(O,G )
ii; the first .function o.f a '.ialsh aa wel~ as of a FoUL·ier
aeries. The signal I'"( e) of (25) , however , conta.1.ua the
fu.notion wal { 1, 9) , which does not belong t.o a b'ourier
aerie a .
Let i;he time base o.r the carriers in Fig . 51; be increa-
; . C~~"lR~ 1 HA:lSc'.ISSlON
Ged fi'om 250 µs to :;oo µs O L' 100C µs wHlout. ch'.l?Jging the
sig,11al F rt( e, . 1L1wo more nxam1 les o.f ~equency Dir.Fle side ..
band system;- are obt'1iz:cd . Tte cnrri••rs wal l;J, 8} are permi i Led only for T ~ 500 µs, and wa1(6j,6J cr.l;v !or T _
1000 µs ; j • 0 , 1, 2 , .. . ... Advantages and dL'liwbacks ot
tbese many pOtH;Ji':>le sin~~le sideband eysl;err.s :.uve not y&t
been investi 0 ate<i . Q-•<1d-acw.'e cio<!ulDtioi:. appea~·" as the
SJ ecill! <:a~e of "he class of !llllgle sideband systc1Us ,
wnare Lhe tii:ie base of the carri"l'S i.s equn I co chc time
base of tho ~oqueccy towpaas 1ilte>'s .
A c!:lar·nctcristic feature of ~requency :1:ultiplex systems
co.,,par9d ·,.itb tice ;nul~i;>l9X :;ystcn::; is tl:t> ease ..-ith
wtich signal:s in cotuwnication n~tworks Ol' radio signals
cw1 be combined Md eeporated. Tltc reason la oha~ 1'roquency aml&i1>lex •J.gnals n=-e inLerently 1tarkc<i by theii:'
{r~quei:cy, ~ta ch is :independe::i-:o of delay - irees . Ti::ie multiplex signals, O!l the ot.Ler l1and, lluving vaJ·ioue unknown
doln.Y c.imes need sowe fiddiLion nl murkiL1g io order' to be
separable . Since sequcncy multiplex signaln ure also inhe!"'C!'n ... ly aarkcd by t.hc~r nequ~ncy, one will cxpec~ them
to laa6 to comr.iunicatior. net~o:.·kn that aJ.'e vo~y similar
Do those for
rrequen~y
o\
6\
f-ZS011<
Fig . ~?
U1l
mu l "iµlex si;;nals .
~!
1!6
Jl'J
,_
Ml
1,1,11 ozpllll
Oc:::upe•irm of <.equ•mcy banJ11 by multiplexing 4 k:.ps
wide base banda .
fig . ?.? r.bows s. possible sequency ulloca&ion for groups
;. 12 J1ULTIPJ,EY. SYS'rl:.MS
125
and supergroup' liJ a coll'Jl'.unication nE>twork . 'l'hia allocation is chosen so that group , surergro11p A or supergroup
B are cosete or the oai;benatical subgrourt of Walffh functions wal(j,9) with ,j sn:a1ler t,:e.n 32, 121' or 256 . Single
sideband modulation 11nci a tae base of 2~11 u .. !Ire assumed .
The individual c~.arucels occupy besebands 'rom 0 to It kzps .
Sixteen b11eebEtnds wnke a 15~·oup that occupies tho> neque ncy
blllld f i·om 611 to 128 ,tzps . The carriers ore wu1(32+2j ,a) ;
j = 0 , 1 , .... ' 1 5 .
Amplitude modulat i on of Lhe can·ier Wf:ll (')6 , a) l' Y a
group shii'tr tho oequeucy lland i nto c~.e intl)rvnl from 128
to 192 kzpa; the carrier wal(64, S) "hift~ o grour into the
bnnd from 1q2 to 2"1t. kzps . Tiles~ 52 chn!'l..'lt!ls "re carked
super~roup
A in Fig . 55.
Moduletion of 11 carrier wal ( 192 , a) by " su1, eri;;'C'..lf A
shi!t:s it into cbe band :·rocr. 256 to 38'• kzpF; the carrinr
wa.J. (128,e) shlfts a supergr-oup ;. ~n:o t . .e bend :·roci 38L
to 512 lczps . 'rhe resulting e=11 cLnnnele Are drr.ote<i by
supergroup B in Hg . 55 . :'.able 7 sl'..ows to wh,ch position
of the soqucncy bon d tte c.l.iannel s arc nhifted . 'Ihe 16
<lh'1Jlllels of n group nr e shifted by che carr·il'rs wal(J , 9 ) ,
j = 32 , :;If , ,,., 62 , from t b.ebaseband . }'or ins Lance, ul1 e
signal in chnnnel 10 is modulated o n Lo tile c9rrie:r
wal( 50 , e) . 1ha sequ (}ncy o!: •ral (50 , a) iu equal te> 2x5C ~
100 kzpn and the sigual occupies the b9ncl from 100 ~o
104 kzps . The carrier wa.1(50,9) becot.1eP wal(82 , e) hyu;ul tiplication with ·.;a1(96 , e), or wA1(114,9) by mi.;ltiplicatioo with wal(6ll , & ) in suferg:roup A. Chnnncl 1 rJ occur ics
the band from 2 x82 = 181 co 168 kzps or ~!:.e ba."ld f:-oa
2x'l14 • 228 to 232 kzps . Finally , the carrier ....-~ t ( c.1 •, g
becomes one Of t;he carri ers wW.(11•6,8), wat(4?8 ,9 ) ,
Wel( 210,9) or wst(244 , e) in supergroup ~ . Chsnn<'l 1" occupies one of tho It kzps wido bandt> with lower liait
2
X1l~G 292 kzps , 2 x178 = :;56 kzps , 2•2'10 • 1120 ki~+ or
2x21111 • '188 kzps .
Consider t:he cnse of a supei'group B, a s upergi·o up P.
!llld s group being transmitted . The oignn.l. occ11pio s the
126
; • CARRIER TRAJISlUSS!ON
.;equency bond from 61• to 51 2 .<zps accorclim; to Fig . 55.
O::e u;ay extract ct:.; £l"OUP by u;oA.ns of 11 1;equoncy lowpass
filte1· wiLh cur.-off noouen cy o! 128 ln~ps . A lowpnss filter with 250 kzps cut-off scquency wi 1 extract the group
t1-'ld ;be su;iergro"p ;, . :1us silple kind of filcering is
possillle o'1ly if the cut - o ff sequenc;y in equ•l to 2 1 xtf ~
2'~ 2 kzpa ; r • 0 1 1 , 2 , .. ..
Consider as o rurU:.er ex&11ple •be trar.sai,.sion of 8
supargi·oup B. Either ~lie 32 chun.nel s ln the band from
256 to 384 kzps or ti.e 32
~o
clrnru:.('l~
iu Lhe bwd
rrom
~84"
512 k,;pe nhall ·3e ext::-acted . Su;iergroi.p Bis a:ultiplied
by th<' car·rier wal(128 , S) . 1'a1>1e 7 :;;ho·,1;; th9L the band
2>6
< 38'1 kz.ps (j • 128, ... , 190) is tranaµosed into
the bnnd o < "' < 128 k7.ps (j
0, ... , u2) ; the band 384
< 9 < 51:> ;.czvs (j = 19?, . . . , ?54) is trsnspo~~d into tho
ba'1d 128 < qi < 256 kllJ'S (j = C,I• , • •• , 1 26) . A sequency
lm.'}'aad filtet !rnvinl( " cu-;;- o.rr ,-;equency of 128 i<zps can
ext>racc &he bar.d 0 <'I' < 128 ·<,,rs. A :i:ulolplication by
wal\64, 6) co.n sli:Ll't it to the ·o=d 12il
< 256 kzps ,
which. .i.r the bnnd 1"'or ~ suµer·t:;!"oui::· .A .
:.et supe:.-t,-rou;> B be irultiplied by wal(1C:2 , 6 J inst.sad
ot: by wnl(428 , 0) . The band 256 < q:> < 384 kzpo (j = 128,
, 1 1)0) is t:-nns:Jo,rnd into t< o band 128 < iii < 256 kzps
(j = <>' , •• • , 1~" J, tbe b1lr:<! W• < "' < '12 l<zps (j = 192,
... , d'''•)i'1tothe ·oand 0<qi<128kzps(j co, .. . , 62) .
A seq1.umcy lowp1>ss !"I.Her can extracL t;be 11!1.lld 0 < rp <
.. 28 kzpA, h"t.ich c.o·.1 cont.!li:ls. the other Cl:a!l.llcls of super-
<"'
<"'
...
~~rour
fi .
Ar.y Individual chatll'IPl i n ~ho band 2j < q> " 2j+4 k:r,ps
by D'lltiplicacion wit" wal(j ,e), and
fi.leerin;; b;; a ~equency lowpase filter )JuviI:g 4 kzps cutoff Sequonc:y . Tr,o fil tcred sigllal mey then be ohifted f)O
""Y po::i~ior. .i.n t!le sequer:cy spectrun by u:ultiplying it
,..,:!..t:: ·bf' proper ~al:;b carrier . The extraction of indiv1tlunl clrnnnel" Ol' f!;.COu1 r of cllo.nnel s wi ~hout need to demo 1.ulaLe nn<i. l·emodulate all cL.o.nnels is very .!.limilax' to
~ha• ccu: be done i..n ticn ~ul~iplexing. It may be used to
•'.an
br.
eoc~:-acr.ed
;. 1 2 111JLTIPLU SYSTE:'lS
1 27
I ble 'i! Tran!lpoaieion of the corriers 1rnl(3 , 0) .. wal(62, 9)
~ a group to ~Ile carrier.:; :.al(64,9) .. >1al(1~>,9) o~ a
0supergroup
A and the cai•riers wal<128 , e) .. w-11(2?0,9) of
supergroup B. The
sc~~cncy
~j kzps for T • 2<.o us .
a=
t~c carrier~
is equal to
SW?uqro-.1p A· 11;'1\tOS
tn1op· 16cPl.a.nMIS
C.lltr•tt wa\($5,ll
I
(&tflff
1111-1 I •l•"f
cti•11n•I
I
•'•
I
'T
••
I '
....••
..
•
".."
..
......
......
"
.."
.
....
II
" I
,
'011·1~1)'
I
'"
""""
"'""_..,,
".....
""
"
ll"•J(llU
100100
11(1H~
JO' coo
1Qt(.llf>
101 U»
JQI U(I
UOC'O
.."
UOCU(I
110100
llttUU
.."
nto;..
~
IH4't.0
Ill¥-»
UIUO
••
,
I tnn1uy
lc.l1i ll11il
l t •~l lllll
I i ..!11 llMI
l•l1W ll\I
Wq li64,!I J
..... . fl.llflllllO
I
l~n.&.r'f
""'
11\JIJlllli
Iii ~
I 1•111 1110
11\llfllil
I llill llhll
110
I llJI OOU
1 "4!1 lllU
'"'
I U)I 11111
I IOI Hiii
lWllll!
I llll !Miil
I 100 OOll
1110
I toll llii!
11111
I 001 IHI
'1111100!•
IHI
11::
l'>lll l+l (i
I (I.Ill 11111
II I
I 110 0 10
1 IH.!Ulll
'"
'"
I Oll+th•
tf>llf•11•
I IUI l l t
I Hl OUll
1 111 cut
'"'
"'
t:•
...
' " ' ' tjl•
I Ott 1•
IOU U•
11111 . .
I 11111•
t•Jlt'tQ10ll, B : 1 !;tr,H•-·groups 4
w•tl192-.tl
c., ""'
1t:-H111Mtll00
'
...""
I ' ,..,..,., I
Ill lliMJON
11l 1100 0IO
...
Ill 01.1111(1)
Ill 000 111>
1&'l
"'"'
111001 CO>
"'
'"
.......
....
.........
lfl!
t b1~.a ry
' I
ID 100 flOO
10:
II> 1'~'1)1 0
1(11
II> 100 100
'"
'"
:oo
,., 1011I1(1
IUIOllllJll
HI 11411 4il0
ltt•
11) 101 fll fl
lill.IUI 100
"'
'"
II> 10111111
II> IOI 1111
I
bon.lt~
n!I
1110010
'l'.11'
11111 l 'IQ
11 1l>
'1.1~
!\Ill )IQ
till (IOO
•:,11
11111110
~flZ
):fl!
11111 11 ~
11 0011110
11 •llU l!lh
:1111
II 0111 11!•
:111
1111111 •+11.1
111111111•~
l lo•t
10 11011111
~l!
ll 0111 11111
111010110
"'
l!OOUIOI
toOll 111
...
>OUOllll
:u
HMll 1111
l"IUOOO
e111
II Gii ll'lO
191UOlU
11111 ?1.IO
t111
HOit •'Ill
=
II OU l!U
ltlll 11•
=
b'""''Y
JI 111111111!1
11 •, 11101m
"'
"'
'"'
J
ll •1111! 1!11•
ll 111111 11!1
~II~
10011 ll(lll
tOOll Olf
I
llMU(l(l
JQltO(l(lli
10HO11111
1111
.,
i:·: •
ll 011•! 111111
1111
Ill OllllOJ
lll 01il1110
I
"''
'"
'"
I
. ..,,.
w•ll2fJN
·~-..-.~
IM
Iii 1.1111 ll(.I
,
"'
'
(.11 . . . ,
UOH ICIO
route individual channels t;hrougi: a
tion network [14 ) .
:::1e
1111 JOO
1111 llr\
"'
: 10
"'
:u
111111.•J
I lbbll.I
111110)
:••
; 1..
...'"""
111• 110
I 111 tKrJ
ll 111 llJt
I
11111 IYt
II 111 Ill
m<i tcho<l cot=UIL.c:i-
lt has boon asawned so~= that the cbnnnel~ nnd groups
combined into supere;roup A or B are aynchi·onlzed - Thi s
assumption hold" t r·uc iJ' all ehi:uincl" ava comt)iruld i n che
s!Uile e:icchunge into groups and $Upergrou pa . Now cons ider
the cnse 1;hat channels are combined int.a g~'oups 11t rJ iffe1'-
128
3 . CARR I ER :'RAKS!-USS1ou
ent. e ):ct.anges
group~
these
that
a.nd
tLesc c:::.·01.lf!: a!'e co1:>.t.Jir:nd into supel'-
at a bit";LOr level e:xcha.!;_[e . O:..e cu1.:1ot aS.::;tll:ie , that
e:. .rour:s
a:.'e :::yncnronized . One tJa:y, :r,011ever, assume
cLe~e> groups llfl'.re i;lle ~rutiC" tin.t: IJnun 'I
=
250 u a .
w1Hll.8)
..111),6) ~
.__11?1,SI
•1 1".8.~
-~1...9'~
,._, • I ' I z f v,·t..,,i. i
.l".\f1 • ~
~
I •
1 1 1'1'f'41 , 1 •
........
·~~~~~~~
cri 'I
t
g:o.pt t I ' I
1.
e·o~p'
rnT
r·rrrr
~.___
I 2 f§f;f..·i. ~j ~ I k 1 1
f§lffi.\.:4 s I t. 1~~11' ·
1"0-PI r&.~1 1 I 1. t.::t"l;."'4: fTJT
~rou~ 1 , t:>.,'§1 t 1 2 t§J}'t>... ,,..1 5 1 ,
pr
~1tn,oi l
~
• pn>a+
ssss,sso,,"u
ssss
I
s.
e
rm
u• o N»
t11
0
ltQf
tUi5
V•m ;-1 it~
r.. no,,
:F:.g . 5f; .frinciJ.lc :·ox· tht co:no~nation o!' rwo non- syncnronized groupl.i lr.eo a supe,-grou;:i A ~ccordir.g ~o ~ig . 55 .
The combinl.iLlon of wisyncln·onioed i;p•oups wi th equal
~illle base will be dtAcussea wion t·eJ:oronce Lo Flg . 56. Thie
figure sllowo Oll ~Op ~he ::tade:nachlH' .CuncoionG wa.J.(31, 9)
.. na wal(127,e) ln ~he inLerva l 0 · & " ;. The cultipli-
cacions
wal(12? , 31'.<al( 3 , 9) • •al(&; , ~
wa_r127 , 9)Ka1(!1 , &) • wqJ(;b,B;
:1ie~d
the functionn w!il(G11 ,a ) and '1al(CJ6 , e) . These are
the ca rriers req uit·od ror trai1sposition of two groups inco one s upcrgroup .I. ncco r<lillg Lo Fig . ,5.5 . 1'hey are the reference for synch..ro 1tizati o L.
Lines a of Fig . '.16 "how eyobolically the signals of two
non- s:;nchronous groups . !hese signals consist of sums or
che functions ~al(32,9), wal(33 , 6) , ... , ~n1(6? ,6 ) acco~
ding to ;.•1gs . 54 and 55; ohe amplitude of these fUJlctions
• 1 2 HUJ.'l lfl,J:;X SYS:'ENS
7•
depend~ on t.t.C rn:-t.:'-=.ula!.• Si~.fl8l trar.~r..1 t tcd . f ~g . ~? S~O'tiS
chat
8
sigr..al contti_:nlr..:~
::!..9
_:-e,nc - ior.s j-;a l ( 3 , ~
1, .,,..,1 { 33 , 9
1 ,
i"" ste; funccion
"'=h st~rs 1/(4 .:i:le . '.:'he
odd runctiou~ ·,·a~(.?~ .~ ), ..-a.:.{;S,8), ... , wbl( ;'.,9) He no•
shown iu fig . S7 , du!e tLe:y oi:-1·~r or.l;; \Jy ra .r,.cto1 -1 in
••• t
wal(6j,6)
;he intervul -i ~ e < 0 ~ro!ll i:;ho even !"u.:~ct. ior.~ ... ~l ( 32 ' e ~ ,
W"$.l{}LJ,e) , •.. 1 wnll1::.2 , aJ . l i·ie :1:lgnal!S iu t.~~t! lir1en a of
Fig . 5b Qre dtvide<i i nto i i; terval9 1 /"'' witle . 'PJ,,.i.1· "myli1iudea tu•c conntnnt.. iu these inte1.. v;:i.l .;; . Tllo .i.nclivid.ual in11
ure
dcnoLe<l li.Y 1 , 2 , , . . On" u.o.y i'ui•ther· nee i'rom
Fig. 57 ~l:lat l'I o;ign"l cont9 i.nl. ng the f uuc~ion:i wal ( 52 , a),
tervals
, ,a) will hoving r;hr a:I:i!•l-t..iUf:' v ir1 t.h(' ir:tc~rval
-f ~ ~ tiUSt t:~v~ the runplit,;U<l.Y -'./ in chc invc r·-
... , wal(t:
-i
~ 9 <
l:
val -t +
~ 0 < -t - ~' . '!'"niE :-es:.ilt. t.old:: gr.r.<>:-<>l L~- :
Che amplitudes lltWe - c" s:me a':>So~:.ite Value 1'1-1°! OJ'f.O:l.Cle
sign in nny two inter:als
2l:,. e<2~
1
bii" • ;;ii
bL -
90d
of b'ig . 5t ::::ho\" s·..cc.r.. in;-cr~rnl :::! of equul ~t~o ·.;::c
value of the Gi"£;n9l altAT'rin ti~,.;~)y Lt:1lCU1:>1J "Hd r.ot JuJtcJ1eC .
The a.t1plHudos buve equal ab,;o.J.ute .,,.lu~ &.:Jd OpJ'on1to t:i~L
in the inlte.r·valrt 1 anO ? , ; iil1d '- , ~ aud "' , C"'ec .
Line s
s
Piig. 5? WaJ.ab fwiot l ous
j
wal (j ,9 )
wal (?i,9) • cr.J(.,9) .
2 , 4, .. ., 62 .
Saot-!..lng -he 3ignaln of 1 lne~ a ~~ tt-.c t111t::s indica.t.ed
l):V tlce triggM' pulse of linL' 11 ruld .::o J d lll'; Lit.: sampled
volLagPi; il11 1•i.•1v ou1 intr r VHl of· a u1·o t- jo n 1/1_,, y .:.elds t.he
signalt; of line:: c . The.y }1.L'e ~:vncht·or.i~t:"d wlt,.h the carr·ie.-:-s wal(1.,.11 , e) P.:-nd w~l( H , e J . I his i;;:ru.;hro:11Za tion ie
no=- yec GJ.ff 1..:ie::c . \i-t'Ot..I 2 .!..!'.. J i~e C ";.1e("iu~ n- tioe 8 •
0 ;;ith ~he i.1.i;~rvals 1 und ? "' :;hie" the narl i i;udes have
equal nb~olat.e •;nluro . :;r~l.i.J 1 "ncgin~ t-.•itL L1...•o interval~
for 1,.,1J1ich t!1 l ~ is not no . S!Jlfting t l1e 1."1·0 111ls by a samp~ in!5 ru1U ho! c! i n~; c ircuit: yi~ld;;: tl:r· ~,i ~11Atlts 01'
lineJJ f .
G!'OUp 1 i:. now :.;,yr:c !::iront:-.ed co.:.·rcct L~· buL ,... r·<>up 2 i s not .
G:::oo~p " n.ri:; t:o be taken t'roc 1 ~ !!e- 1' ;i.:: -l bi·o~p ? f'roz line
c ~·or a.o<.lt.:l7tlia:iof tte carri,_r.: wa~{~ , 9J ancl .-;al(96,9) .
:~ate ~iJ.'.1t..: t.l.r:· prot:eJJ o~ :.;J'lll":t.ojr~1ni~a-;ior.. •tfft;_·s _f'rorn tbat
of;; i :me .tJvlsfon , sine" e:ii·ou;>n 1 "nd ? :nay 1JC rhifted with
r<:f 1;:1:ence i.;c
f'~ch
c-c;l1a1· b)' nr.y mul.L.:.ple of 1/:'t2 • .Such en
:.1 .:·li-'; r·:-1r·,v ::i ll l..f:.. .,.~oulct .. ~H rie on .!.ntercLl;l,_u;1;t!
ti.me
~:
che
<.;.i-.;.L.
on .
.. ~ t,..n
!~y·:r.bolicull;; ,
tl.r .- lt;Ltl
t
rhannels in
or c 1p..:rrroup A hris
fcl~o-,;ing for-:i :
(&=o·.ir- _,,.
!'IJ•Jl~c.: 1 1 , 8
Vemodul~Litt.Ll
:;i~li.;.
~ht
•
'"'J\·.'~l
fgrou 1
..L.
to;' :--:u pe;:gr·oup A by
c:..#1 , 9)
.,.,ul
011 1 9)
O!' wal(96 , 0 )
£olLc.wing t •1.io .... irn'll .c:;: :
:(t;rou1 1 )hnl\•-11,e) + (!'l'OU? .?).;~! (9E ,a) :ltnl(6'• , 9)
= (i:;ro:.ap 1) + ( •'1-o:.ip ? ),.,al ( 32, ~)
[, fl-OUf' ~ 1".i«l \ d , e. ~
=
Oae
t Lt1
('If
\ gl"" ·~
le;r ou1
Lr:·!"f:l.~
·~:--
">wa_ ( ;>- >@ ) )Wal ( ';•o , a)
"-') "' (p'O'JP '< JwoJ ( j;> , 9 )
~.Ju:
!'lr·h L t:..Rnd .:;1d':.1::'1 1:1ust be nup-
µr·~:::1).J 1 ir~ r·rdf'r co o"t.1t•1ln e-!.·oup 1 or ~--ro11p 2 ncparate-
1:; .
:t
e&:8i€'r~ to
i.
G'.lfl'l"t?!iS :;he-
te.t'.llS
(group "')
or
) ·1,ar:. the a::hcr~ . 1'his is er no ~:·:ictical conse~ue::~e . ~~:.ee t!:e te:·o• (Group ?)wal(32 , 8J "nrl (group 1)•
(..-::·o·~r
.rir•
·,,r ...il\32 , 9J
ol tai.::J~d ,
wb l ch 1!1..2.Y bo
'.Vf.ll\ ;2 , 0 J :
((group
h:.JIC2 ,9 J]\·;al(;2,e) • (grour 2)
[(f"r'<lu;• 1 wnll' , e1 wa1(32,9))= (grour 1)
• d
d~mod ul ave
1'",'''
"lfl J,.,,,J~,,.1.::,
:E'Y '""""'l"-1"''"
"7i.1 2MUL,J,
151
'
The i;erm~ \rl'ou;• 1) !lr.d (i;:-:-o·J.r 2) c9n 'Oe su;i;:.:-e~r:Prt ty
a sequcncy low1 ·1:-;t: filt.e:· ";h<::.t .Lnt.et-·~·~tc::i ovu1· t.t:e ir.ter-:alS 0 < 8 < 1/)2, 1/52 < 0 < 2/;2 , e:.c . •hQ •. .;! ir.t<'gr~
t:!.OO i.n"tervnlr. are ~Lo1:n i..:_ tle l<::.St l i l l l ' Of ?'if. ·-'J· f:-1.eir
. • nu
, .ro1t.ion
.
. eql..a_- co '/ . u• ,~ _, µn .or
..
....
non-norm al i:te•l
_:_s
n 1 t1c:4
cha.Mel ~ele:mony "Y"~e:n . '.:'hese intep"ttion inLt:'rvgJ::; ex,,,~
tend oVC!' two l tltr :-•.rn.:._,::; o.f g:·o~~ 2 i.r
!. itt':! C (1nd Of t::;roup
1 in line f , ln which Lt:e t1.ilijJlitttUes of tllO ri..Gn<' !'lav.-:
equal ~bsoluL•.• va lu e and OIJf!OSite sif•lL 1!;,nc~ , rho inte -
gration yie lun zi:ro n11d
~JJe .si(';n~l :'l
are mipprc:10r.d .
J1ulLipllcucio 1 of gi·oui: 2 i.:1 line c ttnd (,;1'0UJl 1 in .ine
1• by waJ.(;2,9 J mo«es Llie si~-;z;s in t.!:D int<'rvdn 1 a.nd 2,
3 and '" e~c . t?qunl. There .is .;.,o ClinCclla.t1on h,j irctq::rlltion and the sii;r:nla
(frou;; 2)•al (;:.?,a)
'~rouF 1 )t
or
w•l02,Q) J'>8Gll th!-0°..t.;h L!Je sequer.cy -O'<J'""n !'il~c~· .
Fi£ - 56 Osci~!Jtrra=S of :;eq·1er.cy
a.ulciplex.!.!"i,,. oi a lt.:lf•J h)•-1y S1b1Lal . ,·l : ir.r1:t; ::1ltt1F'I P(SI ·' i b : ou-l t-:ul F"(e) of El C• qu(Jn.:y .... o·"'::;:i..::;s
r"il~c-r ; C : i.:nr·:·J.tljt'
••.,.t1l "' ,e J ; L :
-~irs.t
mod'-1-11L-OH
[
"''~1 ( •.:
motl.ul.:.-d.. ion J•' 1~~s )wtd
E: :::arrioT'
11
',0J\'.',-~L , EIJ ;
,o , i
.F : :::eeo::d
(t-,e )•,1al
';! , "! /
=Fn {e Jwa:{1 2 , a i ; llor.;.i.onte.- .:;ctle
50 µs /:liv . ; ~COU-L'l.•.•!:l,V :1 . 1.01.:.:.: H!lC
R. l•:.& .ll.E al A..3G-1'•-·lL!lU:1k1:in A1. ; ; .
Fig. ~6 ohowi:s eome oscillograms 01' n sequoncy rnulti1>lex
sys~em dovt1loped by LUKE aud NA CLJ,: . •rhe "" r1·1. e" ••t• ll ~ , e)
ie shown f"or cllil'i t:y instead of one oJ: LLo cw.Tier,;
•·
152
wa I { ;2 , a>.
. ..
1
;,;al ( t-2 , 5 ) in Fig- . 55 . ?er tr.e :;arae reason I
the Cal.'"l'ier ·..,al ( 9, '3 ) i !'l .;f,oT,m fo1· t.hr :-t-conc modulation
r'lt l1er thru: Ll1e c11rri1~r wal(So , a ) or "nl(1.1+ , e) o::: Fig.55.
!'he crosstulz<. nt.t0nurition obtal.ned i 11 t.ris equi p:oe11t wae
About - ;,;; dB i f nn cx t: r" r;yncliro nir.nt·I on 11.ne was U'ile<\
ond rtropperl Lo obo u t - 53 dB if Lhe synchroni zaiion signal
wnc trans:uitLl'tl with '!11> te:ephony d1-nril ~· a1id extracted
t.y ~ '.-ial~U .functinn r:rqcAifit.; filter . '!his attenuation
>:o, c be hh-,1! ~nough to 1:1eet i:e:ephony :.t,.,"l<la~-tls i1' signal
co.,ri;11Q.o:> '> ~1:e used . llnweve:::-, sequcr.cJ t1•1l ti!'~exi:!g is
rr.ai:-:ly of iz,t"'"'""t for rea/. po>:er :imted di...-Hd signals
nr. t:t":(: p:-es~11t., aud - ,:, an cr.osstu::.k nttrr:uation is more
tHAn enough in t.bis cann . An advanced t.nq 11(') nc:y .:nul"Cipl eJt;
system is beint-'!; <l<~velopod
fost Of.rice
by HltBln::R of
t: h(" We st
German
Gepul~trneat. .
3.13 Digital Multiplexing
:t.
been p l :l""t:<l ot,;,~ t~for" .... hat sequency fi!.tera
brt!'H?:C or~ ~-ial.:.;L fur..ctior:s can Le i.:npl~n.o::.tf'!O easily as digi t:1 .... fl 1 ~e1's . 81uc(~ tanJ);aec fil L~rr rC'-ql1jre sequency
sl1C"ting o{ sig1:.al::# ,iunr. ~n rnul LiJ;lex ts;i,·r.t.~r.i.n do 1 one "''ill
,..xper:t that !'lecp.1en~y r.iu:: tiplex system:'j cnn no i:npleccented
i~ttsily by cilgi ~(lJ. oou L:ime11t . ConfiiOer Lliu u.ul Li plexing of
l:Wf) - -:ile_pi.100;1 uler.A l ~1 ACCO!•di.:i5 to 1''i €' . 5~ fOl' illus~ra
!:ior.. .
'I';,:; si;;r.&.l<i f,(e) "''~ Jc,,e)
~o be 1tultiplexed.
Tl"".,.!'!c- 51.f:nsls :s..re rcpro!;cnceC. by <..lu~ curver a and a• in
F!.g· . r-0 . _.!;!" neif 11 t. .. odes .L:.: a pa_rti.cu\91" inter val , sa;-; t.he
i1:tc:rval 12:1 un ·:; t < 2~.:' 11s ) are trannfor·11 ed into digital
J'Ol'JI' by aL nnnlog/<i lei Lal cor..vex·t~r . 'J'oblo 8 lists tbc
rl t;:!:ital £·~pL·eccnt1 ,ti<>1:
~·101 1 0 fo r .F ~(O) a nd - 011010
!'or I•' f (6 ) . :·Iul tiplmd.r.<: or cileee Lwo value" ·•ill be diecu"co<1 •-:ic:n <'dfe.-cncc to ':ob le & . tt it< asmur.ed tha<: J!'z( 9l
nr.1 It' 1 e) {;l.1.'~ £i(J~:.i1:1 o.:" gr.. e - cha.tJ'..1.el inul t::i!'lex system ·
'rte~ ·(~-~1=:-. fur~ct1or.~ .,;al(O ,tJ )to .rnl(4,8) are used as ~ar
riers . Only ~-..-o ol' &he cha:aiels carry signals . '::bis c\>r!'"6~-or::i!' <:O an Ac tivity factor o:: 0 . 2;, , "IJicll is reprcsent .. as
,.,.e
- 1" DIGl'IAL liU!.'l' n
; "
.
Table 8.
Digi<:a l
1.i:;x:;:c.
segueccy "'"-: t.iFlP.~ing of
two
sii.;nnli;
Ff(9 ) nnd Ff(6' •1ccoi·<lil'f5 to Hi-;.'·u . c , c ',. ·· b ' refer
to tlle L'cspect i ve lin~ in Fig . 50 . !'(a) stw1ds f o r l;ho :own
of .F;i'(O lsal(1 , B) + 1" :'(3 )sal( :;. , ) .
c c'
J'
<l
- ~·rt
+1
+1
+1
+1
.•1
-1
-1
..1
- 1 -1
- 1 ..1
- 1 ..1
- 1 -1
('
a .,
-F:(a)x
• 1
atl.1(1 1 0 ) nnl\3 , 3)
- l't"
1110110 - C1 1010 1.0011100
+11011 0 1011 010, +1010000
+110110 -~1101(1 +1('40000
+110110 - 1101(; +OC1110J
-110110 t'.)11111 -OC11100
-110110 -,11 ·1· _,. 1·
- 110110 - 11010 - 1G100u<.
- 110110 I 011011.} -(;0111
~
•f( ~ Jx
::nt(1 , b,
•0011100
~1(•10 ti00
•H
~
)•
set:\3
·I
a)
0041100
-1~•10000
+101GOC{1
-1 •.o1 ::000
+001 1100
+0'>111(){,
+1• •1 1(,.• . . J
1101'.JOJO
+O<J11100
,..__..J111C~
+00111
_-;.:.1
- 101
r Xl1110:.
1111111oooc - u11 J1000cl
-110110
-01101
tativ& for telephony :ou.lt.iplcx:
!"ic.
chnr_nel~ .J ~rr ir.g
pn:ik crtt..!'-
'l'bo two carriers -s'1l('>,5) a:id --nlP-,~l ~!tltbO Mprnsented by 8 digit., 11 o•: - 1 a~ cho•,m i:. ~ne col11urn <:
•nd c' of c'able 8 . The oega Li•1c nigr: of t~.e ~ard1 "~ is
C>l' no importance here . Tlte ca2•:.•iern - .oru(1 , 3) and - cnl (~.,e)
1llllphtude modulaLed liy F ~(9) ~nd F~(~) :•~elc C, tb.r, th~
number• +110110 and -01101C "1ulLir>l.1• <l lJ;: +~ or -1 ""
holtll i.. the columno d unc d' . TLe ai:.lLi;..:e:.: .cip: 11 -1'(6)
or column e is ob~ni!..e<l. b:.· aC.<ling chn t~-10 nu.cb€'r."' ()J' the
snrne ll.ne in colurcn c and c 1 • Th" 111.ulti1 i c·.:-: !:iigt.al ic represented by numbers i1aving one mot·~ tliti;lt thr1n ~~~ o; 01·
F·~ (e) .
The 8x8 digi ta - incluQL:}F r..l.Jc :ii~" - of l .. " riit•Jiol
-F(e) 'lla;;~ be tl...8.L.BJDittc:d in many 'lr:a:1!1 . For iLSt911c~, or~e
could use 6" block pulses ;:i=h =pli~udes +'I or -1 . ::.
this cnae , sequency divif>ion would ht' us<?c :·or :tullirleXinc, en(! tin:e divinion for trnnsmis.,ion . Th., C,L pulses
llOUld be bhe sumo number es in timt' multi:riloxine; of 8
if one parity ch<?ck digH were added to U.e 7
eacl:: channel. Such a
chock digit would pori:u.t
1;4
si nEle er.:-or rler;ection l>ul no e=-ror COl'.t.'ectJ on .
De1:1oc.ulaciou or -Hell~ don~ b:f rulLip:yir.s - f(e) '"'ith
-sal(~,=; 9rJ -dil(;,~,. '!he ::-es'.Jlt1"'..; 3:.".1.1r:vn=ters.u.••
1
sllO'IJ!l in col u:urs g a:lrl
Intr!gr~· ior_ of J (!'! '~I'll 1_ 1 , e) nnd
•
~·(e )s"'-C~ , 9) meows ~ddini:; Lhe 6 mu:rbern 111 colwms g Wld
g •,
\•;hie:-; yie!ds +11 0110000 and
8 yields &!".e O!·igir.al values or~
!:ici;.l
·.1ay
8j pl;(
t;bc
1 . c:i
le~~
to obcain r: 11ese
fa.:~L
-.""l11•J"'i"~:"~_c• .
F; e
!lll!Lbt.::
1in:!.sh-f<~ou 1:ir.r
Divisiou by
n:;c f~ { FI) . . he pr c1J'.lul;: , •.
trcirtd£or:nariou
...:ourse be to
OJ'
section
Lo che rnulL i plox s1.,-,nlil -b'(O \. I!'.clriol•Jr. of this
ti!t.e-consu.nl.ng oe·hoJ woul1
ob~cu:::-{'
tl".e
~x;>tauat- ...on
of tt.e princirl • .
'l'he Sif'"U'1l -1"( 6 ) COil t •ins Ouly <:llf.1he:·,; w1 tt absoluLe
voluti !l011100 nnct 1010000 ; two of .,net have uep;ntiv<'! aigna
and t:wo of
~ocL
_ro::it._v,,. s:gns . rh.i= is t_ypic:tl for two
ac:.,ivE c:-i.anr:ul::: . Ee!"J.CE;!, ~r c::e nu.:r.bcr i~ <,,;lu:in£'eC due to
ir.t1"ll"fb .l'enc~ it Ca.I: bt• r !'l'ectcd 1.y coir.pari.uon witl1 tho
tihI'f'" lUlchwigcd numbe:·u . I:J most r.1;2es i-c i:; Also possiPlo
t...o ccrre<:::t Lwo e:....;;·oL·o, rn ...1 in tLo.u:~y "":ase!:i more then vwo
can be correc-:..ed . ..;ere i:. ti ~ s 3 defini1;e 3.d•tmttt~e 0•1cr "CiD.l: di vi :;io.u. t•:-:.e under:yir:g: re:aaon is chat no
uoatul informt1Llon ir. ~r·(lrtsaitted chree four~hi; of che
l;!;.:ror!"
ti.me i!' -che ~ct.lvie}· fuctor is 0 . 2"" and r;icle <livision is
u::n(l .
A condd.erabl<' nu:nl.rer of variaUon"
or
thtt digital se-
qu~ncy uultipl~xi:1g r,choure o: 'l't1ble 8 have been investii:;nted . Howcvor , the roesible nm:il~1· o.f vari atl.ons is ao
g1·er:c , that no rt~fi.nitc cor.c:usionr. t.ave bee:1 reached yet
abOUL' ::heir 1·elr:rive c.icrit!i .
3.14 Methods of Single Sideband Modulation
Amp: itude :t1odU.:atinn of sine or r:osine cai.·riers yields
a dou>:>le sideband oodulo~inr,
due
to the 11ultiplicat:iOD
Ll..Clore:us o.f these i'un•!tioi:n . 'l'hei·e ~re a nur.ibcr of :net:liod•
me el .unlnatian o f' one sidG~und ~ha~ eWI be analyzed
£01·
vei:y
we~-
by ox·LLogona l runctiont:i .
Conside.i• two t"rans31.itters 1 both rndiat:ine .ninusoJ.dal
135
'·, 14 m;1•uODS Ot ss..1
functions o! frt.:11u~nc~: !int ':tut !~~1-..·ir.r,-
'.J
;.h3!l:& <1!.fl'C?rc:i~e
of ; n. 'J'he ca..rrif'rS l !t:t~L!--uj-e- nodulated Ly t.1.rr.e f1:n~t.:.ott.~
:• (e)
and c•(e 1 , £~.all t.a•:e ::::= ror::i i''(e)'[2co~o 0 e =d
n'(e) '{2sinr. 0 e. I~ is '>l'sunec! th~t the frc111.-ncy '.l 0 can
be rer1"oduced r.->:r:.c_.l:·' a:: ~J-;.;: rec-.=iv~1, bi.;t. t.het t:he.:.-e iB
9
phase
~if.1."ur<"n<.:e
0-
\f2 cos Oo0, \r2 :-ir. O~B
Lt~
tet"',;;:..-..n
..1.·~ceJVr•d
and i:nc: loco... ...... ;;
cnr:.·ie1~s
! ·l'O<;.UC,...11
cnr:-ie1·.::>
'{2co a (o 09+C1) , 1[? sin (n 0a+a) . Multiplica t ion or.~ r·ecei·;"<l
signal s( a) ,
(27)
5(8 ) f2co1' (r. 0 8K>) =
''loJ c·~sa
1
•
t- : .. (3)cos'2G B~
5(8 ) '{2 sin (r.,e+o )
=
o•(e) ~ir.a...
(2BJ
~ IJ .. eJtsinft1;9+:. J
- i<'{a; sino. ... :>'(a cos o.
+
::
·J
• F'(S) i;in (2n,il ><1 J - Ii'\~• cos \2!'l 0a,.,. )
I'he t.e1'tll8 on vt..e rigtit ha..":IC slde:J o""
(c.tJ
J ·u.:J
(?
t1
mul -
tipli ed by cos(2n 0 6+Cl ) or
El11 (2n 0 e •Cl l co 111.nill v<=iy nigh
!requoncy componl'nts only; they sh0l l ho .a.pfi·esse<l by
filters. 'rho right bllld s ides contain tho u F 1 (0) or u•(e)
onl;y i f tba plwee uif!erence a vunislt,.~ . ll!'11C• , rwo cu=·riero or oqunl frequency but a. pha.se di fferrnce ;11 may
tra.tJSllli~ ~wo in<l"Jl<'ndent si"""l~ F' (e) 1u I .1'~6
"'Hltou:;
mutual interfe1·ence . P.Jct.irog it <iif~crcntl;; , eucli I:re11<ency Chtw.ncl Caii be subdivided into t_.o ph:izc (:t~OA!.oe~F w:.iich
~11 be denoted t.t-rt: as sine and cosine chb.Ar~el . Syr:c!l.1'0r.oUE demodulation pe!"ci"t;s u::ili::at!.ou 0£ bo!-11 pr.qsn c!:!Wl-
nel s .
A certain Cime fw1ction a.a., alwa.,yo tc •rruie:nitted
through the nine cna."lneJ , but nc·v er t!:l1·ougb t.ue costr:e
channel in order to make t h.em disUn~--;ui S\habl e . One ID'-Y ,
fo:r ineto11ce , r·epluce F'(e) in (29) l.i,y 1 1 '"(6) And :ro qui,,.0 Lhat F ' (a) a n<l D ' ( e ) hnve pL'aOtically no ene r gy be low ~ oertoio froqucacy . The s i gnal 8(8) miy r.h~n bonelllOdu.lu~ed by tile circuit of Fig . 59 . TJ1e uignal t 1+Ml' 1 (9 ))x
( 1 +cos 20 , a) 1.n obi:ainorl at; ouLp:it 'I , a.nd
i .. oLtai:iec. ar ouLput: ., _
v• ( s )( 1 ~ cos n0 e)
s.:.i.t:"le ;:i~elib..UC :nodulat.!on i5 6.!l i:xcell·-,:-.: practical.
cei...• for t l :uismi t~ir.t· ch.rough ,.11.e tu::l cor,11.1> channel.
To i;how i:Jli:; , lot a r-ii:,t.v.l F( 8 ) bij c;xpanded luto a serioe
o! r.:inc aud co~ine pu2.sc!: accorainc to r•'l t; . 1 :
f(gl at:)f\ , S)+.'~
~
l: [1.c(i)cos2riS+" 5 ~i}sin ~ni8]
(30}
"'
:o·it"te:- ::r3L..:forlls g(O,v), ..-,(i, ... ,, t:.!JC £s i,v) of thE-se
pU- ·c~ ~...!.'E:! t.i.VCn by {1 . 24; . 'he fir. t five t.rnn:;.foros. arc
.:.l:cwn in F!.o ..·1 •
:.,t:-L us · aenot.e the ji.roduct.i::i .!'(0 , 5Y[2 cos flo•3 , f{O,a)x
\+'C. ols1nc.t• 1 ~'co~ 2r i i3 conn 0 8, 2 coe 2ni:i s i r: n 0 e, 2oin 2tiiax
c
~1 11 a.r.J ;'=~nC.Tii1 tiln :l 0 S by U 1 c(~•, , j 1 !>(.JJ, a.clc
),
.
.
.
dc 1.~" J 1 dsi,< {tll ?..Ild tl~i.• (';') . 'Jlle Fou.r.:.er t-·unstora.s
th~;r p.roC.uct~ ;ire C.t!HOtflld Oy ."1n,clv ) , •• • , "il,s Cv) ;
~--·.. {,\'(J
~\·~
h c1,c
, ' J-'IJG i•
,.:, g(O,v1v 0 )]
(;1)
4 ~(gc i . v-v 0 , .~ gc~:,v+V 0 ))
\I
H ~v· f ~; 5 \i,v-v 0 .' + g 5 (i , v+v 0
v0
o.f
)]
0o/2n
... t ~ ... ·.gns in >tL!'enthesis: !".old :or th& Fou=i~r trans.forms
137
- 14 11E1'HODS Or SSI'/ •
•
(v).
t..,,
(v)
•
wi<l . ,,_, (v) .
. Tho Fo\;.ri~r trar..sfOrJ)S Ge~ v) :-1n·:1 ~s( v) of the rw~czi-
_j..00'
ons ?(e )'{2 co:i 0 0 a =C. }'\c j 1,'2 si::i ~1,B "·~ obtiuned fro:r ( :'0)
snd (31 ):
Gc(v l = n(O)i.o . .< v )
-f: ~'1 , ( i )!,,," {v) -
n,(i.Jh.,,
( v ;}\~·2)
i :I
G, ( v ) • u(O)ll 0 _.(v) ~
~
l; [a, (i)h,,_, ( v )
+ IJ,(i)L., ,, ( v )j
j: '
Considc-r the
'::9Bl'
u,(1)
~
a,(1) - 'nncl all oLlt.,£· co-
efficient:o eq ua 7.~:-o to ge t an w..de1·~t,,;ndir:e; o.C LLe sh~.pe
of Gclv) and G,( v ) . The i•esult'ng F:inriPt' ~•·an.Jforn.;; ol'
tile funct;ions
'{2 cos 2n9 •,r2 cos n 0 e, '{2 c~s 2n6 '[2 sir. r: 0g ,
'{2 sin 2rr9 '/2 cor o 0 e, 1r2 s~n 2rS '.'2 d'° il 0 S ,
lil'e :;hown in L!,e first fo = :it'"~ of Hg . & . 1:ote ~h%
'{2 cos 2n9 and '{2 nin 2r 8 are .cosir.e ancl :an.- r u~~e., . that
equal 2oro outnide the i.nle~val -~ ~ d a t ·
The fol lowing ain5l e s ide'ba11d si~nnl~ n~u.v 1.:e r!.crJ.vetl
!rom the t:x·o.n'-l!orn.~ in t he f i.r.tt ro u t' llt~t':'.l o r f ii:;. •50 :
• coca 2rnJ al u Dr.a " sir: 21a cos n{la
fee ( B) • cos 2n B COG o.a
sin .?na sir. it 0 9
fsc ( 9) • cos 2re .. in 0 0 e - sin 2ni: co!l ~ 0 0
fcs ( 9)
(53)
-
fss (9) •
cos 2nB cos 0 0 ;; -
s.in 2.,.a si.!J :1 0 9
The Fourier transfor.r.s o.f t ..csc fwlct.~ona .csrc rt-:.o,.:c i.:i
hn~s 5 to 8 or Fig . 60 . !n« :u...,::,;~01.s f cs ( s ) anJ r cc, 9 )
~.ave almout oll oi cheir e?:e:.·g;,· in tl~P ur 1 er s::.dPo.:;:..uC.
Oo/2n . I 5 c(e) and 1'ss(6) Lave mo~r;ortheir energ;; in
th~ lower ai<lebuad " < o. 1 2n . l>otll phone channels !il'e
t.tsed , Ginco nll four signAls (33) contain ~It~ ait.e c..rrier BLn 0 06 U.nd the cosine caL'L'ier C05 0 0 9 . !le ;•racti <:al implomentation 01' singl.e sideband mo-iul ncion <>ccordinl' to (33) is uauolly called second met.hod or l'h~n., :;hi ft
method of SSS modulation [ 2 ]: A signal F(B) ic modulaeed
">
F8
/ . C:O.:<R:EFI 'l'l!iJJSi'ilSSlQJl
onvo the
r.:ari·ier sin
n0 e ~
u;Jtl the
n:irr:c s1gnal wit h all
osci:lation~ 90° ;;t:.e.s~· 5ti..fted l.~ rr.oc'l ·-"la.i;etl on Loche car-
rier cos f/0 6 ; sw.n or -dift"erence of the
riodula~ed car·rie.r~
:,riel:iB ~inglt: ;;irieband S i gr~s . f'!',e fi::.,s: :nett":od Of
SSB
ucoUulution ot:tai.cs Li!e ~u:r.t: resuJ "t. U;r
si<l.eba.r:d by m11an.3 o.r a Ji_ tr;~.
one
CV\
. . --t--- . . -
\)J
· -- -1-- - -
ty\
J-.J'
-- - - 1 - -- -
v /\
I
;
' v
2cos 21fllT
./\ - - -+-----~
5
\)
----r----
(\
6
A
-- - -1--- -
(\
COSUot
2 C'Ol 1 ITU f · lin u 0 f
2 sin 2 Tr t! T · cruGJ0 t
2
s in 2 'ff I/ T· Sin c.Jol
c os 2 T ti T. 5tfl CJol • Sin 2Tr111 . COj Uo
- -- - ! - - -·
A
CO.S 2fft l f sin c.>0 t · Sin 2"t/ f· COSCJQI
/ \ ----1-- --
A
CO$ 27TL/1 COSCJol ., 5in2Yt!T · 5jn ~t
7
v
4
I
( \ - -- -+--· ·- -
9
~·~ppressing
- fq i
0
IT -
(\
r,r
cos 2 Trt lf cos (..)0 f - s in.2/ltll ·co:. GJ0t
Fih - t;(} Fou:.ler tran::::·orrnr of s it.Lt>ru'idccz inc carric.rs a-n rl i.tude u.odu:ntcd t:.y sine and. cos.:.!l~ _r-ul ne .
Line :J lr: ?ig . 1:::0 show:~ wh,y n~1-;a t.i 1.re f'rcq_u~ncies- cannot
l\O: !l..l~regar•ded . TLi:;; tI•nnr.r oT'O looks like Lhe Li·t1.11sio:rm~
·~f lines 5 cu1d G fo!· po sitive v·alues of v =- IT 1 neve;rt;helc!ls , this i s ;iot a single o i deciand signal.
Single l>ideuand ar:d double sideband modulat ion pe:J:'<ni'ii
the S8DD llUJ!'lber Of Cb.rui.nels in a cert9in f'requcn cy band ~
lf tl:e c•,·.10 phase c.::if!nnels oi- <1aeh 1'requency channel are
nsecl . The explo it ation ol' double sideband modulat ion i n
t.hj s
..n:iy -
u~ually
re!"errea to as quadre..ture modu1ation-
is handic,,pped by hig;h Cl'asstalk in t he case of telephollY
; • 1.ll METriOilS CF SS~l
tra.naraissivn . $1r.slc cideba.ad rr.<'ldulnt.io.:i , or: t.!1e other
hand causes morP. dirtortions in dii:;it.ul sip;c.:cl t.r·~lllnrr.i:-
sion i f SSl3 filter,,"'"' used . A d,.,11.ole i;idebaml ~t.,LJlnmH
ter transmits all <1 11nr£~~ rit:hPr 1.'.:·ougb tite siJH~ ().:."' the
cosine chan::c- c! :.1 certair.. frflqucncy bane; n sing I e ::!i<lebnnd trans:ni:::;:e1· li~~s.ait:.$ "G..: l on~r~· chroi;r:- t.l . .i... ~::_e
as well n.s \;he cot..L.Ut: chaa"-...""te~ o !' ... frequency Can·J ,,:,s} !" as
wldc . 'l'hermal nolHt.< .i.:.l'lucr:c.-.:: ~oth tttt.'..J.ods equr.lll.1· , ~·ro
vidod of course
~L~ l
~hase-:ocnnl ~ivr
filter'.iu':': is
1rned
!or double sidi!land nigr.!!J.~; -ti-,,..rwi::e oce wouLl rnce•v~
the signal fr•oa ono 1;han<> on--re_ 1'ul ~Le noi~P frore toth
phase ""llllllels .
Tb1
investigot 1on o!" lillplir 1Ce :noC:ul at:~or: oy et.cans o.::
sint o.nd cosine pu._$es su.:fe!'!'! froc1 the fa.ct thuL t!1ese
!'unctio ns a.re not.. .CJ.•equency limitnd and are i;tun'L-1:1t•Sor.tf! to
plot. i'h" resulLl! UL'e simpler to obr.iir. witl. ":oli:;l. fLU1CtionL. ConsiC.eJ.' the 1,.1aiGh f•.Jnctiona o.;"' ?ig . 2 u.F f'r'~quer~
cy J"unctions ~al(O,v)t c11lf..i , v) • wai{2i,v) Ct.I1C ::;t,l( i ,v =
wal (2i- 1 , v} i1lf'.'9;eAdOf•1 .... e .:unction~ . :Lilt .!'o:lo-..:i.r,J: tice
functions are ottninP.-d ti:,· a Fcuritti· l !"!:tL.S!'oro.nti on :
w(2.k ,9 )
""J wnl{ ?k
-oo
s
-
w(2k+1 ,9 ) =
1
v) cos 2Tf'J0dv
( 7;-11 j
\•:nll2k+1,V.1EiT1L-v9j'.I
'
The functior.!l •11(:,e)coe ft~!J U!ld 'lf(,1,G) si:. :lcO, j
or 2k+1 , have tl°'." f() LLotiin.g .?ouri _.1· t:r.·a.n;::fo:·r:t! :
...
-.
c:.
..
...
2fw(2k , 8)coso 0qoos2nv9d0 = wal\ ·'l<,v -~ 0 1 - w·11 1."k , v+v .•
.
0
2rw<2k.,e)sin0
1 9sin2r.vod9 ; w·l(2k,v- v, 1 _,
~wl?~+1, 6 )cosn 0 esin2nv9Li::
..~
=
·,.,al(2k•1,v-'!,
~al(2k,v•v 0 )
+•·a1(2k+'1,v•~J
W\21{ 1·1 , 9 )sin() 08cof.12!TV9d0 =-Wt1l ( 2>t·l 1 , v-v0) >"ILtl{ ?k • 1 , v 1V,,)
(;i5)
3 . CARRITR 'll<A!l5;·lISSION
'!!le follo,;ir.g !:i g:oale 1:,..,; ng nll <:<.•' rgy iu the u;iper
or lo\\er :::ideban Is on!:t rna;,· be ae1·ived froa:. tr.e time fu.nc-
t i.O!JS
( jll) :
•'O,a conn0 s ~ ,..(1 , 0)son:l 0~ , ·~(O,O)cosn 0& - <:i.1 , 0)sinr. 9
1
+,,•(C , S J~u.C. 0 e - w( 1,C:jco:sn 0 e ,
h'(2 , 0 Jconn 0 a
-..:(2 , a)t:ir.n~s
1--
w(0 , 0):; ln n ~
0
.,.- •...•(1
, C)cosn a
0
\>:(?t ,'> ls:in:i 0 9
".-(~ , O)sinn 0 a, W'~2 ,0 )cosn 0 & -
- ~<; ~ ·J)co:i0~9 , ·.~· (2 . ::l:)~ir:U.:'3 , ,·.1 {~ , o)cattn 0 9
(36)
i>o= • 1tu·ier r;r:ms:·o1:i.s o:· Ul• !'=cUon:; (5&, &!'e shown
in Fig . •'1 . Tt" ~rrows ir.dic:itc in ><hiclJ JirecLi.on the nb-
aoluoe vnlue of the fr·•"1unr.cy of N>il(2% , v) w 1d wd(2k+1, v)
ir:-::rea.aet! . l1!'.e o!.rection o!' t-!:n 91·.:..·ow; r~ar=in:; 1.u:chan d
.fo-
~lie
UT'fer
dir.c·~8..!lds
Ant.i ie rcve1:se1 fo.:- the lo,,.,cr ones .
wa\QO)Q)'IS}cg• ...·~ 0.6l$1rtl0 9
...F=i
____ ._F3_
Fiy . t·•'i Poui·i c':" • rn.::sJ"or:ns of solte
1=Cc:.uen.c:r _j:i;i1,..ed .si.r..g!e sideband
~ i;nal::;i v 0 = Oc/2, .
w1"t?ls111011Q.. ·,.·i:.P)co• Ot.tB
, _ r = t ____~
0
•!'
;\
~lock ~iaJ.,.i' :l!C
::·
fo.r tf.O
~econ<l
=nei;nod
single
01•
Sit;.8-
v~Jr:.d rno\;,.ulq-ioi;. .i.:-: sho""" ir: ~·ii; . 11?A . 'l'h.e frc:q~enc:y limit~d
rii~·nal f•'( 6) is fed ~hrouf•'ll t;\o.•O pbtroe :>!lifting no,;work& ·
rwo zigr.ul!:! ~pp~·1r a~ l!..~i r out.puts , ;those oscilla- ~on
0
C0)1!JO!lcnt . ..:iav,..
pha~•C dlffercnce of CJu
buc are Qth• r -
wise equA .• I 1hc r.&l"l'i e .c·:;
rt icuci.e
cos 2 rv 0 0 and t.:Jit~ 2rrv 0 9 are ani-
!Q()ct.Uletl,~d. . 1'!'.e SU.Ill of
t" e prodUC"'°'":S ) ieldB
eicc:>a~c.i s~gy:al , t.!~e tliffere!lce a
0
!lll
upper
iower ~ideband signal ·
- 1'f i'!:ETl!OilS Of•' mm
,.
Fie; .G2 (left) OuL,Lac-:ing act-t:od (aJ "DC SA.ll.\GA ' ~ tour~h
method o.f sine;Lc e.!.'1eban<.l mo:lulatioc (b) of & r•~r:i.Pr wit!.
frequency v 0 by a :·rc~ue2cy l!.ai ted !>1t;nnl 7( A: • FS f.h9~r:
shi.fting network, r-; !f.u1":..~1 :..er , S adccr , !!? bOL<lP•~s fil ter, F' ( 8) i:ingl e si<iebsnd "ienal.
Fig . 63 (right) '1iEAVE!l '" t!lir'<i :netnod of : 1:1,., ~ ,;.:.cebiuui
.modul«tion of a cnrrie.:· ·,;it=: freq1.4~nc:; v 0 +~ li.)' .u freque:1c,;·
J.i;nited signnl ~·(o) . CS oscillator , M riultirhe,· , Ll lowpass filter .
A
very slnrl 1 lil.. a:ntr.od is due to SA.HAGA L3 ) . ...,he car-
r iora cos 2 rrv& 1.mc ,;iJ:1 2nv8 ai•e adoerl to Ll:e f h t<$., stif ted sisn£Lla nccordlnt to F'ig . 62b . 'l'ho t wo aurnii a1"0 rr.ultiplied togetller . An Ltpper side'01mrl ::iic-uul is ~;enM·•t~d ; i!!
addi tion , eignal:; are produced in Lhe uur;<;btond nnd nrow1<i
double the carrier i'rcquoncy 2-..: 0 . A sim~ 1 e li~n 1!'' il tE- r !::!J.l. -
presses those undesirable signo.ls .
;. further :i.J:gle siC.euanti. modula-.:ion cict.t.oJ Lo .l"e to
'.,'E&'.'::.ll (4) . l"ig . 6.! l'l:o>1~ a l-_oc;: diAgt'alli for it:- implel:lentntion . A :<igm1l F(e) :-:ith r:o cneri:;y oui;side the ::ru..d
0 li .f i 1/'.r or -1 ,; v = o·T :: ;.A i" modulnt.oJ onto tr.c ca1·- i or
riero sin ne O..'ld cos nS t·ri ~b ! .reqUc!.lC;,· v I • f,
i 1 = 1 /2T . The fr·equency of the cru·i•iel' is in t:-.e midd _e
of the bl.Ind uned . 1'he 1>0dulated carricr!l ps~:i tl1rougl1 lo•;JPasa filters 1'ith cut-off i'requencies \lg• i 9 T • ~ - The
filtered aii:;nill.e a.re moduJ.ated on~o t h o high rreq\lency
carriers sin :?"( v +1)e and cos 2 n(v 0 +t)e . 'l'l:e a um ,yields
0
; . .,;AP.R!E:ll 'rRJ>J;s :-;:ssroN
rt~
..l.J.:f'e-: · s1Celwu..t nlgnul ,
~he rll.ffe.r~ncf"
a ! ... h'•::- .sideband
~ 1f.!t:l- .
Hi:; . •_II Fouri •er t:r3.1Ulf orcu o: tl.e t:::..ra :r.e-
or
~hod
single sidel:,and uoi.'!ul F.tl;iOL .
E'or an e.xµlu..:.nt l :):. n :' ·. .'E.~:•rv'3. 1 ~ :r.e t : ..o(.:. le:.. tr.<' lreque.ncy l:oit::t:j ill,?l.lt: tsi.Cne.- F(iJ be f:A}'.ar:rled 1.!J,t.o a .series O.f
-11= :'unc-t:io:is i,.:(;,k , 9) and w(2k+1, ~) jcrlvdd by tfhe Fou.::1~r t!'an~f:,;;_"C: (;11) !°:.'ca· t~.e '•htl!i!": funclion!1 .
~
L: r,. c~r.;..(-·k , 9)
l"(8}
~ !\(z,;•.1 ).,(?1,+1,o )J
(37)
~ ~a
fi; ~u!'fic c!'l t o L t·ac~ one> cv~n and one odd funct i o n of t he
•ei· ' "" ~?7 tcirou1•h ~he ci rr.ni ~ of Fit; . .,3 rnthcr t hrui ~·( 9) ·
~·he si1q:les!: ;·11n:t!ons, "'(0 , 9)
_.,, (1 ,P.), ai·e ·rned . Ihei:r
Fou::-1e1• L:-.s.n<!'om:l wsl(O , v) =<l - zo!(1,v) Are zhown :iJl
"":!
F.ig .
i.,
line 1 . ~h~ nr:ro .... ~- roint. in tr.c di!·octiun o~ in-
creCte:i:..& !lbEol~t<" V8llJ• 8 0£ 'J . !':odula~iOD OC COS n0 SJJ.i:'tS
.? •14
f·!h"!l!OD:J OJ• SS!'l
tile Fourier t.:·anr:·o:-ns of line 1 ·o:; t
to
1.11~
t!le loft (l i:Ju 2) . 'J• e t:-anE:':o= shl~t~u
s)loW!l bntch••d for
cl~r1t;; .
i·igi,t. gnd to
ti.e -c~t i<'
The two sl!:.fted ~ran ~·omc are
t.o
shOW:: superiz:tf.O~~d. k!J.e-J."8 they 0'.~crl a;• "..!.d l.::t\'C <:quol signr,; .
NodulnLion of s.:.u - q s:O.ifi;:; rr.c trnn~fo:-r. by ' to -nc
right '1!1\! the ;r·nr..!:ifo1·rr. in.·..;:tiplied tiy -1 ~:; t t ... t.he l~f7
(line ~) . LowpR"'"' filt.er·s su_
p1-'rer-.:; a.1.1 comront:~.L.J vu~side
bhe b!Llld -$ ~ ' :i ~ i liuee 4 $.I'ld;,, . 'l'l1r 1·e~ii-L.Lt.v •. i~ nal:;
So ( B) and t. 1( 0) o.ovc odd ~r'~JlSforo~:i ( l in "" II ·ind 1. LineG
6 to J :ihow ~he q>.•cn L1·u.uefor1:is o" t: 0,(a J, e;02(~ 11ml L11 (q) ,
ii.,( 6 ) which O.E•:V llc rn;>eria:posed tc ;r c 10
also showu 'ire 'the odd trM:;fc.,...m!l ..:if
er .., a) ' 11.j
L I ( B) ;
9)
iJ
(
I))
ru~rl
,{
0
1
02
g 11 (8 ), t;11 (9) Which ;ne:d Lile tra:;::;fol·l::J OJ: ll;(q) t•Ld g 1 ~9)
superil>poncd .
'.!'he cran!'fo1us of li::.c:; 5 to 9 :.&v.. the cnap' of
!1
wal( O, v) Md -ssl(1, v J . :ien;;e, :me 'lOC&1r.r. tl>e 1.1·:wGf;):·a:i
of t;he follow ng !'un;;tio;iS Nit.. ;h(' t'Cl f OJ' (} 3 :
g 0 (9)cot2ri(v0
+rla.
[g0 /S) ... ;;:01,o:Jcor""'"•~' H
(~o)
[ n,,( e ) + 11,,(~ 1j •in ·•\ v 0 .. u~
OJ)
to
h 1 (e ) s i n 2 n (v 0 ·~ )q
a
shown in l. i.non 10 and 11. Tl:e trar. ">fO""-· :'> f liue:3 L
and ;, ui-e uhl!te<i b,y v 0 +t to t11e ::-ign t am: l"!L ; Lhe r-loi.!'ted trans.Corms B.J.'e m>.tHi;>lieil ·:iy +1 °1· - 1 1.1ccurdiw· r.o
&$
the four µOB'liblo ;.roouct::: or nvcn or- odd LI'WlAI on1r 1-1it:1
Sine or co3ir.e carrior as zhc\-.1!:. :..n ( )r-) .
The Sllll! of lino:i 1() =d 11 yield~ ~:.c Fuud er tr-11J1~
ot 11.D UHe= sideband s~i:nal (lii.e 1 ) . :"'" c!ii'fer-encc yields the tra.nsfo.z.!:l o~ :i ~011.·fJ.t ... J..Jclw J ai ual .
~e genernlly uned .oe~hod of :i1'1gle :11i!c\>r.::c :i;:>:LllaHon is the supprcnnion of one ::;ideo•n·j l.:V ·1 fl. l tcr· . Tl:is
~ilter causee disto L·tionz ·h·t::..ch a-r<"' p.nrticul o .r l;v o·o.lecti::i nable for the t.ransmisdo::i of digita l di:;niiU:: . Fit . t.>)a
a.hows the J.'roqLtenc,y uower specL1·w:1 of u r i1-~Hal "·1 lt!1 i:racCically no <?norgy ou~n idc cbo band CJ ~ v
v 0 • .It ; .; c.onocesntU:y J:o:r the .follo'1ing investtgacion Lh~ ~ ~h" e>o'•e r·
SJ>eo tr·um octuall.y he rectangule.l' in ~he bw;d 0 ~ ,; '- v0
fo r::s
1 J• .u
3 . CAllP. lEH 'rJUl!iSMISsro11
a:3 shown . Fie; . 65b to :1 show~ b~n shift of 'Lhree such si gl1Bls iut.o ndj;.ic~nt; bantlti by mAan:- of
i::-ion cf carriers ~, 1.th frequencios v
1
01
awpll't nde u:od1tla'J , +2\1 0 .nnrl v c+'+v
~and -filLers i:avir:g; "Gransmissicr. t'onctic.1ut> , <1S
0
•
shown b:f
the dctslle<l l l r:e~ , :;u:ppress t ile lower sideb=d s . Fig . S5c
shows the sua of tbe uppnr sirleband:o . The osci I .Lations in
G11e hatcJ1r:O f1"equency aL·er:s a:-e partly ntl.enuated and their
µbase si:ift :Joes riot vnri; 1 in early •,, ·ith ~rBquenc,y . This
c~n1ses si~;nfil d.:.stortion:; .
;\t -Che J. ec.ei\:er , che signals
1
ere separaeed t;y crvJ<lpaH fil tr.rn , and udui ~iO!l<ll diSWI'~ion$
a r·e intrnclaced ( ?ig . 65f-h ) . 'l'he po 1..: ez· specLra of
~hjl nemoclulatect sign<J.ls :::Lows .Fii;.o~i t o !:": . The hatched
areas i.nC.lcat..e whe re osclllac-ion.s ar~ im;.iropel-·ly a.t tenu-
al
rt
''ii :110
+----[Tl----...., ,
* ____ ._fTI,,."'.,.
0
l"i;
j
vc-Zvo
L
c
___ __,_ ] /
-f-- -------..
7
~ L ___ _ _ ,m
__,_,''~---·,,,r_--'-rn'--.._J-....-_• L_ _____
rL__
g _I __
"L
t
,,
"'- -"-"-'-"'t....,-~.-1-.~,-g
,_,.'~-·
,:....,.3""'<1
0CJ:::::r_ \..•
.I h l , k
.._.-:.,.~
~~
o 2~0
~-;-,"T""
v..,-V.
..
~,.~.-. ~~G.,.~
~
..,
g
L
h
I_ ___
d
l
-1--------.....
f$a»J
,________
,, ... '~L~"I Z
~~~~~·~·l\u
I
_,_1_,1~~LL.1-
/1--,-.&<~m;i__
....
I
Yf: ..7~
__Jl.....!.,_
1 .L
l
1§1$f -,3--~I
Ii>..~~! J I _
v, ·1·l·11 'i*~"o "c:'61JO
6£W>J ..LI_ _
kti1.1
.J -·'-~-~ z,., ~'ii €·~ p,.,.o
Fi5 . f.5 (lC"i't) Po·...1 r:r, r.:pcct:-a for tno :noctuJ.ntion ru:lcl d~mo
rlu Lation oi' t l1reo sii;,.,a.s oy singlo sideb(Uld r:iodulat:i.on .
E.!lndwidth ol' tti_.=. slc;nals ls ? v 0 ; lowest frequeucy 0£ the
sigr.i:t l :-.: is 0 .
J:i,t; . oo trii;t;i;;.) l'o•,,•01" ,;pectra for ;;he OJociuJ a~ion and delll<>Liul"l.ion o~ tlu·ee si;;nals by transpose<! sideband modul"':"
rion . bandwidth of the signals is 2v 0 ; lowest frequency
of the sie;nal!i i!; 2V0 •
• 4 ~'TllODS OF SSl'l
3 1
ated and pha"e ehl!"ted , a::d thus cause Hi..:u1sl diatortionn .
There ere :;·"o ""-Y>l to keep ;;nc d:s;ortions sr.1!111 . One
~ shap~ tl.e r-ig".....8.ls , so t"ha- ~os:t c! their ene:-g:,-- is .:ocated i.n !":cquency bar:ds , wt.ere the singl~ sidebar.:!. filters cause littl•• distortion . Or c".<' :nny loc:i.te the ,,dges
of i;he si.Jlglc ,,ideband filt.ers fa: away l'rott tte f:equcncy b!llldS which con•~ir:moet of che s"ign~ l ;.nrt·F,;v . 'fh"' l"ic-s t
method is used i n vestigial sidebrind r.iodnlat. i.onl . 'Phis
method iB particularly useful , if digi tnl c..Le11c1la a.L"e t o
be transml.tted by time division v'1roug~1 exist in1< telephony channl'!lll [ 6) . A detu:. ed account o l' LLu; )Jt!tl1od is
given by BE!I'.<h'! ar.d :JAVLX ('/) . -'he second me:;Jio l ic used
in trnnspooed si<lcband :nodulation [8] . :t!I vi·inciple wil~
be diRcuseed •11ith :.·efei.•ence t: o Pie; . €::& . '" .. e :;ig:ials :.~ve
practically all their energy in the fr~qu<> ey hnnd 2 v 0 "
v ;!i '·"o (Fig . l:><?n) . The wic.t!: of tLe e:i:rty bnnd 0 < " < 2v 0
is neicher zet·o DO.l sma-1 co!':!.1=.a.rea co r.r.,.-. bnc.Jwidt.!... 6V =
• 2v 0 of cr.e signal. It is :l<>t importnct thu~ 2" 0 equal:o.
~'II · Ii; is only necessary that i;h,o "<tply bnn<.l 0 < ' < :?v,
be W.l.dor thnn the fi·equency oand , l!. wnic~ <:he s ini;le
aid eband £11 te1·a cat1se distortions .
J.'igs . bbb Lo d show the nhi:'t of three " " ch i;igrwl,; into adjacent J'rc(Jucncy band:; by an:ri1it;:ido modulatjon of
carriers with the I.t·equeucies v ! - 2v 0 1 \le and v c + 2"'Jo ·
Bandp1rns rilte1·s L1'ving the <:rans:tission f;rnct~ons zho'•ill
b:y l:he 085hed 1 ines c.upprcss t!le l ower sidebar.ds ' ref" ~u:r.
C>f the tllreo upper s~:ieb=ds el~ws Fi;; . v.•e . '!'he zi.,-:n...ls
s.re not diutorted , since there is no (lo:lflrgy i11 t.t~ frequency area& , wLere t!.e b<Cld;>ass 1"i l i:e-A di::to1·L. At the
rece!.ver , -:he &ignals 8re separa-eci ':ly tncdpnse ! i l te:-;:; .
Dis'tortionr are introdltced in t11e frequency nreus aho•.,'!1
hatched (Fir . b61'- h) . The power spec trn of tt... de:noautat'!d
signals show Fig . 66l to k . 'Phe non- dintortod power spectra or tli(: demodulated si;yials ru'e i;~;si n loc11ted in ..,,,e
band 2v 0 IE v:; 4 v 0 as in Fig .. 66a . ·rho diato1·ted and fol -
1
V""t lgia.l eideba.nd codulai;ion goes buck to NYQUIST (5 ].
1
0 H -.....1111'1. Ttal\1-"'lll••Ol'I (If
lnfQ1-,'i~l•Oll
le..J - ov.:;:·
0
~
'-
o~,.;.:._lationr:
a1~e
I ocat.ed
ir
th~
uuu.;.ed bonae
:! c:Vo ar::i v ,. h"'o .
fit; · ' i't !:no~. :; u : 1 ;nal \' 1 lt1 tha.._ c;:u.: he
'.J.Y t.rnns11oucd eidt-"tJnnd. :oorlu I 1-.J ~io:l :
TLi-· biu·1ry r:httr:lct1
1. :·rJnsm.tted
+1-1 i·• t-an!'in.it.t~C. ':Jy -.,i..;. signol .
tiin 30n-t/J ;JBS 1 o:::i.:il_atlunn i n i.-Ul J.!1Le1~VDI or duration
r nn1~ si.... 3hor;/T
A.:?. 17 o. ci.l_;:r':i...,tJ."' . !I.
fo lows f1·om
Fi.: . ol L!".r;- t:.P ~n~rr:y :Jf
I "'t .:.:.. c:>ncer:t:.=o.~ed in tho
'Laud {1 -1) s v • f'l' ~ (17+' ) . 'l'ne lowro:' :'r·e~11i:o11cy li-nit
18 cq_11,.1l t.o ~ ..... c. • 11. i ~he- lJUJldw.:ct-h lp A'; = 1 . Tl:~ •11itlth
1•C lhf' '3-J..~t;,- ban:J. () ~ v ~
... dt'r t.
9.!J
l "'I iLCd fJ.'Om
111t>'.-tnz c•· i
'JU •.. c"'
1l
A
i r.. }'if"' .
v 0 is ... L6" ·- e;._n~ i . . hu.s LUch
. It.r
.;;.ffi ~j ,..., !" -1 - 1 C9U be re1
i;t~iu ::tl-t.Ul FtC'.CO!'di~:1·
f:in~L.Lon
.•'i.l~·:;;; . 1-?H uncl
111 by
·leL=-·=t "' :is :i o·...·n in 1"lg . .)b . It is
t.<J
nnt:n1 or"t"c.::t~ wl.cat ~t .~l· ... i511.!;1 ... ._ tl!"C? ou~sidc t!::.-e band
1 . ~'it.!,;S · 7i:: , 1 ~ c:--,d .; !" 1 i()t-. t.hat u ·,.'ror;g eiH-
-!!: 'J ;
/ie; . '/ Dct"c ... io:. f"t lifi'=nl .. tt~ril6. A:
·; .,. i!. y,g - £in :2j.1.r.-&); ou-zpuc
r.'I -a:;e:.:
"Jl
:..1~e ::· ~ ...... t:.o!l d• 1,t-ctoI.'s .!'or
~;.>~ ?• 1
b) , :-i11 3~ ·t(.; \C) , co~ 32n9 (D) ,
.. 1 ..n.:tl
8
,r-.
slr. 32•& ( l:.J , cos ?i•n~ (F) 9nd sin 3'fn8
1 1 i .• . Du.L·ri I nn o:~ t.h1 LL·::t.cen: 1r = 150 1:1a
( :",''t·c·,,
c
r . .:::~ .. n, • . r:owAK :ind R.mm:s 11
...,·· .\__ ,-- - r
j_e:1
>· ) ..
D
E
F
G
F 1 1 • l:·8 .i!'t·~~uenc:v ;.iowc;r spec ~l't< of thC
l'o I I o•..J.it.. i:: I·~ l ses' ticco!"ning to fi'ig. 1 '10d
" : ~·::wnl(O,R] (A), sin 2.,e b)coe .,:
:-'in ·Jn6 (c: 11
C'<•.: 4,9 \e1~ S.1""' ,.,
1
.... , .::::-:;_ t nr. \.f:! ' i
sin BnO ~h), cos )ne
, 1 l . a = t 1, v = rl', -t :: e ~ i ; r 1 ~
WJ... i Li.eu ir· flert-~ for •r = 1 2:> µa .
nal produces vc=-y liri;lc ou.t;puL vol r.rtP'i'' at.
t.=.e
samrlir~t:;
tillle
. 5 Correction of Time Differences in Synchronous Demodulation
31
Consider a rreql.'ency :11.!.nd 1- ni1'ed >1i~nel F(3 )'{2 co~ .. 0 •J .
lt aball -o o sync!.i-011ouo:'..:y <lemoaulatec by mu Lt:iflicacior_
with a l ou"l <:1<rrier >{2 con (0 06+o. ) Which boa the rhni;e
dit!erence
whh refet·ellcc to
a
~!'"
l'ec"ived ca1• 1•ior
'{2 cos Oo9 [ 1} :
?(& )1"2 cos o 9 a1r2 cos (nss .cl) E'(e J[
co,, a -
"o~ (2o 0 a~a
•)
'IO)
Let the signul be !'T.'cquency- st:i1 te<.I Ly an nuxil l ory
l ocal carrier '{2 cos (n.a >o.., ) 1md t li e r, \JP <le:codula~o~ :i:mchronously by tlie locol c'u'L"i r :cos [ <n 0 -!l, )a 10., J:
·r2
( [F{e 1'{2 cos 0 0 aJf2 cos r..9 •o.,. }2 cos [ cn 0 -a. >a+a, :
( n1)
F(e ) (cos a +coe(2( n 0 - n. ) d-o. 0 J •cos(;:>"~ •" 1 o., 1- cos(20 0 9 •a •J
O. • a.h .:.
a., ,
CL 0
•
O.n - as .
Equations {110) tmd ("", ~untidn ;he <le: 1 rr.d si~:nsl He)
nul tiplied by cos a and hit·• rreque::cy tc=.s ...r.ic:h can
be suppressed by l'ilters . hrre ~e " nu.m'ter of :r.ecnod."
t or tbe rernovn.1 o!" co s a . One :nay derive , e . g . , r. . ino
oscillation \1'2 ain ( n 0 a +a) r r'O:o Lhe I ocul codne ca 1'l'i<'r
V'2 co s (0 0 0+<>. ) . I'.ultiplicntJ. •L of tJie =·tc~'vcc s:i1.n~I bj'
this sine oocillation yie:ds:
F(e N'2 cos 0 0 6\1'2 sl..ll { 0 0 9+a '•"' (a J~ sin a + "iu (2n 0 6+a. ! ]
•
4
LeL ua assume F(B) may l>o written as cun. F(0 )=1+M.F ' e )
Whore F 1 ( a ) ie n sii:;nal thnt contains 11i·nc~ica.ll y no c•uerl!Y bi,low a certain frequency arni :1 io the 1•odul<tti " index . The right :and side of \4-2 asmneo the for:t.:
oina. • Nli' ' (e) ein CL + (1
+ rJ" ( S )}
s i r (2;i 0 0•CL )
(113)
'fhe second and thi rd tet'm c:an be BUPIJl'OGGr.><l by u f1~e
quoncy lowpaas fili;er . Tho terlll sin a rc.1>11ins . H ••'•Y be:
uaea.
..Lil
a
feedbncit loop
to shift
Ui
1 ocal
cnr:-ier
\1'2cos(Oa6+<>.) and thus \(2ain(n 0 9-a ) ,,n :::ucn" wny, th&t
eln a. vanishes . a then equa.l.a ~era or 'ln integer mu.Hit,le
10·
of rr imc coe a. equnl s +1 or - 1 . Let t:r.e f<'ecl1:Dc ..c loop be
st.nble .for a. • C, Z?n, : 11rr,
anJ lnst ab le for a.
:rr
'
1;1 t . . . rae \'n:ue~ cot:a. = - 1 nre 1.t:en ur.!1't::sb 1c . .Fig .59
.l:lhO'.·;s- f.1 bl ocl\ dlW.f3'l'El.!!l (')f a J·c ce:.vnt\ tC-.rd.. c or1·ect a the
fh&Se ~if !ercnce in t:.is wz;.;; . Cl :_.::; !l:SSU:3Cd to b~ zero ex-ce~t i.ll t ile feed"oack 1 oor. , wi..&!'e vai.ues :ioldi.ne; for Cl - 0
nre sho.,..·n . A very detni leU. treatoent. o.: .-y::c~roLous d~ .;od1;.lstion of ninus,,1dal c•trriera i s 1'1vc :i 1.>y '/l 'l'lmBI (2) .
Conr i.je1· t be correctior.: 01'
A
rii:.!! U2.f!'r·""e ncc , it Walsh
cat"rie,..sare, .. !'"ed . 'Ihe z1ft.tl !" .. \ 9 )w>.Al( j 1 j)
o=-
~15) shall
mulLi;•licati.on Nitli Lr.•" loc!ll carrier
w!l'~j,1-Bvl . rhe carri~r~ WD fj , ~
"W:l ,,al(~ , - 9 v ) ire
P":-iod l" f aucr.ion., nnd do not vcmi slt out<"id<' Lil" i nbe1·va1
-~ ,; 1 a 1. Thr de:r.oc\ulotrd slgnal hne tLo ."ollo·,;ir,g form :
b '' d.ecnodul"L "'l
by
r,,.(" )'1101 (j , s )w<L:(j ,o-e. J
(44)
'!'!..e prc:.luct. of \>;a_I {0: , 9
9.Ild "tlal{j , :! ) i
1'..r:o~:r , bu- not
t:l1at or Nnl ( ,1,6 ) i:Uld 'tH.<l ( j , :J-9..,) . Tho _problem ls similar
t
that of ou~ti;>ly1r..r 1/'2. cos :"l0 ~ "'i~h ·,t2 co~ (Oa9~ in( tiO) . 'Illis cul t iJJlicai;ior~ cannot Ue per.formed Hi th the
:uult i i; l icntion theorcir.G (?) alone , one need,; ill ~drli.,;on
-ur shift \:heorer.1:- o~ "O.i:.e and cor:i11e fu1:. ction~ :
"o~('.1.-al :
~o~ a. con ~ + ninu uin
s
"tc .
(45)
be tle~o:npor;ed ':ry tbl.~ shlft theo1·om ,
!'1H<l t hP :nul tiol i c'lLion l '·.~orcme ~ 7 t!l'lY rnen_ On 1lpplicd.
•[2 co:;(0 0 &+a ,
!T:U ~t
'· . . . _ti;tlication ur:d i:?hi!"t tr.cor~1~ a:-e ~dS<?lit.ia.lly the sime
fnr •in o and co si ne .funct;iom1, s .i.nco ( 7 ) ore multi:;ilicetior: t:ieoret:Jfl' if rPati frO?ll :_ej't to right l!1d :-bift t ~0 l"CmG if read f1•on. r11{ht t o l".ft . W
ata h ftwctiono ltave ve1',V ziuf-" binnry ~hi!'i; ~ .• eore:n~ (1 . >,QJ ,
i<al(j , e•l>0 vJ • wal(,i , ~)i<al(j , B v ) ,
bu~
,i.11,
coot:>1ic.s -he 01'<iinary subtractioi: sigl'
end
col;
:nodulo ?. «<lditior. or subtract ion a i g n .
Cert.tain sr ecia.. cascn o~ tbe ehiJ·r theore;w of Wn ~sh
fur.ctior.,. "'ny be d"ri ved :readily. Fig . ? <lhowa ~hat the
:<
l 1•1·io dicall;v coocinuod
f u.nc!,lon" snl(1,S ) and
ca1(1 , 8 )
- 1"' coR.REC"10:1 01' ::II-3
,.
/
DIFFEREH~ES
149
axe tr(.Uls!or:u~d iz:.to cue:: othe:- by a shift of ±i o!· -::i !
<o unnomalizcd ooLuLou ; t!:e s:...:.r:; cq'1u!s tA f"- ,al 2 , ~)
~d cal (2 , a) , t ! for :;~l\~ , a; a."1<1 cnl(3,S) , c~c . ~e! i be
a power of 2; tile ro:lowiog gc~eral ;'omula. hold~ :
cal(2',9-2"'"' ) • sal(2', n) : ~: = o , ' , 2 , . .
consider· Ll1c er.ore ger.,...ral ·:!ns:e h:J Ldlug 1'o.t·
ger value of i :
cal(i 1 6.;9 0 )
onl(i , 0 )
•
(11&1
~-U./.'
inte (47)
Table 9 s hows vulu~a of 0 0 for i - 1 , ... , 3'"' . 0, i.; detertninod by ( 11 c,) for i = 2 ' . 'C'1<>so v1>-':los ui·e in1.n·kea b y
a star in Tuble Q , Onr ma-y t;e;=. th~t 9 0 • t fJL .l : 'l: iF
equal 9 0 • - t for i = 1 'Dit!_ tl..e .'3i1~n """"V~r~~d . 60 i'or
:. • ~ may thus be c a !ll!d the • ~~i:e" of o 0 for i = 1 ·~itlt
re!'erence to line i = 2 = 2 1 • C.•ne :r.a.y rr.-1'.j,dily sec t;:iat
30 !or i • 5 , G, 7 in the ir:a:gc of B0 _f..,r - • 3, 2, 1 ~i~h
reference to line i a: L ::J 2 2 . '!:hi7 1..-·,; o!" i:r.n;es :r.a.v be
~'l'it ten as J:ollo«s :
cal(2'tj , a+90
)
•
c(ll(2 • +j , 8 )
oal( 2'-j,9-9 0 ) • sul(? '-j ,9 )
k = 1, 2 , .. ; j • 1 , 2 , . . . . " 2 11 -1 .
Equat ions (46) wid ('18) are the •reci· I Lld.n u ..,oi·e:u of
tne W(!lsb tun ct.ions . It co1·respond1? to t 'J"' !'' l 11..i " r~ill _x
= cos (x-;n ) for nine ~JlJ cosi:..e fwicLio11 . .
'l'he following rr.lat;ior.s llolJ fol' i!.· :;fl} .!'.wet ~oar l:l••ead or (40 ) an<! (1.8) :
: ..... )
sn1(2•, e - 2· • ·') • - c&.1(2 ', s)
oal (2••j,9+9 0 )
•
-cul(2 '~j , e)
- cal(2 ' - j , g)
I:. =- 1 , 2 ,~ .. j j-= 1, 2 , ... , 2 •-1 .
Equa tione (48) ~ o (50) yiald :
Cal(i, a r9 0 ) • - cttl (i , 9-9 0 )
aa:C(i, 9+9o) • - onl ( i , d- 9 0 ) ,
(
-
;; .
1'
,
Cl.Hit] .·:h TRA~lS~:1ss101;
T11t1te () . Somr V't. ' lt0$ of Uo and S1 ror t:·.a r:pr·<'! tal nhitt
of '"'i:€' pe rio l iC ':/ rt~~h flUl<.:: ... iCL;
c·1l(1 1 9) and
s111(l , e •.
~ hCOl'e!l:
-.
u~c .
i
t;.ina .i.~1~·
..
Bo
.
-
OOOC01 • -1
OOOC10 - 1/::
11/i;
' ccoo1·1
•• 0001~·() · - 1/1,
OOC10~
-1 /L
(
000110 I I/&
"( C00111 I ,1/1
I';
:·;;·
OC100C
- 1 ...
001C()
J
1
001C1C. - 1
11
001011
11/~
12
001100 +1/1 G
-1 /~
1Jjf1101
11
00~11(
t1.B
1';.
C01'1'1 11 /4
1
C1CCQO • _, /0·1
1
.
~
.-
"
9I
dee .
--,,
-1
-1 "
-1 /:'
"
-1 /t
-1/:'.
-1 /1.
-·1 1<
- 1/1·
- 11;:
-1 11•
-1/2
I
-1 /d
-1/?
-1 IL.
-1
- 11 ·-o
-
1&
1 '
20
?1
22
-~
~
.1
:=·;.!_
;>•
2.(
?7
2<'
2')
~·)
;1
:·2
1J.1.11"ryl
0100C1
C1C01C
0%C'I 1
0101co
010101
0 1 011C
01011.-:
0110C-O
(;1'001
c 11o~c
01 1011
01·11 00
01'1() 1
0·111 ~c
01111'
eo
91
-1,<1
- 1. '!
- i/2
- 1/11
- 1/;_
- 1/li
- 1 /2
- i;1.
-1 /1
-1 /il
... 1/~
-1 /'•
~1/'
-1/?
- "i/1G
- 1/'"1
- 1 /q
-1 /2
- 1/E
- 1r
- 1/4
11/~2
-1/·•
- 1/"
• "i /l.
,1;1.
-'I/'•
,1/b
·1/11
1ti00CO · -1112e
-1/2
- 1/64
Of'
c•tl(i ,6"~ 1
3
- col(i , :IJ ,
sul ( :.: ~ a ..
e,)
-rnHl , a )
(?2)
- "'e,1 .
i/·-ilU,.,~,
1
vn. ..
o"" 9 1 UJ.'IJ !1!~01•r-" 1.H Tnble <) .
L 1.vo:..ild Ue 1.:umb""r:-.01ne to obtn l n tla lil.ld
~es
cf i
a.u1.:. f9~1:':'!.
l:y w1 t:):tAnric.a o!
ty
-..:rit..-=-1
1
~qi
n.
for l lll·go
e J . Or:r· c nin Ob\;ain 81
as ·c·l.t:ti..l.'V nUJJber .
e,
~quals -;,
lf t.l.c lO\\'c:Jt b .. ne..r;$ d 1e-ii; is a 1 . A.u lnt!pec t ion. of Tn<;:+ reailll~l nllch'!:i !>l1n1
· 8 1 i s -i .fo1 all odd v~luec oJ'
i . 8, is -~ , i.!' ~hQ lOWe-t bin:;P''.Y .!ip;it is · C lft.lld the SC-
1.:l~
COr.U _o-.ie:;t. D 1 . fie-nt.~:·nlly- !:.O .. C.r: 9 1 equalr
- ?· lc·l 1
i f tnc
k lo"""" bi1.u1·y 1ii;it~ •i·e =cro .
'rho a1Jnu lH Lfl v:i lue ot €1 0 i :J dot." i vetl i.u the same wny
frOJl the Mn:Jl'Y rep:-r·"~~tc,t ion or J, .
I 0ol equals 2··-~ if
tto k lc>:P.~L t-inar:; ~fH!' '9.re =ero . a0 equ:lls - f90 f, if
1hc d.t;j< ' · . " i s CJ ;
Cle ·'~'..lals +IBo l i:f th 1· ligit k • <'
ic 1 . Co11 s i1fo1· o::; exu "ll•lF t he mcubors i n 20 nnd i
28
ir1 'P•-ible g . r .•e two lowE·: t. bin~ry die: its (.t: • 2) are zex·o i
thi£ yields le 1 1 = 2" 2 ' • ~/16 . Tl:«• tou=-eh bir.ary d-1 it
(k_.2 = ~) i:; 0 !'or l r 20 1:tci .,~ eq 1;;~l.
1/1• ; tor :_ = 28
the fourth. tllgil ... :i "i ..!11C. r. 0 ~qu~l!i 1--1/'& • ~ proa:· ,..r -h.-•
ruJ.ct: ro1· de:.erc1nn:;io:l o.: 1) 0 ?J.Hd '3 1 tia given Ly ?I...:Ji-
LER [}] .
A ci!•culc £<>1:
t
e
corrcc~ion :;~
n t.i:ue ni.!"fcrcnco b1-· -
t;ween received C.al:'l"iJ?!' ~nd locnJ L'.at·1·lt!r Wt.'i.'( v~ bu.<.?ed ut:.
Ghe npecial ell lft 1; <'Or<>a: o f ;.1a1"" !uw.:t ior.s ( l"if· . U:) .
Let u u nuaun:<· Lb'l ;;ii:.n~- [ 1 +M3' " '9}jcal(i , ~) in l"C'coivo··d .
f lf(0 ) ia H t>.LJ:J;Jlal &nni; n al::i _pa.1:tsr;;d t.11..1.·o u1.t1 !J .ie'i.ilency lowpass .ru~c.t· . A local ct1.rri e r eol(l , ~ - 9.) ii; f l'Odnce<l in
o;he fw1c't.io11 rcn~!'at:or Fu . Tb.e l oc :.1 1 Corri "l" pnsse~ u
vu.rial. c clulay ci[·cui- R'l . I.be c:irri~r 1.~nl{t 1 1)i~ol:tai
ned ut the outp
1~
of ?.1,t once t he cit·cu.i t ls
!.oc~ed
or:. to
t.he received ccrt•it:-:· . lt. fu::-~i-~rr d'"' .. ay ...:ircul.t. ·.,•ir..h fixed
de~ay produc"s Le C'il'rier cal ( o, 3- •. ·;, • £"1 (: . 3-9v ) .
lhe received Sil'.O!<l. i~ l>Uli:iplied ty sa!(i , 6-l,) end r. c
product is int -:rAted C1~r:.ng tt~e ort!'.oi..ohality in-FcrvP._
-i + 9v ~ e .ii ~ + ~v of ::Sal( i, ~ - 3"' .1 • TL0: ou1.11ut vol r.nge of
t.he integrator
i!1 erin:plecl al. t;Ilf! ~iID• ':'" ~ ; -9\. , ~ +- 3v ,
f + Gv, ••• U;v Che .sample.:· AT :) tciLl ir l'od t;o rut r·•.rc.u: ar.:.11r,
circuit Tl' . This cir·culL aveL'a!{e'' over mr.Lll:V ... our-~<l or.1plitudan . The .1'ollot\'i1ij.i; ci.veroi;e i ~ o·otninr:d •1t t~1c ou l put:
oI 1'P d u e ~o Lllo f><C& l;hat C.l\e ' 11~r;gr·11r.0 1· J '1vrt·nu:e: ov1:<r
the in1.erv111 r. -tk + av 'i e " ~" • a, :
Pi~. r9
Co:t'reccio11 ol' a t. ime dir re.t·enco ":.;ct. 11)<Jnn received
"" local corrl.c" cal(i , d) «Dd c<il(i , d+9 v ; i " """''~r o r
~:i. ,11 ~1ultipl.ier , l'G f unccion genel"n t or , RV vu t•itil.lle deH.,y
.l'CUJ.t: , D fixed tlelay by 9 0 ., I illter;rator , A1! 1 Am 1'll cud.c
6~8.IQ}:t Ar ,
'e~db11c1<
l 1P
~Ve.rage r· .
loop .
9 \, is
pUt
eq_LtaJ. 0 , t!'XCC-f•ti
·
th
- / 11?
7.
1,2
;J .
,, (!l.~I
-
-
\HIM/
I
o
~111.~1 -
-
-
-1il~
#
Vv-v'H
VV'JIA,J
C1Hti DI
lv~Wo{
- --
c:11fS,lf) - - - O{V\,./Y.t
"""" - -- .JWvvv
<OllL})) -
c ~.o/J)
-
--
~lt3,0I
-
--
"'·~
V'V'J'.I•
.;'V'l"\~
v-Jv.rt•
'Vlf.J'iojJ
~f
\1..1yvJ .J!.M/\-
~
"'{vJv
'
\tvy.-J
vl..At
~
.........,,. ...,.,.,,...,
.,,,.,,,.._ ...,,._,,..,,
ra10;tl - -
'"1H2,-0J
....,.,.......
:1.tf
~illfol --
oloC!ll'1,&] -
"'f'i'.,,./"""f>
--
!i:tf.;.,e1 - - -
cc
TRAlfSC'J!SS!O!i
W*IW./
(flll7)J1 -
wt£_~,
w..R!l.I.m
F,.,AiJ:.!
,,.,.,.;-
--
,,......../•v./J
W'"V/"' </"'v.A
..~rtel
.,.__,_,
----
..--.,.
mi:t,\l:
'®
.Jv+
'VF~w ~
--
~'°.~'
--
e~U6-,(iit
~.:II ~jlll
<);d,&l
"-"'""'
~
~$~5))1
~~!j);
]'ig . '?O r:orrelotion funct io ns l 'oi-· pe1·iodic \.,'alst f\ULCtions .
(C1
'M~""(e)Jcei(.L,G)s,;l (i ,a-J.))
Le:: 11s "ESU!nt;.· t!:at tlte nv·erage of tl.e sc-conc tern:.,
(MF"(e )<,,>l(i ,S ) s,,1(1 ,e-a, J),
('A-)
increa ...~e:..; :r.or'2' sluw~ y with increa;;in!'.': svcragine; t!.oe than
the avnt'af;e
(;>5 )
(cqJ(i , e);;al(i ,B-9.,J)
n f e:i~ ·'ir" t r.ero. . ':'1e ~e::-e ( '.>?) uo:nL'lates then in "he
012t;>ut voltail,'e o~ ;!Jo: av":r-ager . :t 1:1HJ" be u;;ed to .sl 1i't
~ he locnl c!l!ri or cal(i ,9 - 9v ) n.nd thus oal(i ,e - e,), so
",;)
~hnt ( / ) and {",a) ".'ar.isl1 . :'l-.e vslue" o;' e. fo = wllich
,
w1d ()4) voni Bl1 1.4'0 obtainod f rom ~J1 e fol towing i ntegral :
(cal (::. , ~ )sal \ i , e-e.
1/1
)) = Jcnl(i ,a) sal (i , e-a. )dB • r;,,.;\e..J t,6)
.,,,
!'i>,. 70 shows so:ne f=~cion s F,,.~,(9 v ) Wld ?, 1,.1( 9.) in the
miin dl."i':"""l. Pc1,,.(e v) i.a show11 ji~st b<'lO" i;hem11in dio;;;on~l ruiJ ~; ••, (Av) just nbove . The in~ervru. 0 l! 9 v l! 1 is
; .15 COJiREt;1'ION Ot 'l'll'!E DITFER3JWJ::S
--
&Ol($,6j
Mv..y./"'
--
~
- - - C(lll'J,01
--..,.,., .,.,-.._ - - - 'IUll7.Q!
1
- - - m 1 1iDI
--
a.ollfi&l
""""""' '"""""" - -
" ' 1\.9!
- - - '.lll lt.,01
~ ~ - - 1•l lJ,GI
~ ~ - - - lolliJ,11
--••lfl
- - - 'll':'.11
- - - <!llllJJI
lll:ll!'QI
ohOWUi --be Iunctior.s
have
to
be
continued
i.,erio~ical:y
out 1dethisi.nte!'Y6 . Fu.s•(ev)is:"hown er..largedi:if-£ . ,·1 .
The do shed line.< r.i1all i:;i ve some ir night i11to tho~r "~i·uc
bu.-r;o; a mor~e detai tc<1 diccussion of the coi~relat:ion £unct i ons of Walsh funetious <-IOUld land ~oo ciee;Ly into llilstr1.1ct :nathe:netice . One n.•Y see , !:o'.·1ev .. r , from !:'ig . 71 u,,,-:;
F'c1>Aev ) ''"anishes!or i-= 1 1 2, L l 8, .. . , 2<r , if &v ttqualr..
.ero or an i.:otege1· multiple o:· ,,z~.
,,1 ;2 ~ . r1.:s re~u_t
ma;y nlso be obtained from ( "6 1 and \ '>2 J . ;·i:e feeJbac .< loop
in k'ig . 69 may bo mnde stable £01• 6v
0 , ±2/2i , ±1./2!. , ..
and unstable for Ov • ;:1/2i , ,,,3;21 , .. ..
Consider the w.~lsh l"unction!'lo:· ;.•ig . ? conlinuE>d J f'rlo-
-
dica.lly to the left and right . A ohift. of sal( i, a), 1.2 • ,
by e, : O, :=2/2i, z4/2i, . . . yields again ~he fl!r-ioaic
function sal(i,S) . Thlni:;s are more coa.pLO.caleJ i!' i i;o
not ' power or 2 . !•',;.,Ca.) vanishes ror certai~ vn l u<n av
• e ~,
but sal(i , e-e ~) is in general not l tlentlcul wHll
Gal(l, 9) . tlence, thi:> functions cal(2' , 0) and sal( 2 ' , e) nrc
tl1c most suitable for synchronizution . This r•esuH has
; . t:A.R.Rlrn
TRA~rs~:rssror.
Firr . /1 Cros!'COl'l'e .Rt ion £unction.- ~d,si ( 0, J cf some Walsb
!'u.nc tior..s .
nlready been -"~ 1 ln tt.c rtit1cu.ssio~ or the •elephone au1tlp1 -·x z-y~t:~:r.c!· ... 1.T . 1 . :.. ~.f!:tlsh runcLion t:-acking--fil~er
llCC" r·:ling 1:0 Fi!; . ~ 0 lin:: t Ct"!: C:evol ored 'oy LUK;; Hnd r.AILE
t)f JJ..:- -e: e.:u.nkt,;O Jor i>~CJJ ;i cultiplcx :;y91,e:n .
to t,cn0rul l.~e , improve or
.l:l i:nr1l:l f..Y L::in <ii~cuu . .·od mel:l..od for tJJe coJ ·rection o.f time
dlf£<>!'Onces . E'or im1Lnuco , LLe signal lcnl\r , B) +l1Ftt(e)Jx
c1.1 i,& ) lll1;.y bo tni.n~uW.Ltod inst e ad of [1+1·tF•(e)Jcal(2k,e)
i! r~ '""qu.:L.c: ~ fO\o,.f;!t' of 2 . ':'lie ch.reP L ocjg I, ])!] and 1!P
ln Fce; . ··9 ""Yb<> co:sbl:.ed ir.to one . T~.e feedback voltage
.n a rn Fi;; . "9 n.'l~ (col(i , 6Jsal(i,9- 9v)) in Fig . 69 oey
t' ft:d i.ct:o Tit:e oscl.llotor or ~·unctj on generntor rather
~!1ru. _nto a pbaee ohifter PS or deh:v circui r; RV .
-!ie.i·c- n:re
CJ
nu:t.:;.tf't'
~ ways
~ . 21
'"'" B'h~,. E L'··o-"ll
' · •-o....
TII·...:.
..... "
3.2 Time Base, Time Position and Code Modulation
3.21 Time Base Modulation (TBM)
Any cart·ler :.;an
\'l,..
aJnJ,J:itul(! modulnt<'..i ... f
it c1..in h<'
wri tt:enas l;imc func~.1.un ·.,.·~{r.:,01 8 0 ) = V~·(?. , t;/'l+t; 0 /T) . One
wil- expec't t.hat t;.l"e"' 11101 i= ..:. r:divldual :r.ndiJ.luliio11 J.el.r100~
cun be defined , s.1..nce t . ..iis cnrri•
coni;n...:..ns :;J c !lO!'IJ&' 1. r:<l
1 .:.·
sequency k , tr.c Li "
t"~" '1'
nM t..1~ d"lny t. 0 bce _J~•
t:.c
wl\plitudc V. C".od Ltle. ~ >O ll of '1' in c"l l cd u Lime oast> i.~dJ
lation .
Tue be sic ioee. is to replace • l·.Y a fm:c .ion i;( ~) . 1'ncre
are severt1l ~..·aye t:o r.lo f;ni~ .. ::.,, t lt'\3J ·oe 'the m.cdulnt.lr.g
signal a11d M a mod<JloLion i nJex . One mi.y 11 nc the dnf.i.nition:
~(k,e) -
( :;7)
~ [k , g'9' l
g(S) = J[1tMF(S)]d~ = ~;[11~lF(t./TJ:dt
(~8)
This is ~he appi-oacl. take•: ln f:·eq;icncy tLOllt:l.'lt l on or
sinusoidal car.:·ier!"! . ho-,;e'V~l , the aC.var..t.u~t•. of t.tiis !lp-
proach are ntrongly connec~e:~ to tne t::.ct l.?.a:: :...ro:1~·1cr.cy
o.nd -cime a.re combinl!d a.ti product. ,
(59)
O(k,8) = Gin k9,
for sinusoidal f\t.."lctioos . The co1m.a Cet."1ee:-. k n!IC 1.3 in tl:e
ganc-ral case cake:; the fo l'-o"'ini; de.f i.ni ti or. of ;:;( 9) :to<·e
advantegeour. :
g( 8)
=
il!FCeJI<
8 [ 1 +i'l.F( 9) ] - t 1 +MF( & )
•
1
'!'he modulated parai:iet.er· is no.i clearly tho ~in:e ·""~' I .
Fig.72 allows , how " sine func tion and · 1 \falsh .rw1crl.ou
are changed i£ the ti:no base 'l' is cnaiweo i::to '·f/~ <u.d
!l/2 .
The modulation index I·\ :nay Le positive
Larger vnluea of P( O) reduce the l;ime blloo
increase the time bsse for r. < 0 . This i s 111
o frequency modulation, where an increased
or '"'gnLiv~ .
for I'! > o ano
close 'UlOlogy
voltage ot tl'.e
, . c;..PJlIER ·1•1<1..:1s:·1rss1c*
·111
"' -- --- -- ----~=
• 1-1"!===!--l
e_
p
o
d ..1(],8'~
-111
o
s-e:(~r.
9
_
1n
;7
•.,
o 8. _ 1, 2
1
Ti.r.c!'
ba~~
c
d
• .-'-'---.;..;_;...-'' ----'-.:.....;'----'-;..;..;.:..:..::.:..:.:
-~
?ie . 72 { le:'t
b
modulatior. of :. Dine
~nd
~
\ia.l sb
i'Gnction .
Fi g . '/3: (t•igh4oo I B.ock diagratl tor t_rie ~P$e :s.o1ulntion of
/.'el:;h carl"ie!'"!"" . A. a:Dj l it.ude !;ll)tiplr-r , l :.:itl3gL·ncor, SV
vol -zac;e compn1·ator·, SP ::;torugr, Z counte.r • fo'G f'uncti.on
gen~:-a~o.1: .
n:od:J atir.1_.1 :Jl L:J:ial :nny i.11c:-easc o.r ¢er.;.rcase t:·1e frequency
of the c"lt~_riC't' .
r··1g . 72 snows tr.e.t tlll'.! required baudwid~!1 increases with
t:hi:>: ~odu~Y.Lion in.lex .1 . Ibo st:or~e...;t ti.a.e bu.!:e aho·,..,n is
l1alf as wide as the longo.ot . 'l'he .frequ.,ncy or eequeucy
bantl~idt.- occupied :.\y tne sllort functior.s sin 2 :18" or
!Jal(, , a" } is ti·;ico as tn.rn;e as Lh::lt occu1-1 i ed by the long
fur ... \.~or. sin 2-19 o:· sal(~ 1 5l . A netniled nnalysis of
enei•gy H'1 t ributi1Jn a3 func.:Lion of sequency for various
v3lllt!! , f H aLd n1gnels F(GJ i~ still lucking .
;. p:rnslb l e circuit foz- Lill:<:> base modulni;ion of Wnlsb.
fur1ct~on . i!i s~owu ir Fig . 73 . ~et the signA.l llni/e &he
slnpe shown oy thP firr-t line of ch~ pu-se d.u1gra&1 . I t :a
M.:i;pl ~ci •t ti:n~ 6 = O by Lhe lunplivude sarupler AT (a) i
tr.~ ca:i:plec vol;:;ar;e in stored
(b) . An in Lei;;rator
r
it:
producc>1:i "
the holding circuit SP
L'llll' P vol oage . A voltage
3 . 22 TVIE POSr'l'IOrl t".OUUL:,rro1;
1'.'/l
coc>pniator SV CO!lpnr~s thi" racir vol toc;e with th• one h<'ld
in SP and resct3 :nt.e6.:.·ato= l •,.,,·fie!1 both ve l tn~cn bcco:nc
equal · ;.. nnwtoott voltage lc1 .re;.ult:: . The nmpi.itudc G.::.d
duration of tho =:f!wteet!J is _propo_
r t;ional to the Y01 tngc
storca in SP ·
The puloes (d) 1'1'o:n the co"'f·fil·aLoi· S'i wl!ich re!le;; ir-tegratoi' I IH'e slno fed into the coULLe1· 7, . /.. pulnc (cj
is geni:>ra1;nd l>:Y t'. if A. certain c.1.JJt1beL' o! pul:1e:'l ha.n Occ11
r ecelved from SV ; c;hi!l nU!C:be I' i s S l r. I•'lf; . '/ £ . 'r'he pulse
(e) el earo SP and Gt orcr. a nei< anr l Hude sample of the
signal vi.a ti"Ullplor A'r . Hate that the J i st<wce lJetween sanpl ine; point; a dep<'ndn on i:he stw!pled a;aplHu:le . Counter Z
is reset, wben t~.c puls<' (e) is ge:'le<·aLed . 'l'f lr i-enPtti?lg
happens nL i;he rimes 0 , 9, 33 , an<l. "!! . '!1.e Alllpl'cudc of
the signal llt tir.o" € is t..-ice a,; lbi·ge as st tioe 0 . He-:c<? ,
t he sawteeth :nrc-> -,.ice a.s lons ~s be!o.:~e . The 8 pulse~
(d) generated by 1'he voltage COJJfiH'1<tor S'i ill tl:f' t:Cc!"
i ntervo.l e .-:; e ~ 33 }:ave twice i;he di:-tnnce ~8 int.he interval 0 ll e .. 3 . Feeding chePe rul~N• ir.rn ., funcl.'..on
generator for Walsh funct ion,; gonorn.tfla time 1.;;1ee r::od•.llsbed Walsh runcti.on::; at its out;>ut; Fi['. . 7'?1 aLo>«> ~lte mod:.lated carrier ~ol(3 , B ) .
3.22 Time Position Modulation (TPM)
The varinb l e 9 ol' th<' carrier 1:qk , 9 .. e,) "us 1'ec>laced
by a .function g( S) in the c<ese of Lime bi.Se 'J;Odlllacion .
I'he parru:1eter 9 0 l<> replaced by !! .!'un ... tion h(O) ir. •he
case o.r tiJ1e position 11odulation . Le~ F(9) asair. t1.;note
the modulating signal and~ a coduln-10~ index . Tte following definitions nre in"roduced :
J (k,B+9 0
nCa ) •
0
+(k,S•h(S))
)
0
,
(01)
MF(a)
Tile moduJ.nt;ion J.noex M .m.(ly b e 11osit;ivo or nei;ei; i ve .
i[ k,8+9 0 +M.1'(0)] wil l be shifted tow<1.!'dl.'I l<.11·p:er vnluos of
8 fnr luL·gor vnl.ueo of ]'(a) i f l1 is !lep;e.tivo ; the OJJposir,e
holclo if M io pooi i;ivc . 'i'.b.is corresponds to pnoao coauta-
3.
CARRIER TRMrs:nss101;
tlon , ~..1hore the J)hase or t!:e ca:-;-:i er may be adv·a.uced or
reta:«;! ed by a 11-u·i:;er tunplitude o1 r,Jte :>ig.:>al. Fig . /'l shows
a sinusoidal carrie1· t(1 , a) - oin 2179 f OI' ~::a l; ln·ce s.1ifts
MF·(e) ~ 0 , - t and - t . Belo·..; is shoh·r1 the 1:ial sh carrier
i( 3 , e) = sal (3 , ~) ~or the same tm:e" ,,:iifLa . Mote that the
seetion of a funct l or.: •,,r~:i ch projeCt.::i b eyond o:'1e liDics +i
or -~ rlue to
.fU!.1c1.i iOYl .
a
shif1; in adcle<i aL
Llt<> or.hor end of •he
,
.}
-~1
1
Vl
111
J
~]
--,..
-1/l
0
Q·lf l -
V1
I'
~
111
~
1TTTITT
1111111 111
'"TTTl"-
IJI I I I II
9 fht.9 ~- li9a;-8
0
ee
~-
.ii'i e; . 7 11 (l r.·ft/ l'ime ?Osi·r.lo11 n:odt;lation of" a cine and a
'. ·ial.!:i!_ iunct ior: .
r'ig . 75 (L'ir:;Lt) Block Md Lime diav·aru for t:oe tim1> posit lor.: moc:iula-cion of ~o/ul.tJlt car.r iex·.;; . A'l' amplituC.e sarnple-r,
I i.nLeg1·aLOL' , SP !Jto r age , $V voltage compar•al:or·, TG trig;.~encrator\ ~TA gate , U divide.r , FG fLLnction e;enera.tol".: ..
1;:e1·
Fig . 75 sho>.'s a bloek d .iagr""' and a pulse diagram for
time position mod11 lai;ion of Wu.lsb carriers . 'fhe ampliouda
swnple:i: AT Brumles periodicnlly 1'hc araplitude of the i.r>put SiS!lttl ~-~ ~he Lirues O, e, 29 ,
and ~he reoul~ing
voltae;es arc held for a certain ciwe (b) in a hold:ing
1 ,,c-.0.
circt:it S? . AI. integ:r9t-;:r - pro<:1lcr•!l " rar:ip vol t'>g<' ( c) .
A voltage conparato- SY gene::-atca n p1ll~c (r.) n~ ~oon as
the ral:IP v~ltsgc rcncbe5 ~he value of Lhc voltage stored
in s . . '!'hi!· pulse clea.:-es SP and re~e-i;:; 1ntograto::- : .
Posit"vc pu Lsee (b) are ob;;ai...::led at tt_e outpi:t o:· SJ-, '.·:ho;;c
.;uration is propor~ional to cLe a:r.pli tuJe o t' tho simpled
voltage . Tdi;gor pulses (e) mn;y µass fN1:i tbn trigGC,. generator TG throtq:;h flUCe GA1 as lon;:- a~ pul r~ (o) is present (I') .
A dividal' U1 produces vi:i g;ger pul!'le~ (g) fror.1·bhe trigger pulses (o) , Lilat have a much largrr rer·iod . Tl.e;; pass
th.rough go 1;<' GA.? t.o the ftuici:ion gonnrnt.01· FG , ·,1llicb produces poriodic Walsl1 :nncti""" , c . g . , ~nl ( ~ , 9} . TJ,e trigger pul~cs (f) are !<dded tr.rou;;!: t''"'e GA?. to th<' ::rie:Ger
pulses (g) ia:.:Lediately a~tcr the ti:J:e_ O, 8 , ?:;, • .. T!:P
ou~ put o! the functi.on gc:ierator F~ :..!:! u ti:te ror-it:.on
nodulated Welsl: runccion, i.f the periorl nr tr.n p1lr"" (e)
is small corr.pared with tt:at o..:· the ~·ulse:- (gJ . i.tl.f! div.: 0.er U2 producon pulr.es (hi .f=-om ·he µ111 ec~ (g \ tnat Lurn
on the ~amr t ing Cir·culti il at r;J1e tir.iG:"': . , 9, 2®, ...
D&modulntion circuit.::; .ro.c· timi:i bane tind t.i.:ac position
madulate<l >lalsh cu,'t'iers have been dcviced . They :..re ·o aoe<i
on the sun1e 111·lnciJiles used f'or thr inoduJ.,1Ll1w Ci J'cuits ,
but depend sti•ongly on t;hr trans:nl s>liou l ii1k rrvi :a god .
8.23 Code Modulation (CM)
J1odul'ltion of ~..ue nOr<!:s.lize~ sequenc:-,· ,c o:: !J cnrrie:-'"
VH ;r, 9+9 0 ) is called code modulation for Lt.P followir.,;
:!'eason : k di:itingt1i,;hen the i:mction~ o1 " Eyste1:1 , whic!l
!.s evident if a rarticul.g.:- :::y::trom o! ;·1L"lct.ion.. is substi tut<>d .ror t (k ,e ), e . f: . , "nl(l< , S; wiL!: ic • o , ·t, 2 , . ... .
Tlte 128 signals Lnat o:ay be coneLr·uct1'<.l fro« 7 bir:ary
block pulne!l 1"orm such a system with k • 0 , I, ... , 127 .
These Aif.)Onls nro used for nanr,misnion o f' t o•lPphou;y Bir;ne.ls by menuij o.r pulse code modul "~io11 . Il1 i.i :;uggests a
connideration or modulation of Lhe uo1.,nall:i:"d i;cqucncy
an ,. generfllizntion of pulse code modulation .
3 . -.:AH!H SR TRAHS'.':I SS!Oli
CoC.e oodulaeion Of fw:cvior..z S'UCh !.l~ sol i 1 ij) and
cal(i,o) :neans a discontinuoc1s cWi.ngo o r the fur.ct~ons
aince i can aaaw::1"' lr i;ep;er va:ue~ only . Thls .:r: in con-'
1..ra.st to amp I j tude , tl:ne baso end t iue l)O~i t i on !hodula tion , which permit continuous chan~;es . Ho wr>vo:r., the functions sal(u ,e ) twd cnl( µ , a) are d.,Jinn-1 f'<H' i.l : real va-
lues o f u 1<i;;h the excertion of Sal ( 0 ' a ) . Hence ' code l!lOtlulti.t i on rr.ay be cont.lnuo~~ , ac least in th. ory .
Ther-e is .:io es~en:inl t'!il:-e::-ence uctncr-n code o.odula-
Lion and t:i:re tase mOd!.!lr.tion :'.>r si:.ucoi1t.1 f\bctions,
si~ca i a.~d a are conn~ctod as µroduct and no~ separated
by a comICa as ror
sin ts
=
si n
ulsh ll11d
1
·1..
o~her f~wct.ion~ .
;}.t
A Cl00\11 ation of
J-c- ho ld s;
(62 )
l
lllAY bo i n-;;erpi'eted 00
fl l:IOCLtla~ion
or
1/':.' and v:..ce ·vernl:l. .
-::e re are r:any i.;os.slb!e .:noc:::.i_atc-s and dtHtodu.lc:.tors for
code ciorlulation . Uning 1-n"&ef;er vnlt..eu oJ' i:he
nor.:nali~ed
sequenc;; i only , ou~mo:r produce all f..wction~ Hi ,a) and
connect t!ie proper oue though a ~witch. t.o o cotLmon line .
·;·he den.cdulator
n.~y
be ba.:sed on c:rossl!OJ.'.L'el ncio!'.l of the
t'cceived fu..r1ction ri wiLh ~ll possib l e on~s . /~wore ingeni o~s
doo:odulato r f or· W11l11t i°WlCLions :nay uso tne fast WaJ.shFou ·d cr tl'Wloform or ~ec~icL 1 . 2;; av won done by GREEN
WJd col Laboracon5 .
3_3 Nonsinusoidal Electromagnetic Waves
3.31 Radiation of Walsh Waves by a Hertzian Dipole
'Jlhe so:ution or Me.:x-we:l 1 s equ ation:J
:·or- the He:r:·tzian
dipo l e ma:y be wrHtron by !<vector po LeuLJ.nl A ( r , ,,) and a
~cn:.nr ?O-centia.:. cp ( r 1 t:) 1,,•it;hout reference 1..0 un,v :par·biculn.r nystell ol' .:~unct1o:ito ;.11ch :i.s sine o.t· izosine :
A\ r, t
p\ t - r/c
L,l"
_ 1_ ( rp (t-:"/c !
i~l
:;>' r , +- J - ~ !£-;i
1
rp ( t - r/c))
rl
(64)
; .;;1
"•JJIA~ IO~I
nr.
O? WAI.SH \.lb.V3S
r is the Yccto1· .from t he di role to
Ll1e obse:·va.tion
p0i.nt alld , . the di st.one<> . p ( t) ia chc dipole momenti :
p(t ) : Ill t ) s '
d p'•)
-:t
-
P. (t
~
(•) s
CG!>)
is the dipole vceto.t• , which !It.IS r.he di.t'f ction
1
6
o~
Lhe
dipole and i sprapo1•i; i onnl to ic:i length . <J(t) is ·t he vuriable charge o.: r;hP Ji po le ant1 i( t) t?"1c cu..r·1~e nt in L!Je
dipole . lt' is ansUJ..ed a ... us'.Jal , t.~t. s 1"" uo sr.al "- t .... nt
q(t ) and i( L J do rcot dopewl or. s. 'i'he l·et;n1·ded arguri~uLa
t-r/c of p and
P indica~c
th" L ·,,,.., d<!l ay letwcQn a chMtJ;e
or p at the di.pole and a change of A and '1' uL th~ obs.,~·
vation point . c 0 is the dielectric cons"?;:int oI e:r._p~.: space.
Electric and magnc~.Lc force.-" E1 r, t; • =<! H ( r ,' J :t~y be
computed J"roru A and '!' b,y mean• CJ! the rollowi ng f ot'inuln" :
E r , t)
-u.~As~ · · 1
H(r,t )
rot A( r , c)
\lo
-
(66)
s;Md '!' Cr, v
is the mugnotic permoobility ol' empty "fince.
The followicg solutions for· E nnd H holului:; in the >H•ve
zone are obtniLed i"ro1:1 (.:,:;) to (C•.J
4~~1rx[ r x p(t-r/c
E( r , t )
1
H( r , t)
"• =
~
EF' *
U.
1
}=
~ 5) :
4 "~~.J :Ji(:;,-:;,11
\ x.r x s
1
cl i( t-~/c/sx r
4n cri
ot
1
c • -~Cc -j..Jor.,;
• 'x10 • ·o•/•
-•
P(t-r/c)x r
"577 Olu ,
Th.e wave ;i;:.one is dc!.i.ned ar: n .t'et)ion ,
Wht-L't.
(l·l.l)
r is
"du!" -
t:!.eiently" 19.l'ge . A more rostL<lCted defiJ1i hon will be
e;iven below . 'rhe usulil dc!"i.n:i.tion , th"t. ,. must ll e l ;u·e;e
compared with i,,he wavif lcn&tL , H!'"uu..:oes
rent i{t) .
&
.11nusoiC.al
The neru.· ione is dc-!ined an o rce;lon 1 Whf'lre
~ur
i.~
in 11 ..;u.f. ficiently" s:nell. The followir:g .foJ•mul.as fo1• E and H "'"
be derived for th: near zoi:e fl"O (63) to (< >) (5] :
(6J)
Hrr 1 t )
=
P(t - r/c) xr • i(t- r/c } 9xr
4nrl
•1anriw~ T~""llatOnol fnf«IN;t;OI'
'~nrJ
(70)
3 . ~AZ!H rt:R 'l'R!..CISViISSIOll
:he t:ave lt>n~ .:ooy now be d.efined bj" the requirement
.hat E and H oI (t>7) nnd (&a) a:::e a.uch lnrgcr tlan E and
H o:' (1'9) and (IO; . !!:Jo OPJJOSite requir.,rnont defi!les the
1.ear zone . J'fJe follo\-:lng co~di tion8 w'P. 01>1; al ned :
!'
»
W9V'e zone f or E( r tL )
( 71)
wave 7,one fo1.~ H ( r , ~)
(72)
Consider a sinusoical c·1rrent l( t) - 1 cos 2;ft . The
i ntegral equal:;
siu 2n:'t rmd tile diff~:-om;itl - 2nfix
:.in 2nf~ . 1'l:ie cor.uiLlonu for tile •.-ni'I~ "orH> for E and H
boc:on.e identicn1 in ~1 1 Li; cose :
;4J'
r 1 ~ c ' /(2rrf) 2
•
>. 7 /(.?n ) ',
r »c/2r ~ - ~/2n
:t
is np:parenL f<·oo • 67) to (7;1) ~hat a r.inuso i dcl. current i1c) 1d.ll produce a dnusoidal v•1riati.or of E and H
\l,'itil ci..::ue in thn n-&.vc i.o::i::- as well as ir:. the- r..ear zone .
Thi3 is due to tho J.iOC:ulia.r _:"en.turf} or Rinusoidal !"unctior.s 1;0 reoa.ln uinu:ioitla.1 if ir.tcgrrtted or uif!erentiate.i . 'l't.is i snot&ofor or;her J::uictioms i(t) . E ru:d H will
\Jol.h ·, ary in the wuvc 7.one propo1·tion nl co
according
Lo ( 67) and ( 68) . Io the near &one , nowovor , E will be
rro;;ortioIW.l to j'l(L)clt nnd H proportional to i(t) . The
time lependence of E e.nd H is tl:.us a flwctio::; of the disLZL"'lc~ bet:weec dipo c and otservation point or bet.ween
Lr•ansa:i'°"er a::d receiver in engineering ter,.s . One may
Coreeec a:-. "¥Plication of r;his effect to aircraft coll.inior. warning: . J;ote tl:,;t i(t) nous;; hovo a :l:nall di!ferent'l.al quot i cni: 1 iJ t.:lle t;ransition i'rorr. near zone t o wave
7.one i s to be r nr [' 1:oa1 the t r ans:aitte1·.
1'he po•• er nowinf" iL1 Llce wave zone th.t·ough ·the surface
nf a !'pher·e witl:. rndiun r is obtained by int.egrating Poin&1115 ' s vector over t ':c <urface of the sphere :
*
F(r,t) :<t:i£tr , t)xH r, &1dO" b~~l p~t-:/c)
_
Zo (dHt-r/c))' s>
- 6, c l
dt
81
(7?)
•
( 9 9)
163
tntroduc-c;ior. of chc r:ns-currcnt,
yiold~ the radintiou rcnietance II s f
ation power P :
p. (P(r,t>)
=
( </>Ji'
\-ct•
>- ;6~~'
(P(r,t'1
,
Za 1
Rs • P/l rms= 6~cl
1·0>1
,t
L'
the avernge 1·a.di-
':?
(75)
';)
(t •>
)' >
<(<.litit•
(
i' (t , ))
As an ex ample , cor.::iidei· ;;!.._e rndlation c.:'
'hal~h
wnvPi:: .
There are two ca~e:" tuat. !J13vc to be dist'ngui,.hed . Ont:msy recd currencn
i(i ) • I
l
· Tl'l
sal(k , e ' /T)tlL ' , i(t)
,t
I
I .! cal(k , i; ' /T )dt' , /61
• 11:
into the Bertz.ian dipole . E and H will Ll!en vory at a cor ;aJ..n point in t!:e wave .:oce ;>royorti.o:i~l co sal(k,t/':') o~·
cal(k>•/T) according to (67} and (68) . One :nay ..1110 feed
Walsh-shaped currents in~o the dipole:
;l\t )
k
Isal (k ,t/T), i(t) , Ictl(k , .,/'r)
C'i'l)
H will then va:ry proportionally to "ol(k , t/r) 01' cal( k , t/T)
~ certain point in 1;he near "on•: «ccardini:; Lo l7C) . E
and H rill vary proportion9J.l;y to tile di:· !'ered ~ 9tr:l ·..:.,1 ·;,
funct1ons in the .far zone ; a?: iI:Leg:nt1o:J o.r c:.c :-cceive:i.nput voltage will yield Walsh-s!:."'>ed volt.age~ . . hi> ~e
Cond case requires thllt deviatione f1·ot1 tte ide·1l ~; 'P"
or the Walsh function~ tll'e taken in Lo ttccount .
Fig . 76 showE integral !'unct i ons of the Wal s!: fus:ctlons
according to (76) . Table 10 sbowr, rraJr current I, r11- CtlrY.en\. Irms and r,;a;dintion ..resi~tance Rs fo:.· ~ 'SertZlilll
d il'Ole into which currents according to «ie; . 76 o.re .fed .
at
For comparison , thG values holding for sinusoidal cur ·ent.;
nre also shown . It ia assumed thut ~ ho aver age rntliuLeLl
POwDr is the same in all cases . One may see that I, I ,ms
,,.''1.ld R,
are about ehe same for sine and Walsh fwictlo ns of
164
CA<rn IER
3.
~=:::::::::::::::::--- jf..i(l.81
~ ~/ulll.9\
..
~ 1-J 1(1.a>
~ t/ai{!.9)
"'7
Fig. 7(; Integral ::'u::lctions ot
l!:e ·.,'al :.h fWlC!.ior.s .
J1 cal(i , x)dx
Jcal(i , .)
.. 112
J' aal(i , x)dx
/':..<>.. ~ !Jso1(;r,e> J.sul( i , e)
............. "'7
."'
,.......,_..c-,. JfceliJ.Ol
~ +f"'Ml
~
f<>l!Ul
~
f,.r:sm
~
f..1f5,9)
~
J1al(6,8l
TI!A.) JSJiISSIO!i
9
=
C/T
~ftnl(~&)
~
f..1(7.91
~fa11(7,0)
~J..i1451
.i''1"--~o~
eTnbl r· 10 . Pea..t.. cw."rnn'I" I , l.'1s-c:irr1?nt '"'' nnd radiation
rcr:iGtance 3 5 !'o;- ~ Hertz iun dipo_o . l'. 0 ~ 377 0.bn: , c velocHy of light , n leut:,th of the C.i;>ole, - period of the
1.'a•llaLed f1mctior..a (Fig . 76), ? &V~'1't1ge J'r1<liAtcd power .
~'
~.,,,,. "
c'fl
.::;j.n
;:•n ~
Jc.,1(1 , B) ,
f !;a1(2, 0)
nin 2rr:I
J
[cn1 (2 , d ) . . sal('- , B)
Jcc~\ 4 .
0 ) • •
sin 16r.tt
f sal(S, <
6:rc2 I 2
n s1
"'•
Rs
¥n=o . 22;,
i:co .2;
(~ 0 .112
m-0 . 11111
1
-i;:;:rO . 079
16n 2 =156
.J,;0 . 072
6iix5=192
0 . 056
-1-o.O'
•O
o-
.,,_,.,, =631
,f1;;--o .063
~0 . 034
1
256x3=768
~0 . 035
1
:r-0. 020
256r. 1 •2520
1
F' 0 .1'2>-
~~
Din SnB
br.
l "'"
I ci: o.2s9
!:o.c
"
J°sal(' ,3 )
vz·1 ~
..
~n 0 . 1' C)
':.J
""
i.x;:.:=12
'in 2 =39 . <;
16<3=4B
165
equa). sequency . One may further see Lilut I , I •ms 9nd Rs
are exactly eqiw.l for •,/alsh r·unctions
cal(2',e), stl("'+1,o) , ca1C2'~1,o), ... , sal(2.. 1 ,'l ) . (78)
r,
lrms and Rs
on Lbe frequency in the ca~c of
sine function~ . While tho l!e•·L~ian dipol o i8 no~ a CT'n<'
(frequency) wideband onteJJJia for simrnoidul currents, H
io a true (sequency) widela.11d antcnnn £01• current" huv.we;
the abape of inti:>grated •'1>l31J :u::ci;ioLS . ~e !':Cqu"nc;v
de~end
blllldWidth is detcrmir.ed by the ctoice of k.
1~tP
~Cl
·I;
L-J
L-J L1111
~- A-.
141-zr\..J
\..J
'-
JfE O n
d1
00
D.M
Jf
-Vi
Hg . ?7 ihciacion of a Wal.;!: :... vc
~ert~ian dipole .
by a
T/i
D t-
<:onsider now the case thllt a Walsh-shaped CUJ'rent is
fed into a Hertziac dipole . Fig . 77 sho·.,,, the ideslL:ed
current Ica1(3,t/T) end beloh a currcr.t i(L) with finit.e
switching time At . 'Che difi'er·ent;ial
it. ulso sho•,m . Con01der the general case of a Wslah curreu~ Ical(l<,t/'r) or
lasl(k,t/T ); the same approtl1:1atioc as for i{t) ir: Fi~:77
Shall be used . One obtainc the !"ollot1ing averages
iJt
((a-f-)">
and
(i' Ct>) :
AU\~) ;
<\iITJ
(i' (t ))
(2I'? kt.t
r;t; T
= !
1
-
• """I'.!L
Lit
~ '"
[ (
!: J
2
dt
( '19)
•
a
1
1 (1 -
~l
(Bo)
Radiated powor and radiation resistanoc follow f'ro1> ( 75) :
p • 2l' k
~ • 21 1.!L ~
fil 5nc1
(1;11)
<!.t jnC'
2~1
- <plltr•
~·"!
llt
b
lTC
(82)
166
GA.'lR:illR
j .
TRPJISMIS!;!Q~
The sinusoidal current I cos ?-tlct/'l' or [sin 2rrkt/T
yields tl1e following values fol' ""'diated powe::o and radia-.
tion resistance :
p =
d5=
TT? ! 1
""'
k' 7.o e.2 = n 2 I 2 i'' z, s'
~1 .3ncl
3nC'
r
k/Z
1
;c• Z o s 1
1
1 z s
1J'T' ;incl = ?n r 3~cl
(83)
(84)
'Pile relal;ions (83 ) and (8'1) fo,, the sine current depend
on its frequency f alone, while ~be relntions (81 ) and
(82) :!'or the Wal.al• cu~·rent tlepend on sequency q:i and switching timo At . 'rheoreticslly, ? and R5 may ·oe r.iade arbitrarily large ror a gi.ven seriuency and nntenna by decreasing 8t . 'l'aole 11 shows the quobie:;L of (81 ) and. (83) denoted by P,.1/P., 0 and the quotient of (82) and (81J) d.enoted
by Rs;;al fRs;;in
fo1· a frequency J • 1 Gllz and a sequency
q> = 1 Gz.ps . Radiated power ::ind radiation 1-'esi stunce are
about; eq ual for a switcning t;tme G.c = 100 p.s . A reduction
of the swibchini;; t i:ne to 10 ps - whict is aboui; the technical lin~it at l:he pi·eseut - makes radievcd power and radiatioJ1 resistance foo· Walsh function" one order of magnitude higher l;Jl~~ for sinusoidal ,:'Wlction~ .
Table 11 . Power rat io Ps;iii /1's1r1 and radiation resistance
r atjo Rs,., /Hs,.,. for a lieri;zian dipo l e . f = 1Gllz , q> = 1Gzps .
A~
[ ps )
100
10
1
...
P ) AI
p- =
2
n' fllt
2
20
200
~S.1J1 I
c
=
r;2
1
f6t{1-<;>t76)
1 . 03
10. 0
100 . 0
Let -che switcf!ir:& timo ~ t in Fig . '?'l be very small compared with the average oscil.Lation period T = 1/<p . Consider a gate that: permits t .he _pulses g~ to pass, but suppresses any pulses the.1J arrive at o~het· times . A large nwnber of independent v:ransmi t t:ers ma.y radiate 11/alsh waves ,
n.ll having the same time base but <1i.i'!erent noI'malized
di not
seque11cies i = q:i'l' . Ar; tl1e receiver , the pulses (it
; .;2
OF ',.,'J°J..iSll '1>.'Ji,\."ES
J'R O•VAGA"'ION
•
167
arriving at tte correct ti:De will i;e SU)'Ji1·~ssed by the
gate. Tho ti.lli:ig of the gate cuEt be correct jusL l 1ke
;he phuse o! t!".e local carrie::- oust be correc~ in r:ynchronaus de:todulot ior. o;· sine ca.!'rier,; . ".'t.!' desiL·eu tl·w.s:ai t ter is recognized , ho>cever , by -i;hc patte«H of the :;iositi.ve
and negntiV<' pulses, ,1usL like "~-" trll!le:<ittt!L' in syochronoue domodula;;ion of sine carrio"" i e t•ecoy,nized by
t}le proper f l.. equeucy . :'he r·ecei v~r fo.r· t\1ul eh ca.rricrc cannot d i:iltinguish 1Jct1<een a s a l and a cul fu.nction o.r: tllc
s!lllle sequency any a:OL'e i;ha.n the rocc i ver !or a foe carrier•
can distinguish between a sine and n ~o•.i.L:e !unction unless a timinb :.aigual. is provided . Hence,. W&lih functioni:;
can be usod at least in t~cor;~f" as ca.r1·ie1·. for mobile radio communication. ·~·his is tl.:e fir'lt new oxiu.ple o!'. µossible radio carric:-s fo:· ao":>ile co:111:11.rnicatioi; - in cont-rast to point-to-poinL t.ransmissior.. ae or. cic:-owuve links
since the i11tl'Oduction or sinusoidal CnL't'iere some 70
years ngo . It ie much to ear:..y to speculato 0:1 any {'racticel applicationa of non-sinusoidal ehct1•01:1agn"tic •·a ves in mobile communication . Ho\,•eve.r , Utt.Cot·u di.nclaimi:ig
the possibility of any s uch applici;tion one mny well :·emember th11t 20 yenr" elupsed betwee11 tl10 thoor~ticnl pre diction of electro:nai:,'lletic waves by ;·JAX>;i::LL n nd tilelL· exJie,,imental verification by JiER'l'7.. , whil•' domr 11r :fetu·" had
passed when tho development of thr- clectrouic '"'ube r.iaC.e
sinueoidal waves useful for large .seal~ I r:.cticiol cocmunication .
3.32 Propagation, Antennas, Doppler Effect
One o! the '!lOGt important ad\•nt:tnge• of sU.uaoic.hu ><a~es ia the invariance of their orthogonali~y to ti.n.e sid..fts .
Pol' explanation consider a sine carrier '[2 sin 2nne aciJ>litude modulaccd by a signal F 0 ( a) . The signal. }' 0 ( 9) is
Practically constant during c.ny period of 11 cycles of the
CO!'l'iel" '{2 sin 2nn9 . Syncbronouo demodu lation o1' tho a:o dUlat ad onrrie1· may be repres ented b:y the !ollowing integ,,al:
1G8
3 . 1JARRIER
'l'RPJ~Sl'\ISSJ;QN
11'-'fl
J
Fn(0 )\i2 sin 2nne 1/2 sin 2ni:t9 dg
(85)
il'• l/2
O'•l/2
• F 0 (0')
J
6' - 1/
'{2sin2nn9 {2sir.2-,ma 0.0 • F 0 (3')6 0 10
l
Il: ~he case of mo.Pile radi6 co=u.nication a Stun
or many
modulai;eC. carriers wit~ va:'"ious t ii:1e shift.:; is rec.ei\red .
Hence, 1• 0 (0)1[2sin2nna is replaced by
tF11(-l) {2 Sin 2 nn ( 8 - 90
)
n:!
and (8:;) assumes i:;he following :'o=m :
j
I
2:;F 0 (9 )'{2 sin 2rrn( 8-e n) 1{2 sin 2nma dB
=
(86)
"•'
= F n<a ' )con 2rtn€1 a:im
:'!1e tillle shif ts e. introcluce at;t;enuation but not crosste.lk . J'he orthogonality or sine and cosine fl.1.tictions of
i;he s1me freq:.iency is tles~:::-oyed by the time shifts but
the ortnoe;onality co functions o:r different frequency is
r>r"se1•ved . T!:e subsets o.f functions (lf2 sin k(9 - 0 k ) ] or
[\i2cosk(e-ek)) are orthogon>ll for n:ly ··ralucs of ak. The
undorlyi.ng x·ea.sons i"or thin are the sbif L theorems of sine
and cosine Jur.ictions :
sin k(9+9 k )
s i n k;J cos kSk
llOS k( 9+9k)
cos i<e cos ke, - s i n k e sin k ei,
1
~' al3t.
(87)
+ cos k9 si.n k9k
fu.ucLions have ''cry sj,mjJar shi f "t t heorems :
s~,l(k , Nlek) =
oal(i< , 5)Gal(k,ak)
(88)
en .I (k ,eeak ) = cal(k , e)cal(k ,ek)
The essontial difference is that ordinary addition is
replace,-; by modulo 2 addit i on. Consider now the one- di.mensional wove oquation ,
•
, o2 u
c ax '
'uid i ts general solucion
(89)
(')O)
The ortl'logonnli ty of
Wal~l1
rtrnct; l ons is i,1en<'rally not
preserved be~aune ordrnary a<ldinon ru:d o;•~llr-raction Sii£ns
occur ...u the argw:e_-:·. t + x/c Wld t - x/c . :o,.cver, t .c
system [r,a.l(k , O+Ek), cul(k ,S -t9 k) l is inca1'1:1 tndepen<lent
oxcopt for "ilw;ul ar easer- . Sopnration o f li111H1rly inde pendent '"lc:c~ivns is pos:;iblo , l>ut "1ore dF .. icult thar.
separa.Lion of orthogonal fur:c~ions . 'Ihe sy~tf'm~ of tl;c
di.£.fei•ont iat otl sal 01• cal .fw1c tionn , Oll t h e other liwu l ,
remain orttogonal if tl.ll:e .:W.ft1'd .
The W'alsh
!u.:1cu ior1~ o~ ?i g . ~
msy oe cocsiocreC. to t"C-
present linearly polarh.ed Walnh wave9 . 'l'ho f.i.i·~~ 5 nt·e
sho"n again in '.:he firct col aM o: Fi . ;e. T~-" circulu.rl;,•
j)Olarized w-ave~ of tt'.~ secon<i co~~ arc ottu.ineC. !'1-om
them by nol ding fast the "ler·L onds" or the fua<!tions ti.lid
giving their "right end~" a twiat of ~1.,0° ir thet sense oi
e ri1·nt hru:d scr<n<. Tl'e :nird co lucn 1 obtaine\l by turoine; t il e functions of the e;ecou<l colu.-r..r ';0° to tLc rie;l.t .
The 1'ourLh colWllll is obtained by Lwisti11£ tl11 funcL ions
of t!le first colwm 2~ ,:11 .;0 i.: :ne sen~e of a ri&J::t l1ar.d
screw .
Pig .. 78 Circu.arly polarized
~alsh
waves .
3 . CA'!RUH 1'RA:1Sr-i1ssro11
:cr-,e func;;ions of
~ ho fl:·~;;
.r igh"t- ;•olari"t.eC. W!i'I\:::,
11ne of fig . '/!' w·1· I.he
U<1tllll
, ..,1:1.& .. i'~e c.. r" :ll.:. .... ~d .fr lUencieR
( turr:s r c-r w1lt timfl!'t i O, 1 , 1 w:ld 2 . Thr:;" •1;::r·,,-1 s r,ippol:ll:'
!'.ore a!:: the Sff!Ci8l Ct-1~'? of l'i(ht ~oln::-ized 111alG!: ~aves
·,.;it~ t!eque!lcy O. The fu.r..cl:ionz in t.lie :;econri line ?:ave
all ~he no1·mn ize:l ccquefJC:;y 1 wid .:·r.iqurm"i~" 0 , 1, 1 o.ru:L
2; c::iz- ~e 1o:C.!.'. ""o!'" tr.c- .fur.c--::oL. - n l!n;. ,!I . 'l'lie ne-
guency Of thC fLU;.ct l.Or!S in 1:.JJt>!'l .:'OU I' ai1rl .five 1•qualo
the
ar·r- 9gair. 0 .
fr~quencil'~
, 1
a£.. 1
2,
2.
Tt:e J!Htzi!U! djrole :.n tLe l;n,,ls of
ztnusoidb: waves . le is w,.. l
Y-...no~rn
tL .. r
qu~·-r.r w 1~ve-lenf
:n.
01" r..alf ;...-avel 1·nf-:'"tJJ ji_polc:- .i·ad;i e.'tc ::ine i'o'nvcs of proper
.frequency much
mo.~:e
1;f!'icicr:tly . Ec11ca, onE: i·.·ill look tor
better rr.::liat :-~ fo1· ·,;gJ.;;t. f=ct.-ons L":ied O!l t .. e l!ertzinn
1.lir·ole . L!on9ldcr ,. dl.pole or I cmgLh 1,. Let Lhe wave
sl.n 2r.f( t-x/c be .!'ed l;..to it . ;.. r~r ec't:ed ~;;ve si:i 2nf'x
( t. 1 x/c) odll \)9 pro<lucec and thr• swr. of l>o LI• wav oM ;yiol.ds
a ntandin~ \>.'ave i..I n..,l loD!!'2!S r\!'C r..:.et"lcctetl:
sin 2nf( t.-x/c
c/f •
~
i
+ sin 2nft ( t+x/c I
~
? sin 2,J'i; co s 2rr,f x/c
• LL
( "1)
This d:!.p.ole m.ny Ue considcrl'\tl too conB i .:..t oI a:ur.y Ut'.!rtziar: d~,oles , but only onE> po·,, r hl:lJ if1e1·i~required to
fceci th!"m all l;ecmrne of tl!e Lra.nsformation ,,r che wave
sin 2r1J(t-x/c\ in~'"' stan<line wnve . n Io.low~ ft'oo (Q1)
<>nd {87 ~hat chi< -1•an:ifor;nat~on iu due oo tne shift
Lht'o1·em~ o f i:sine !ind cosir.o fun cc-ions .
:'he follcw ng eq~'l-iuu fo ...... ·,;o_sh f:J.Dctions is obtained
im1tcad of ('•1) rJ'om (88) :
saltm'l',t "!'ex ~rJ
-
!!Hl\'!''l',•/I9ic cc) •
rnl(cp'l' , t/'!'1..al(<;>'., x/c'I)
(92)
,"tga:n 9 s-t~ tJCiLf~ •"1 f1VE- ie ~roduced . Jlo\';evcr, 11ialSh •..ia•1es
propagal.l,lg n Long o r.1etul l i.c conduc l, oL' ru•n dnGcribed 'hY
~a o:, -/"r- x/c"":' or sal{c;; , t/'!'... x/cT J ra•il• r t::.un by &he
e:xpr·e,,:;1 oHs co' (:;;T, t/1'ex/cl') or sal (<,>i', t/Tei</c'r) . It is
kno~on how co make n t,;alsh wave p:-op1gate eccording to tt:e
' . ;2 PROPAGATIO:r OF ·.-,u,s:-: ;;A'.'ES
171
argument t/1'aJx/cT , bu·t bhe required ciJ.•cui tt·y is :uuch !Dr>l'e
colllplicated tl!an o :netallic COlJductor. He>nce, antf':mas
based on the standing -..ave princirle do not ar;c·ear attracbive . Sin<:o a pow~!" amplifier ror i,,·aisL wriv0s is a nwitc!:i
tbat .f,'eeds oithi;r a ponit i ve or 11 ne;!;ativo curr e::.t to i:he
antenna, it is mor e ti.t~racti ve. Lo use rna.ny He!"t2ian diooles tbot; m·e indivic!ually ~ed by s1.:c!: a:nfli!'iers . A hir.dred Hertzill.ll <iipoleb r.rre~ perfectly prnct~eal , i f LLe
switches are l mplementod by tranniBtoi•a , whi_e a cllo11 ~=tl
and more ore not t.iLI'Ofll iet i c i f in~egrat~d circuit Lecltoiques arc usec . '..lalsh waves land to accive ante:-...:Jaa 8!1
natu.re.Lly as sine waves lead to re<>oi:l)Jlce an-;cr-=iue . !hr
man;; i.ndividuully fed H"rtzian ol.ii-oles <lo not need to be
ar1•anged alon15 £< l ine but can be arre,.,ged in a square a l'C'n .
Hence , a lons or.c- dllenoional en r-.-.nna .JJ&j' be i•eplac~d b;','
a small two-<iimensio!.lal ar:teru:a .
The radiation pl.iCtCrn of somA simple Wnlch wav~ nm;ennas has been calculatnd . Co11s1clct" , e . g ., o p"rabolic rer :ector. Its bew:iwidtt depenc:: on tbe rotio >./D in the
cose o!' sinusoidal waves, "'here l. is i=he wavclen5th nnd
D t he diwnet;er of the 1•efleccot'. "he two 1·atio~ l./D er.d
Mt/D o ccur in the case o f Wulsh "'av es , where ). 1r. ni;iw
the average waveleng~l1 v/rp , ,(IL is the s.-Hcliint: cimC' def ined in Fig . 77 and e i" the velocity 01' llght . T11e l•eU>Jw:!.dth decreases to zero with decrnasing l''~i tchin1> riir.c lie
whil e the ratio ;./O may remain constant . A na.i.·rowor bNur.
l!lay thus be obtained for a :f'ixcd 1·atio >./D or a Dmuller
diqeter D may suifice for a fixec l:>can;••idtr• . Ac .u •l' .,,.
the parabolic reflector can be ~ovlaced b.Y circu~Sl' c!i:;c
o !: diameter D which is cov ered by cany !!ertzian <lipn I us
all radiating t he nam1> Wal@ function.
Let sueb an am;cnno be used to radiatto from s "l nee
Probe to "arth . The almost: eopr.y space >JOuld :.ave :;o dc' ·l'ilneni;al effect on the waves . Upon hitting tbe atmonpt\<'rc
the pulses would be widened and the width of. the boou. woulcl
bo increased . This o!:!:ect is not important since it occur"
0 :. t:!le lant, re lati vely short section of tho tran,,.,iesion
' . cAFm !:ER TRA..'ISIUSS!O!!
172
path . On che ot her hand , the widenin;i: of -hr beu;n occul's
at the bcgl.nnini; of tLe tranurnission path when radiating
rrom the eart h to n space probe und tbi!l is very ha..~ul.
Sinu~oida.> waves woul<I na·rn co r.nve n very t•i1,;b frequency
to obcsi u clle ,;nme narrow bca.:nwidth with a given diamctel'
ll of ~l:.e re:'."lectoi nnd woul d thu s be absorbed comple l,ol;r
by douci.s . Hen ce , eloctromegneLic i-lnlsh i.:aves <ippenr quite
"ron.l ~ing in certain applicauons, but once more one must
caut ion thnt; no Mcperimen~al Vt «ificntio11 is available yec .
A si:luno ida
el octr:>•111gneti;: «nve E ni n 2nf ( L-x/c ) is
tr<inofo r mod by the Dopp!l'r effect into the wave .E sin 2'1f 'x
(t ' -x'/c) . Tl:.e shifted Iroquoncy hos 1; lte vHlue
f
1 - -.; c
~
t
a
l"i -
v•/c'
(93)
'
where v i. t::io relat:ve ·;elocit:y of t..rw1swit"t"er and rec~ivCl" .
A
.is.: sh wave
E(x , ~)
e
(94)
x/c'rJ
Esal (qiT,t/1
Ls transformed b.Y the Li·o nstormat.Lon e qun tlons
•1ist1c
ot relati-
m~ch.an..ics
x •
x • + vt '
(97)
·1 1 1~
fi -
into the £ollowing !'o :-m:
E(x 1 ,
:
'
)
•EPd(:)''
t ' - x ' /c
'['
~ -
- v 1 /c 2
V/C
)
(96)
in order to bring \ 9v) into tho !om of (9'~ ) one must
do!'ine t.Ue tra.r1~Jor·oed !'lequl!!1CJ' cp ' and time bane T• as
.follows :
<;; •
:
qi
T' • ·r
1
'1
-
v/c
(97)
- v 1 /c'
~ 1 - - v vt~cz
c
(98)
173
3•73 INI'ElU':sROi'IETRY
!t follows:
E(X', t ' ) . Esal ( :i>' 1 ',t' /'!''. x'/cT')
(':)9)
Eque.tions ((97) enc ( J~J Bho>• thar scquency <i> t.1>d frcciuenc,y f lil'O chro1gcd equ~ll.Y by thF. Doprler effect . The
additiona l cha.nge of the ~ime base T ;;<;carding to (')S)
gene1•o.tei. an invai•iar.t of' Lile Dopple1· e ffect 01• of t.or,1n-c"
transformation :
(1 00)
A sine \oi'aVe wi-cl-_ f ,rerp. cnc:;; f rlid:.a.~tJ ty S tl'OJltunitte!"
wit"b relative vel ocity v Cf\nno~ be d.:.s....,i11gut:; t-.rr. !:.·oc: o!4e
1<ith frequency f ' :-a..iilltM b;y o LL'!iliW.itter ••itll ~·.,l~tive
velocity o . 'Phi s is r.i;enernl 1;,· nob so for Waln. ruuct.ious .
One mny readily Bee J'~otu J.'l,g . 2 L r.ti~ a 1•0ductio1> o f the
t"
sequency '!' s 8/T of !Jal ( l:i , 9) to "' ' =
yi<' c B 'll ' • :•/'J',
but; tha resulting :valsli >WV!' wou:c. diffr·!' J'roo. si.l(( ,B ) .
l'h&re have been m:~e:nptE Lo de-f'c't :. ntel:ii::<·nt rigoals
nanamitted from oth,..r _plane La . 'lhe~fl ~tteC!p:t:s werP 'based
on the assumptioi:_ tl:at such nignala "ioul<l be !"l.n;i. ""·nve .- .
Tb~
Doppler ottect of W'als:i. wa.·,,· ec l·ai se:: tt:r. quae.tion
'!llbtlter these waves arc no< more like!.y to be 1lced . -~
tranmnH ter located on o J>• Clliet and tranami ~ ~ 1 ng Eine
wa.ve1 would be rec-e ived wltl1 A. dif~e.rent.. freq11~t1,~y !J.·Oti;
any direction ill space ar.d this frequency would ul:;o ,;epend on tho position of the ;>lwiet in ii;<: or':>il . ;.. •,;,.1~:
wavi", on the ot!:e:.-- har.:d , CO\.l.16 alw9ys be id-=-...z..:i!'icC. as
~ha same wave, regai-dlesa o: di rec- ior: of pl·opagntio::.
Purt:heratore , a •,.'el.sh wAvo n&.n twice Lt.e evi::-ragc ro•1re.r of
a sine 1<ave with equttl nn11i.Hude , an il.'portant 111lv1rntat!e
!01• 1<eak signals .
3.33 Interferometry, Shape Recogn ition
1'iE: . ?9c sllows the principle or ini;crf ero:nntric wi:;el
t.uaauremcnt . '1'"'110 receivers at the point::; J't nnd B .:-ece i ve
><aveo fl'Oll a far away tranSlDiCter which cravol J'l'OCtic~l ly
paro!lel along the rayr. 8 and b . J. a:easure:nen. or t"e p:-oPagntion time dit=:ercnce i:.·r p AC/c yields t h e angle
174
(\ f\ f\!\ 1
:.~. ,
t.i,,.,.,.011
•
.,' fq\7\TV
J ~(\ (\ (\ 12.
-vpyv-
•
.
s.
"/
•
?ig . !" l en .nte1·feron~t.ric inea:;ur~Jlt!'nr of - a11gles . •)
J:esoi.aLlon and rf'l.i;.>lution x·o.ngt: of' ei 11 e 'tTaves i b) resolution ~u1d reoo I ut l o:i 1"(utge of Wnl:-tL WrJVC>.S; c) 1-~eo:netric
?-claLiorir fo!· -wo r~ceivtir~ A an.J E no. ir..ioncd on t!:c stu P
:'l.e.:·iCia..... .
...
Fig . BC (rii;!:ltJ :tetlectl'ln of si!le ~nd 't.'Als!.. 1~·ov~n by two
point.-l ike tar~:e~s .
;
.=.
s!rr 1 ctTi A- . :he sa.u.l l~st: Xt''l!'!urab .. ,.. tice difference.
A-m!l't de; 1i<nd.s -
ror
wel- n;; for 't/.'11811 runcciona on ti.tt a-rAdicnt of t.:-ielt• zero cror.:.siug& . Y!enct:', ~'I'mi,. ia
proportionAl tn 1/f fo:- eicc fu.n<;t:..ons "lnd p:ropol'l;ional
!JiJ-1e IJB
t:n 1/(? fO!' ~,]nl:<!'"J f'w:c~iO?:.t;i lihe fI"'Oporti"lnality £actor i
,1enoted hy z in l"ig . ?':Jn w:id L> . The ri>so l ution , tba·t i'1
t.he Sit"illc!.r'; meaf".nrat:c t .i.me G.:in.1 ::>:-- tht smallest measu:•nt le 8!l•:l ~ Cd! ~ cc T~,/iJi, is ap"roxUia., ly equttl !or sine
r:u~t1 \-.'Al.~i.i l"'unct..1.ons . Jfowovor , t;be rosol u1.;ion range is c:omi leLely d1ff erenL . Th~ lUL·gect rer'llli.'1$ible value of t.'I
'"·"- lie lo<>tw.. en -T/2 u.no TT/2, i f T is the period 0£ tne
..iave , ainco a wavo deluyed by a Liul tiple or T io equal to
the unrlt•layed wave . Heinee, c Tm., equals 'I'. Since T equo.la
1/f iu the case of sin" Cunccions, 6T.,.. equal• 6T,,../C •
17'.i
Certain 1.rJnl~h fu.uctions sa.:( i, €t) Lave tt S!JOrl.~st ,;.-.criod
? = i/'I' and o?~.. equalo then ic.TM·• /c . Such a lialsh funct ion !s sho-..rr. fol' i = 3 in J•ig . '/C:.b . O:;h"r u~abl~ val:ies
are 1 • 2k - 'i accoi«ling to Iail:e " on pago 1SQ . ;,. larg"
value o! i i.:lc:.·euses ;.,he i~eso: :..rti~!":. l"Snge GT"''c v:ic:-1out
inc:rensiog the ett.alles"t meas-u.rab~e timi:;. 1i rreronc~ ll'J,Trir .
The l'o llo «iDtJ: i·epi•ese:itati'le values miiy be computed
from Fig . 7'Jc . ;..;,L /, a;id B be t1<0 poi nc1 At :i
52°
northern and t:iout.he.rn la.tit:utlt3 and H~nume O'l1,,.1n to be
10·• s , The distw1ce ,\B is abOLlt 10 000 ~m "'nrl Lt n usable
observation 1.tngle 180° - 2a = '/6 '. TI~c ro11otution equals
A~ s ~x10·• or A~ = O. OC," . 'l'he V<ol\H• o r j in 01' l.rdn0d from
t lle follo\fing relnhon :
( 1 01)
An angle of 0 . 05" co~1,,s;>onds to a di~cwice or
11 "' on the llurfllCE< of the l!OO!l ar:d of about j
"bout
le:& on the
surface of :·Ju•n ~r.~n :-=arr. is cl05l" to £nrth . For- corx.parison, the sinallest reso-vaOle an~lr of 0 . 05" itS ubout
one 01·de1• of mllgnitude ~maller Lh= th" \J~aL LaaL can oe
done with astronomical te~escopeci .
Thia method or wigle meaStU·ea.tlnt ttpu\la "'' nttracti v"
!or apaco probe tracking . AD Bccurat~ knowtodgc oJ' t he
distance li!i would frequently not be roqui!'ed . Suet. " case
i8 guidance or a oroc<> probe ~o tho .,; nd ni I ;v n!' wio LLe1•
or to a boaCOJl transa:itter . .I. CO!!Sidernnle IUlOUUL of daL«
processing equJ pment is 1•equi«ed . ':!"' p1·-·1 ioualy 8Ssu.:ned.
Value i • 106 JLea.ne t.L.at two Wals:i fucctior~s t.:::>.nsisting
or a periodic sequence or 10• bloc?: ~uls". t.a•te to be cor.iPBred . t.. 1Dinim11.11 storage capacity of 2x1 C.' bi;c would be
r equired for tile comparison . Ad1icionnl stornge cnracity
wou ld be needed to improve the signal - ~o-noioc :-;iti.o ty
a ve:raging over· many mult i ples of 10' pu!ses . Avera[;i"e;
over , e . g ., 1000 multiples would require D ~oLal of some
12x10• bits otorage capacity.
li'ig . 80 shows a radar R and two poil1t-l ike l,ai:get>1 1:11
and B2 that are close together . Linoa a and b ahow sine
::; • GAHJUEH 'l'RAriS:nsstmr
waver. ref I ccted from B1 tL"ld B2 . Line
c
show::: thB su.in.
ot
tllese t;WO sine waves whi ch is r9ceived b:y tile radar . A
periodic sine wave would l ook ·tl: e .;ame whetner ref lected
by twc \;a.rgets or by a single , tr.ore refl ecting t:at'goi; . The
pulsed ~ine ~·.rave of line c shows deviacions at; begin.ni.ng
anci. end compared ;·1ith lines a and b . There at·e 1000 cycles
and only t;wo of them discor·ted , i l i;ne oulse durat>ioh is
1 µ s and the carrier frequency ie 1 (;n, . :Je.n1.>e , the energy
indicating two targe;;s is in tile ord<H' o!: 0 . 1% of the total enersy of -cl1e l''-'lse a.l.Jd is insi5<1ificaut .
Lei; us
consici.e.t Ghe
i~eflecCio11 l)f
\.Jalsn waves . Lines
d ano o show the waveo reflec·teLl from B1 an<l 32 , F.llld line
f' show::: \iheir !>Um . ·r11e difference be ~ween '''aves retlec·t-ed
i':rom ono or ;;wo toarg;eto i.s no loni;er rest1'icteo.l l;o beginning und end oJ" a :pulse . A :;:ieriodic Wals!i wave waulci_ still
tell h.ow a:any targets there a.J;'e and what the <liiference
o.f their.· distance~ nre , ulthougn the Qbsolute di svance could
not be ill.fe;•red .from the sfiapo o!' che reJ:lected signal .
Since lines d ~o f in Fii;; . 80 show thao the sum of se·..reral ~,ralsb \,•aves of equal stnpe bu-c va.!."ious i:;.irne sbifts
may be a a.i:i'c1.,~ntly shaµed i,..rave , one must investigate
the reflec;;ion on the radar dish . '!'he p~·oper approach
would be to solve t!:le \·:a•1e equation for the pui'ticular
boundary and illi ti al condit.i.ons . 'Vhis has no L been done
yet . Wave opt ice has been dominai;ed b:y sine and cosine
.functions as much as communications . 'l'bere is no theory
fo:r· Wa lsh \·r aves or complete sysl;ems oi' orthogonal waves .
It would be '.'II'O!'lg to treat 'W.:ilsb waves as a superposition
of sine a.ad cosine waves and apply tlle known resul.l:s oI
wave optics to chese sine art(l cosine waves . Sequency fil i;ers , sequenc:y multiplexing and ~be l'esults for Walsh \>lave
ani;e11nas would never have been fom1d, if t he i~alsh func tions had been treated as a superposition of sine a.nd co-
nine !'unc"tions .
Li;tcking a 1,.;aye theory 1 oue m.ny use geo1netrical optics
as a first approximation . Fig . 81 shows a cut of a parabll lic dish . 'rhe distances r 2 and .r 1 +· d a.re equal . Hence , (I
1'i7
' . 1_,,,!__
• cos•
l•i1~ - b1
•
11e:'lec1;ion of
·~ave:3
acc-::."dinG to geonctricul o~
t.1cs <>;,· a fBL'auol l c 1tir:·or
a) tu>d ;;>::> j:er~·en<!icular nirrors (b) .
•
'.lal eh wevr. radiated f:-on th(' focal poir:;; D wi:: be dcl9ycc
equelly whether reflected nt :3 or ~ •..:id w.:.1: ad~ •ithout
i;ime sb.1.Ct . Vice versa, a >it1:ull1 re:'locted by ~ r•racolic
dish bo tho tocal point D •.,.ill no:; be d.istori;ed even thougL
;it is uot sinusoidal .
Anothe1· e:;./JJJlple o.r a d1r.cortior;.-fr·ee i-efl<>ctoi· is
recL11J"1gular r.lirror $hewn in ?it', . l:l1b . Jc t'ollo·d: fl'O• che
geometric re•ations sho·.m iu that figure th'1t ti1e two propagatio11 pattls ~ and b !:L!'t- 1 411al ~y locg . E:e:.ice, thcr" will
be no time sh.!.ft ba ...wee~ •..,'ttlGh ilttves re~lectE:d froo va-
rious points of the reflccto:-, and t!... e ~..falt'h t·:.'"l.·1~ wi:.l b·J
"efl octed without ch'Ulg<' o.C ~hnpe . [L csn nc slcown t~.a;;
tlhis reouH also !:olds for '• i;Jtre<;- tli.monsionnC. rector.gular ro.CJ.octor .
Ln general , a Walsh w1<ve ranoct<'d by a L'<!'1·et o f finitu dimenoion will no lon;>;i:>r be u Wal~L wa·.re . 'l'he sJmye
o!' tho reflec;;ed wave will yield in.formation nbouL tLe
Eeonetric size and shape of the target . Cor.sidtr ~le re~,
t.
-- ec·ion of a step, li;.e the one oo sal(1 , &) nt 9 • O in
Pig. 2, from.a spher e as shown in Fig . 82a. A corracL t,reat 1' "'•'1'tlum, Tr11nani1u 1on ot l11lormu.hon
1'//j
~.
(.;/Jl'UEP. rRAllSIHSSION
'j • Rf••.!.••~)
R s<•~•
•
e
•
"
-
Jl'J.g . 8:' Shapes o r otep waves rorl ecced b:y perfect scatter-
ern oJ' various shapes . a) spl1ere ; b) x·od of l eng t h L llJld
d.J. ·une~e f' d «:.. ; c) cylinder ; d) CiL'culcu;· disc of
d;lameLcL' :?R; o) r11d11r reflector (3 pe1•pendicular otlxrars) .
~-''
179
INTERr'SliOMETRY
ment would agair: reqlt' re a solut-'.o!! oI tte wave equal.ion .
A fit·nt approxil!lation may, howevei' , be 0·3tainc<l by ussUJ!ling t llat a sphcricr;l wave io i-t1dl.i1ted from eac~ point on
the surface of tl1c nphere , that ia .ulu:oinated by tho incident wave . :;:e re fl ccted ster computeC. under t!Us assUJ!lption is sbo11r. on the rigllt of ?ig . S2a . Ini t ia.i. ly the
wave is reflecteu by the points on the surface or the
sp.hore close to Che plane S only . Aftei· che ti1m L =
mic1- sin ~ ) hns eh11ood , the wave will be reflected by all
p~i.l'lts on the spllere having a disi;ance ~irnller thnn ltc ~
rro:n plane S and :nucli me>rc power ••i 11 be reflecr,f!d . A':.
~be time t = 2R/c al I poincs on the illu.:ninsted n~lf of
the cphere ref lee-., P.nd th.ere •'>'ill be no increann of r·eflocted powe r for l'l.rgeL' val uec of t . Since a~l w..1sl.
functions may be coosidei·ed r,o br superpooitionB 01· st;ep
functions with positive or ncguLive ampl itude, one mny
conat1'Uct tho shape or reflecLed Wr,J !'lb f=ci;ionn from
1'1g. 82a .
'l'he co:nputstion of the s!:l!lpe ol' the reflected •t<'? ·.•~ve
is ns follows . The ru:ipli~ude dirn to .Ile reflectio:J frol!I
an annular area with disCance ~ct !'row plane S in f"i..,; . 82n
is proportional to Ha a:rea 2nR coa 6 lld l' , bul. only the
l'rnction sin~ of thi~ area reflec~o baci< in to tlrn dlw,ct.Lon
o! incidence . The voltage u diap'ayrd on an osc1lloacope
as f·i.nction of the angle 5 is thu:i giveu by
r
n/1·fJ
'
cos ~I ein S' di3 ' • KTI'R2 (1 - si11 1 S)
~ 102 j
W!l"re K is a facto1· chnt correct!': tbe dir.lension o!ld ·• lle>ws
~
ampli!'ication , refloc~ivity, etc .
Since u is displo.yed on an oscilloscope a~ a func~ion
Of ti.Jne and not of ~ one may subeticute
.or
ot
~ttenuation,
2R(1 - s:i.na)
!'rem ~'ig.82a into (102) :
,,.
180
'-l(Ct
: ErR'[~-(1 ct.}'] ,c:; ,; :OR;
""
u{c;.) •
::nr<,
ct >
2R
u(ctJ is tl:c curve ploLced in Fig.!321t.
Fie; . 82b shows Lbe shape o: a step "11'"' NJ' l ected by a
r·ocl OJ.' lengLh L nnd uiumet.er d << 1 fo1· various angles o. Of
!ncidonce, . F i g . 82c aL01·1s th~ reflec•ion 'o.v a cy.Linder , it
L!le incicenc~ i i; pl'rp,..r.dicular t o th~ nxi: . Fig . '!2d shows
r;!le r·eflection by El circula::- di!:'C of duirr.c t'°'l' 2R for variouz angles o. of .ncid~~cc , and fig . 82Q ;naL or a rade.I'
r~:·1ec;;or consill~ir.g
of tl:rce perpeudicnlllr :r.irrors.
The .queotion arises of w!".ich 'rJalGh fanct..!.onn •11ould be
b .. st ro= sLupe reco,.;r:itlon . Conside1· n11l(ll,0) in Jo'ig . 2 .
Thi" f'llncti on ia oan:y Lo l'ilter and J 'J'OCO~G . However , ib
hus a sho rtes L period of e = fl or ~ • t\'I.' w1c:l thj e caunea
ru.1l>l£;uities , if a t11rt;f:'t. has a large1· llimonriion than fcT ,
or if Lhe"!'c a.re sevcrnl t"a1•r;et.s i...-ith distanct'n larger th.an
tC1 . 'l'he ounction ~al(?,6 is t.al'Of'r to filtf'!' e.nd p!'OC<'C:'.' t!lan siL 8,e), bui; its sho=~.,,-t vu·~oc! fr a
1 or
t ~ T and n.mbigu1 tlt:S will occur• !or tlil'S.l:'t:~ witl: dim~n
sion larger thnn c: .
The function nal(1 , 0) is equW.ly uiu.p!~ to filter and
Ju·oce,;!l as sul (A , a) aud aloio ha~ ti•~ shortont period T .
However , sa1(1,0) har. 2 nteps only, whtlc i:al(7 , S)haa 14
•t<'fO, and iL io th" 11Leps that provide inrori:ietion about
th" ~!'\ape of the ta,-.,:eL, !Oot tee coneLur.t; er.ct ions of i;he
!'unctions . Hence, thu co.re collplicatec! Wolsh i·unctions
s;·e- bctl;er f ::""Oc et-.e t.heoretical :poir.t o! view .
ln-te.z-.fcro::net.ric :crnci:ing o!' .:::pace rrche~ ~tnd -radar tari;ot ailalysis can be a.no are dcne by sillu:ioi<!!ll >iaves too .
Th~ :point !:err in, Lhti L \.,'n 1 !ih ,..·aves beltuve inJ1erently very
difr'crent from Dino waves and thus offer a µromising alLeirnA.tive f oi· a more d1'\ta.ile:.l study of r,..nolution and u seful signal - to - llolae i·otios that :night be obtained . The
di01cussioi:.tas been restr·icocd to ;:ai.sh wa•H!2 although the
dilferC"!ltiat.eC ~.;uleh wa\1 es of Fig . 77 ap11ear much superioi· .
l!owevcr, tl.ese ""avee ~.,ould in~roduce tho nddi::io??al pa..ren.ni:•)r switchir.g ti:ne w1d thus complicaLe the discussion.
4. Statistical Variables
4.1 Single Variables
4.11 Definitions
C)oneidct• u SClriee; tr>.Xpbnsion o f n str:nA I F( 8)'
00
l'(S ) •
)' a(J)f(j,9)
j.D
There are three basic orcrat i o;;s that ~,.r: b•· distingui nhed
witl: the help of tl:is expansion : !'~l,er~c , :-hifti.n;;; and
signal design . A fii.te:-ed signal F 1 (a • is o'Qtein"'1 by :r.ll1.tiplying e( j) with o.n atten:rn1'icn funccion K( ,') anc by
time sllifting f(j , 9) by B(j)o
F1 (B)
r
2'.K(j)o(j)f[j,9 - S(j)]
l•O
Shifting F(9) iA done by substi t utin15 u function k( j)
for th e vorioblo j; the inverse iuoctiou J[k(j)J = j must
exist:
.F, (e)
~ ~ a(j).f[k(j) ,B )
l•O
k(j) equal.11 kelj for sequency sr.i.fti::E; by aean" or =r~i
tude modulation or a Walsh carriez- wal ( k, 9) i i ~ e<;:.t!lls
1'+j or k-j for frequency shiil: ins- by Cle6..'18 0 r ~in&le sideband amplitude modulat;ionof a sinttsoidsl carrier ::ii!l 2 '!k9 .
The system (f(j,S)J inust , of course, be i:he syatcwo! ifalsh
functiooa in tho one case and tbac of the r<i n<'! - cosinA functions in the other case .
Signal design ls t he wost genert1.l pr·ocenr> . 1'he coeffiCiento e(j) are replaced by new cocfficien~e c(a(j) ] =
C(j ); again ~he inverse !unction a[c(j)J e(j) muat exist .
Fu.rthe=ore,
-he syl'Leo [f(~,&J) ;,-
re~lscecl
·o;r n new
,-;yn:;ea. [ g( j, e / J :
Oo
Fd(a) =
L:c(j)f((j,&)
J•O
The transfo::coaLiou o( j) - c(,:) a l one it1 en lied coding .
Exan:ples of coding lrnvc been given in l'igs . (26) co (28) .
:iote tha;; ote coe.!'ficienc a{j) ,-,ay be t.rnn eforn:ed into a
set of coef:icient~ c(j) .
F:.1. ;;ering and shifting have been Jiacu11sed in chapters
2 find 5 . Tl:e cxt:.,nsion of t.he tl:eory of in.Coi•r.iation transnirr.ion by orthogonal runccions into tt;o area of signal
design requires Lbo m0thOd$ o.r mathcmo.t1 cal statistics.
A short discussion 01' these methods wH 1 faci l itate an
understanding o.C Lheir applications in tnc lase two chapt.e1·s . An up to date ciathel!latical discussion would have to
sta::ct wit!: the concept of o-- algebra . ;, ao.. ei.:h.nt less up
to da"t"e approo.ct. i!" used !Jere, in order to :lvoid excensive i::atl:en:atical 11bstractior: . 'Pile der,ree of abstraction required should µrove satisfactory to mont .
?:robe.bilHy Wll!l defined by ei ghteentil com;ury mathema;;icians as tho quot 1.ent o.r the numbr>r of favorable ree ul ca and che nwtbor of possible i·esulLn , if all retml~s
«1·e equally probub!., . 'l'hin definition a:ny be applied to
a g=e o!' cards .-ithout dlific:ilty . Tb.e probability of
11!-a"i::g a certain card rron " deck of '>2 eq:.1als 1/52, i f
each c:!XC i:; rej.ir~s~ntec! once iI: 1;be deck . The condition
"if all resul;;G are equa:l;r prol>ab' e" .A explained by the
stet.ea:e.:ot "if on ch Cal'd is i·epree;entod once" . This deJ:initio11 of probubilicy does not suffice .for communication a ,
aince Gb~ 111eaning of the condition "11' o.11 resuLts ate
equally PL'Obuble " cnn often no;; be explained .
KOLMOGOil.OFF founded
axiomatic ~hcory of probability
vased on the :;heory o~ nets ~ 1, 2) . Consider a large number
of "-easuxe:nente yielding the resul tG C • C1, t ,, - · · ExsJnpl e~ o:· mien u.ca:iurements are the observation bow oftell
heads 01· tails occur wnon flipping coins, or the counting of
=
4
183
. 11 J)£F1NJ'::ONS
iottors in t;he woNln of a te:<t. . t
t ., t,, ... is called
8 stt1tiatical vn.dsblo or a i'andom variDble .
. C. C was " one-dimen sional variable i n t ile exrunples
juat given . Consider aa a ~w::·the:r example " ball that is
dropped onto an irregular surf ace . !.et C and r deuote tho
coordinates of the ;>Oines where tl:.e ball coa:es co re:;t .
Each measurea:er:t yieldc t.:o vo.luus, and t
t (C , 'l) i:; a
two-dimensional v1u·iable . t i s t)enerally a k- cli<Lensional
val'io.ble , i f H is defined by 1< values .
0
Let S denote the set of nll poaaible rcaulUs
~
of a
111eesurement . s i an~ s. clenoi;e outsets ~f s. ·~nc sua s, 'sk
is defined as thn set of a_l el.,aents belonging to S,
end/or s, . ':'"ne product or im;r.!"SeL t.ior. S; S k is the set o~
e.ll elements belonging LO S , a" •1ell ao to Sk . '.'.'he d.rrer8llce S 1-Sk is the 10eL of all e l ement.;. belonging t.o S , but
not to sk .
Consider ao example wtocre the lengt t· of wo1·o.ls inn text
ia monsured by the number of lectCL'" . The shorte1;1t. vcssib~e word bas ;;h.e lengtb 1 , the lorui;est, e . g . , tho length
25 . The set of all lengths 1 to 2: ls $. Let: 81 be the
subset with lengths ) , 10 , 11 arid 12; let Sk be the ouboet wi l:h lengths 11 , 1 2 , 13 , 1 11 •Uld 1 !;> . T;:ie sum S , ~s k i s
tho set with l engths 9 , 10 , ... ' 15 . '!'lie product s, sk is
tl:e set with lengtl111 11 and 12. Lil" difference S, -S, sk
ia tbe set with lengths 9 and 10.
Sets may be mos• easily defined by inte=vals . ~or instance , the set of rcnl nun:be~·s ' between 1 and 2 ia defined by !;he interval 1 ~ ' ;1; 2 . A oct of c ompl ex number·s
t u+iv may be defined b y two i11tc1·vals a ::; u ~ b and
0 a v it d . Sets "'"Y be defined by k iuLei•vals in ar. EuklidiW'.l Gpacc Rk of lt di:ncnsions . Addition , subi;;rnci;ion and
CiUlt iplica•ion of these sets yield further sc•s . Lrt these
opei-stions be performed a fini•e or a denwnerable iu.finite
number of times . Tbe 1·esulting class of sets i s ce.llod the
0leoo ot Borel seto in Rk. Sor el 30t~ are always u~ed i n
thu following analyals .
A net function aedigns a
number to each el1>oent t o~
'' · :; • .;TIS:ICAI. VfJ!!AflLts
t e S snows tM t tl:te "' c·ment t bolonl!;" ~o LLe ~0t S , while Sc S 8howr. thar.. l:i ir. a !lUbset
of S. Let us rte•~ne a ,;et function ~(Si = ?ctES, ;;ic;, the
addHional .'.'eatu.re:; p,S).: C and o(S ) = 1 . ;:>(S) i!l Called
thG probabilHy that C belon gs i;o Li.le se,; S . )!( S ) c 1 denotes certainL,y 1 1:1ince ea.ci:... :r-csul t. C o.f a m.easurf'.lment mur;;t;
oelong to the stt S o;· ell cir;ast...:'<.'lll~i:ts . Le. t
b be the
r~ . u t of mea<1uring tile length of wore,-; and 11>1' S, denote
r..lJ~ sci; with l.:ngths 9 , 1~ , 11 9.ml 1?. ~(S ; )
p(CES 1 ) •
p(9:::,:::12) i s t;hr probability of : hwinp; or:e of th~ values
<-t, 'IC, 11or12 . ;>(SJ i:1 ~ai'1 to ·J~fir,eadis•ribu;:;ionof
tlit: rnnCom vaz:i1blc ~ .
Cor:sider " ::ubnet S o.'.' S Joowing no value of ' smaller
tl.w> k or 11u·i::;<>t· chrui x :
a ser.. S . The n" L" tion
?~;.
p{S
~
~ Y'
(
~rol:ability
:Lhc
~ 1)
of a t .'10-r:!imeL.6:..onfJl va.riab r.-
C= C(C ,n)
ia definoa as J'o 11 ows :
p(S)
p(k 1
~ C •
r, k1
~ ~ ~ y)
(2)
An ~x~2le ir t~e proU&bitity ~na~ u ~ord ~ith 10. 11 or
12
l~ttera
is
:o~d in~ r.entenc~
witL 100 , 101 , .. , 125
l etters :
.,cs) • ,.,c10 ~ ; :; 12 , 100 ~ 11 • 12~J
A ftu1ct.ion W( x) cau be Cefined ,
in \ 1) :.s - oo :
W(x )
;:>\ -.., < '
j
r i;i1<: lowe!" limit k
" x)
(3 )
TJ1e ii:rr-t uiom nu·1y :iow be stated: ~1~-~~-:~do~_!!!'!:
~2!~-~-f~-~ti-~~£££_£2££~~eon~~-~-~2~ -f~£~!£E_Ei§2_~!:
S~~l~-~~~!~~~-~~£-~!!-~2~~!-~et~-~-!~-~~ 2 -~~~-~~-E.(~2
~£~E!=~~E~-~~!-2!~~~?!!!!~ __o_r__ ~;~-!~~;~~E-~~~~~- ~~!:
!:!~!..~~r-~2_§~
The .f'eotur·(}S o.r p(S) are defined by o socond rociom :
~~~-~~2 ~~~~-~~22_1~-~-g~~=~~g~~!Y£_~~-£~~E!!~!~Z-!~~~:
~=~~-~~£~~2~-!~ -~~-~~£~-~~~-E~B.&2_:_2:.
~ .11 DEFIIET IO~lS
185
'l'tiese sxio:n" may be exrre;;~ert by thn fol towing !'orm:.llas :
p{S) ;; O
c
p(S1+S2+ · · · )
0 . W(x) ;; 1
.I(+ .. ) • o
ll(-oo )
~
0,
S1sk •
0
meBllS t .. uL
the ~ubs~t!: s, and. sk have no C()D'JJ:O.:.
elomenl;; putting it UJ.f!"erent ly , thr> reoul c of fl moouu 1·1'!ment muat no t bcl ou1; to s, as woll !HI Lo Sk .
Consider several series of :-1C!'lleu1·~oent.3 . 1PlJe fir.st
yi~lds the results C, the second Lbe re~ult~ ~. eoc . Le~
Lit. postulate th!it wiy coa:bina:.io:: ct t'"n rruodoa: va1·1olJleG
Ct "fl, ••• shall al no be n statistict1 Vf'rio.ble . T.1i.; tt.:ird
axiom micy be forrnuloLcd us J:ollown: -~--CJi_.:..:..:. _~L '.!!:!:_!"::!-11=
~2!!1_:'.~!!~!~~i-~..Y-!:2!!1~~;!~~-::.~!:!~~!£_~~i-.:..:..:..:._h 2 _ !~-~!!2
!..£~~2!!1..Y~!~~1:.:.
Fo1· example consider one series of r.ina.;ure:ne:u;p yioldinf!
tho length <; or word:> in a cext, tue second "<'rif'B j'ieldin;:; the length 'l ot nentencc!: . I. coo.·~ined varinble is
obtained by combiu log the fol Low.!.ng pairs : Len gt 1 or the
ru·ac word and the fi.rs t sent ence , lenp;th of tne "econd
word e.nd the second sentence , etc . AJ.10ther cxamplt.> J1a11
been given in tbe e>:w:iple following \2), wnere Ll1" longtt.
of each word o~ the eenteace wa!l combined wit!: die length
or the sentence . According to axioet 3 t!lerr sll11ll be :>o
di!!erence between s combina;ion 01' t'r:o or:e-di~1cn~i or:nl
random variablos and one ti'lo-di!Den~iouul r'andon. vnriuble .
Consider n combination or joint dlstl'ibut t or. ( t, 'IJ ) of
th~ two variable• C !U1d 'IJ. The di~Lribution of ~ •lLllout
regard to ~ is called a margi.nnl <!l stril>ution re:ut~ve to
t or the joil:t distribution ( L 11 ) . It is i dectici.l ,;ith
the distribution or C. Sia:.ilar·ly, che narginel d~c:ri bution relative Lo ~ Ls idencicnl with. cbe clistributioc: of
'L
'l'wo variables C Wld lJ have two probability funcL.1.oru;
P(S ). p( CES) and p(Q) = p( 11EQ) . The probability function
p(( ES,11 EQ) :epresenLa tilt> probaoilitj' chat "' aeasurement
yie
set
c~
~.
;;te re,,ult t o! tho seo:; S and th() resclt 11 or •he
Two new runct.ions can be defir.~d:
)'l( ll EQ ltES )
p( ( ES) > O,
(5)
p( 11 EQ) >
The function r( 11 Elt ltes
is called tbe condit~onnl pro-
l;abilit:y o;; obtaining 11 of the sei; Q, i f t belongs to the
"ec S . The distribution define d b;; p{'ll EQI ( ES) is called
bho conditional die~i·ibution of 11 rclati ve to r.he condi-
tion t E S .
fo'or l::lll. exaniple 1 l\1L ~ rep1"esent the lent;lh o! l-tords,
1'I the length of tne se:>ccnces contao.nini; these words and
let. us consider th~ ~airs C, ~ - ktong nll possible words
those with lengt.h ' > 1 are chosen . 1his set co,,i;ains all
pairs C > l , ~ . Consider now the pairs for will.ch the length
of che sentences lies be~wci:m L, and I.1 , L 1 :I '1 ~ L 2 • The
froquenc:y of occurr~nce of seoi;ences witn this length in
a text depends on 1, sin~e a very lon~ word can only occu1· in a sufficiently long Henteocc . Tbe cond it ional probability of '1 having 11 value between L 1 and L 1 , i f ' is
lari:;er t!:an 1 , wlll usual~y d.iffe:::- l"ro:n the uncondii;ional
probability of 11 J1aving e value bet..-cen L1 and L 1 •
Lee, on tl:e othe1· l:.nnd , the p6.i1· ; , '1 dcno•e the length
cf cu"' word ir aud of the sontence " iu a texi; . 'l'l:e conditional probabilJ ty or u ceri:;ain length o.f a sentence
l..1 ~ 1 ~ r.,, if ' > 1 , will usually be eQuul t.o the uncondii:ional p r obubUit..v of a certain lonsr.h 1 1 '.! 1J ~ L ,,
!lince the lengtl" of woi·d le bas usually no 'bearing on l:he
length o r sentence k .
'!hl< exrui:ple leiuln to an importa::t special case of co:r.binntion of statistical variables : statietical independence . Let the :ro~lowi.Dg product bold for the sets S and :<:
p( ( ES ,'ll EQ) = p( ~ ES)p( 'lEQ)
(iOJ
167
4 .11 DEF::H:'.'Ior:s
one obtains from (5) and (6) :
p(11 EQI CES)
J'l( 'IJ E::J.),
p(CES) > O
p(CESl 11EQ) - r( CES),
p(, €~) > 0
(7)
The conditional distz-ibut;ior. of ~ is in t;his case independent of 11 and vice versa . C and 'I o.re called •~atis
tically independent varieblea wia t he: rrobubilities r(t ES)
o.nd p(11EQ) nre cnlled "i;atio:..ically independent .
Let us assw.e that (7) rather tbar. (r,) is ~rue . Subs~i
tution of (?J into (5) yit>lds (G) . H"nce , the cqui.~ions
(7), or more JJl'"Ocisely each cue of tne L•,.,io cquationn , are
noceosary nnd nuffic i ent condit ions 101' acat ist ical l nilopendence .
Let us substitute the distribu~icn func'tion '.; fo: the
probability function~ accordlng to(~ ) :
W1(X )
p(,~x)
w,(y) .
p(11~;v)
W(x, y)
(8)
= p(,:r.x , ~~y)
Equation (6) nesumes
W(:ii.,y } = w , (x)W ,(y )
i;~e follow~ng
for:n:
('))
This equation is necesr;n-ry wid ~ui'fic! ~nt for 8totiut.lcal independence of •he variables C wld ~ . if &he sets 2
flJld Q are defined by interva l:J accoreing to ( 1) . However ,
it can be shown that •nia res Lric•ion is wmcce•stirily
narro•,, a.nd chot S und Q rr.ny be Borel ~ete .
Agai.n let the leng•h ol' wordn be meusured by the nu:ober C or letters . Instead or c consider 11 • ~(0 • '' . rt.c
function '1~(;) in ca2.led a functio!l of th<> rando1t variable
c. Lot a gencr•ul func•ion 'll( t ) be B- t>easurable 1 , real ,
finite and uniquely defined for all real t. 'l'hc .runccion
1
.. ~ functior:. g(x) defined for 1111 ele:nnnts ;; of n set S ~n
uurel- or B- meaaurable in S, if tbe subset s of all c'etient a x, for which holds g(x) ~ K, is a Borel set r· r nll
~enl K . Bence , the values or the variable ~ muot be the
alements of a llorel set .
+.
1 88
1
'II \ ~
:J~'A'l'ISHCAL ·~;JHABLES
:~
then a rw:idom •:ar;able haviui>a ci,.tr;butior. funct, ioH dci'iued by Ll:c raudom var·in 'o le C.
l,,.nt: ~ :lr•note ri t::ec: cr-r~aining "' 11:nd S A set conta1~ne; ' . Tho random variubl e 11 shall be>long to Q l,ben , Md
cdy ther.. , E C b"lon.:;s co S . Le't ,,, ($) d~no-i;e t:he probabilHy of ri belonging to s, nr.d p 1 ,>V the proi>a1iility or
<l.
11 bolonginG co
r,
Jt bo>lr!s :
i ) = ~' Q,
or
Subdit utlon
yieldu
'1i(y,I
~hP~e
p
s
2
(
(1J)
the dio:;r i1Jution .runc Lion accordi ng
!'llEY) • p 1 (Sy),
to
(11 )
Sy i. ~Ce r.e~ of all ' !or wUich hold£ ~(=) ~ y .
4.12 Density Function, Function of a Random Variable,
Malhematical Expectaloon
"he He' r·ibutior: r11nction 11/(x) lUtS been defined in(:?)
by
W(x)
= p(,~x ) .
~hot
:;111;: ctcriva-:;ive '"'' ' \.X J t;xini;r:: tor all _points
>: . Thr~ de1·i \tat;i._ v,.. is cul Lee Ji.s-:r·i·uut.ion density or dencit 1 f·.inct.;lon :
Aseu.:ne,
(12)
w(x) = W' lx)
' i ...
c~:..:le:l
o c ... 1,_lr.1:...,~~ ::-and.o:r.
variab~"'
1-n t::U.e case .
A1·undom vliriaole 'with d i stribut ion f unct i.on W, (x)
~t·n:i"for'l!<•d
i:.!to
~
is
·.el< r'!r.iou. C"Ariable 11 - u( - b . ihe diE -
rribur' ou !'t:nction \., 1 (:;) o.f " i s obt.._L,,ed as follows . 1'be
coucHtio!l • " ;1 correspond~ l.o ~ '1 (y-·~)/~ foi· a> 0 nnd
-o ' \'! .Y-b /a !or a < O. '!'r.e distribt:tiou fwictior. w,(y)
i:; obtained froa1 ( 11) :
y-h),
•,I
3
> 0
( 1:?)
a
'1!2 (:1) {
1
-
•,; I ('l'-Q b) t
'I'!-.in .fo:rmula is
correc~
"<
0
for a < 0 1 only if W1 (x) is con-
4 . 12 DJ-:NSITY .i't.rliC'I' 1011
tinUOU9 st x • (y- b)/a .
shall be dcteroined
-ohat
<'-O
ci3conti!.",OUS l·Oir.La, W,(x)
At
~'"'
function is continuous to
the right; .
The den<>ity !"unction
1<
,(y )
in o':>tnine:l !roll ( :5), i!"
r; (x) is diircrentia':ile for all vnluell of x:
1
( '14)
W' (x)
w1 (x)
'
Consider further tho function 'l = ' 1 ' ; i • 1 , 2 .
Ther e are no Jlegative values of 11 and 'W 1 (y ) cquuls zei·o
f or 71 < O. The relation - ;; y yiel<l~ .!"or y ll 0:
-st
- yllli
'Ihe
Iii +y''21
distribu~ion
f;mction ',.' 2 (y • for 11
ottoiIJed:
J.'
must be contin'-lous at x = - y'" . Tit<' deu. lty ~unct:.on
w2 (y) io obtninud lf w, (xj is 1oif~,.renLi1JbJ,c ror al! va-
'tf1 (x)
lues of x:
w1 (y) = Wj(:y)
C
y yields C J!i y 11 <»·1 > for
Hence it hold 3 :
~
\lz(y)
D
,.,(~)
a
•
\II(;! 1112• •II ), - '00< J
1
21 -
.,.J,JH>o-11 0
1~
.
'1 = , ... ,
;
i
<=
1, 2 ,
( 17)
(-llt>:- 11
~
Let 'n'(x) be the distributior: fw1c~ion of '
=d F>( ; )
a certain function . The follo>:ir.F. Lebr"gu" intee;r"-1 sha:l
etist :
E(g(,)) •
..J
g(x)dW(x)
( 18)
E(g(,)) ls the mathematical expectacion of tlto 1·:llluo1:1 v11r iablc g(,) . Equntion (18) ·oecomc:i a IH~1n~nn int,,i;ral i f
W(x) is difforonl;;inble for all x, W' (x) • w(x) , and i f
~.
190
S!ATISTI~AL
VA:lIJ,BLES
g(x) has at 1.1ost a fininc numbe-r of discontinuit.ies :
Mg(C)J
=
""
j'g(x)w(x)d..'<
(19)
- oo
Fig . 83 nho•·s the Bernoulli Jis::rib:.ition a~ ru1 exai:iple
for di"crctc di!lcri":.ueior.s . 'l'he pro·:>t;.bility of ' being
equal to x • 0 , 1, ... , 1 i R giver: by t!:o equs:;ion
(20)
&'or any set S noc contain.inf". on!' o.f the flO.i.llt!l x
0 , 1,
.. , 1 ltolda :
':'he
~stribuUon
The Bathemot'lc1>l expectati<ln
t...-r•
E:g1,C)] -
(~·} :
functioi:.. ',,'(xJ follo•rn from
ln
repre~cr.tod
by
x)q• (1- qj"' (!)
the awn
(22)
g(:x) hos -co ":lo defined nt tJJa points x ~ O, 1 , .. , 1 only .
a
1J
-J
I
I
-1
0
-1
I
..
I
1x- 3
I
';<
~
3
&
I
·-
1
a
Ill
-3
-2
OS
0
lx- J
Fig .83 (lo.rt) Pl'ObabilH.Y riwc~ion and distdbutlon func tion of n llornoulli distribu~ed variable; q •
1 = ~1 ·
Fig . 81' (right) Density func~ion and distribution function
of the product o! ;;wo Gauss dis~ributed variables .
*,
191
?is.84 sl.0"" on ex:mple or a
conlbaou~
di:.tributO.on
defined by the ~odified HU!ll<el function K.(x) :
W{ic ) •
p(C~ic)
=
tf
K , (y)dy
(23)
-~
lt will bo s hown l<itor th .. ~ ~hio cistri!:>ution is obtained
for the pl'oduct of v,10 V!lrl a\:>lno with Go.'~SBi91" rliwLribuLion. W(x) is diffe:-enLi1<"rle MC yields ihe dcr.nity func-
tion
w(x ) •
~K 0 (x).
{21•)
X,(x) 1.1pproachcs +~for x
U, ::ance Lne truiirrnt of W(x)
ia perpandlculnr to ~be x-axi r in thi s point .
Tllo probability of C 11ov i ng n ceL'Ltlin valuo rqunli< "'"'o
for cootlnuous di•tribuLioJI. :
p(,~x)
- 0
The matheaatica! expectatior. EC e;(' ) follo·•s from
1 ..
E[g(C)] • -
f
r; -oo
g( x )K 0 (x)dx
(1':) :
(25)
4.13 Moments and Characteristic Fun ction
Let g(O in (18) be" powei· oJ C, g(O = ~k. l1n mathematica.l. O>."Jlcetation E(' k / 1 ~ called tl:.e :nonent '1f o:·der· k :
(26)
One obtnins f'or the Bernoulli li."crt:>utior: (22) :
E(,k)
...
±xkqX(1 - q)'-• (;)
(2'?)
The moa1ent1> for continuouo di<J~r·ibu~ions f oll ow rrom ( 1'l):
E(ck ) •
°} xkw(x)dx
-
(28)
The moment o.f fi -st order is a.l.so ca!!ed aenn vo.lue "' :
E( O • m
( 2';J)
'I'nc
cioa:~r.t
l:[(;-c)']
a
=
--
'j(x-c/dW(x)
(30)
nrc c:il l•:d no,-,.e:-.r!: a':)oi;.:- ;;t,c roirt
nr"" t.hrt '?IO:tn:n-:s
is
ue~d
~·nout
i;hf' po:.nt::.
~
~
'l' ,e ce,;~ral mo:nents
- :rJ -;nC
t~e notation~
!'or t!le:r. :
:;((,-n.)kJ • f(x- :11) k(l\-:{xl
(31)
_..,
i-l o
• "I
lq
•
~> •
(32 }
E( C1 )
-
;,'
E{;'> -
;~(~')
-
2~ 1
lhc =econ' or:lcr ::;oneni; al;ouL a 1>0iut c,
l.((;-c)') = E"C -:rn::-c)
1
(33)
]
00
00
00
j'(:r.-:n~2 c'I\ • 2(JJ- c lf(>:- 1:1;dW + (,,- c/ ,'d.W
-oa
c
j.:
2
oo
i
0
I·
( ;ri-C )
.bus :. tr• mi nj ~au:,- Jn r c
2 7
~
U
-oo
2 1
1:1 .
co_'1t.uir:ir1e- Letie~~guo intee-.rals are writ!.en C-XfliC~tr::ly ror· J.:_SCL'tLe <.li.: ....L'.i.lH.lCJ.OU!'i . !iet Px denote
tl.e lll'U\.•abEiL;. of ; ;;ssW:1h1,;r. Lilt! value x . '~he :follow1ng
TLe
~qu.::itio::.~
gcnc-ral t•cla;io::n are obtained ln:itc~d o!' (20; , (2") , (22)
nn<! \c
~•:
(~)
00
l::[E(C1l =
L g{x;r,,
)( :::
00
lc;·1•1t.iona (7')) nnd (3'1) yield :
(35)
4 • 1 ~ i10i1Ell TS
19;
The mo1nents are f1•equcncly "'ell suited Cor the di scussion of a Ciistribul ion . 'ihi,; holJs ; rue ;articular)y l f
the distribution is obt1~ned cy :i:easurenents rat!.:er ttr<n
defined by a. Sil:lple 'lfl,l,nical di~tribu~ion f'>mction . 'ihe
moment of first ordeL', m, cbarr.cter·lzes the location of
the dist-ribution, sine~ t..te tto:ncnt of second ordi'.'r hus i~s
minimum about tLe poi!lt 3.1 accordi.Lt.:; co ( :.;3 l . :he moa.ent
of second o.t'dcr , .J. 1 , ..:!u.r!1cte1"izc5 t!.:.e conccnzra.t:.on of
the statisl;;lcal va.ri<1l. l n. a r ou.'ld the mean m. •rnc uccou<l
order ooment ill also cal l ed va:riance or mean oqu1u·e deviation and the not~~ion
io used . o is callod eto.ndard
devla.~ion .
All moments of odd order va.'liSll if i;hc \li.,~rlbui:ion is
syma.etrica.l sbouu t!.e ccan :r... . Btnce , u 1 chttru.ct,erize~ the
deviation !rot1 sy=e~ry. Tl:e coefficie:H;
Y1
u, /o 2
(3'/)
in called coefficient of sr.ewr:es:'.
The mathematical '!>Xf'eCtation o!' the special ,-.._.,ct.io!l
cxp(i v' ) is Called ch.,ro.c, eri !lt ic function or the ra.nJom
variable'; v iB rea l :
..
,...,
cp(v) ~ E[exp(iv,)J • _Jex;•(ivx.)dli(x)
(~)
Let W(:-:) be difforentinblc . <;>( v) le then the ?ou.l'iH trnr.s !orin of Vl(x):
00
;>(v , ~ Jexp(;;.vx)l,•(x )dx
--
(5<J'
There ie a one- to-one correspor.J.ence between a di<.L. l -
bution func~ion W(x) Wld its chru•nctel'istic CWlc tion o( v ) .
Two identical distrib1ltion function:; yield ~wo fdenticoJ
Characteristic .functions and ~... ice versa. C&lCUla;:ion!: tt.n-v
bo done with characteristic functiocs rnt?:ler thw: >.iL~.
the distribution functions; this is sometimes eeAlor .
'!4cre is a complete BlHl.l.ogy to Lue use of the Fourier
'r lnsform in communications .
IJ
~... Trantn'lflelOn of fnfonnat:M
4 . STA".'IS!'ICAL, VPJiI!dJLES
194
4. 2 Combination of Variables
4. 21 Addition of Independent Variables
Consider two random varia.bles '
an:i ~. having the di.ffe-
rentiable dist~ibutio!l t"Q..11ctions i,..i , {x) a.nC. W 2 (y) ~ ':'he ma ...
themai;ical cxpecr.sti.ons of the function,; g , ( O and g 2 ( '1)
are definerl by tihe fo.ilot·: ing inreg;rals :
=
J g 1 l :< )t< 1 ( x )dx
( 'l-0)
"'J g,(y)w , (:;·)Cy
(41)
-=
E(g , ( n)] =
-~
j,et
from
C
!l.lld
n
be
~tetistically
indeyender.t . Lt l°ollo»•s
(9)
crn(x , y) - w(x , ,' ')
axdy
"
w 1 (x)w 1 (y)
The ftmction
g(C , ~) =
s 1CCJ - g,lnl
(43)
yields the uiaLheme Li cal expectai;ion E[;.;(', n ) ) :
E[t;(C,T])] =
__""J ..f..
(4.fl)
g(x , y)w(x , y)dxdy
J J ~g , \J<)
-oo - co
~~
+ g , (;·)Jw,(x)"',(y)dxdy
1)(1
(IQ
00
(:II)
-"'°
-.:~
-co
-oo
J;;,( x }.> ,( x )dxf w2 (y)dy + ,\:: , (y )w, ( y )ayf w (x )<be
1
Tbe expectation ol' the :-ium of the r·sntlom vai•iables g 1(')
and g,(11) equrr:s the s= of the cxi1ectai;ions of g 1 (i:) and
g 2 ( n) . '!'his re,,ult st ill holds if the asswnption of .statisc-ical i ndepenQence made lte?.~e is no;: satisfied .
Tle fur:cti o::
yields the expectuuioc E[h{, 1 n)] :
1 AJ)DI1'ION 01" VAI<!AllLES
4. 2
19;;
coco
ff u(x , y)1<{x ,:; )axd::
--oo
....
( 41'.;)
E[h{,,fl)) •
• J J g 1(x)g 1(yiw 1(x)r.1 (y)dx<!y
-00-00
...
~
• J g,(x)w 1\ x }dxJ' i; 1(yj1.- 1(;vJd;v
-oo
-oo
• E[g 1(C)]E[g 1 (11)]
The e:iniectr.1L ion oJ: the pi·od uct o f t;iie .C'tJJldom V<l!'i3bles
g 1(c) and g 2(n ) equal~ ~he p1'oduct oJ tl1e expec1iaHonr of
g 1(C ) and g,(11) .
The results :ibout sw.ns and J-·rocurts or l'AnOom vnriablc"
derived here 1"01· t.wo co!lr;inuo\.\r-' vari9bl~n nl!lo 'li'T•l;; to
.more than two va.riab~es and to non- continuou3 distributions. . The expectat4on;:-; for di~crcto r!i.stt"ibutio11s htive
to be calculated according to (;~J .
Let q>, ( v), q> 1( v ) aJ>d <0\ v} dcnot<> the ci:a..::uc Le~·istic
function a of ' , - and C+TJ . Substitution o!' g 1{ C)•expl iv\;
and 61('1) • exp(iv111 into (46) yield8 :
q>(v ) • E{eicp(iv(,+ri)J] ~ E[exI>(iv- )exp(i \I, ))
• E[ exp(iv11) ] E[ eT-p(i v, )] =
q> 1 ( v )w 1 ( v)
The charactcristlc 1'uncti on oo· th~ sun of "tnti:;Licall;v
irul.ependont r·undom \'ariables equal,; Lbe pJ'oducl 'Jf ~lrn
charactaristic functions of the va:l'.'ia.bl<•:;.
It is koown from Fouriel' anal_y~ls t:Jlit tl10 invcrnr: o :(39 ) is t~e following integraJ :
1t(x)
=
..
~ S exp(-ivx 1<0( v1a,,
-..
(11a)
Deno'te tho distx·ibution function of '+n , ' wod Tl Ly 'ti( z) ,
'il,(x ) and W1 (y) and tho densi•y J:unctionr: 1;y ..-{zJ , w,(x)
and w,(y) . The in~egrals in (38) , 09) lllld ('18) may br.
integratea fo!' oimpl o r unctioni:; and yield t.he density !'unction w( z) . 'l'h.is ret;ransl'orma·tion of tho cl1oractori stic
fw:rction ('1;7) into the diatri bu ti on function 1;an ulLlo b<>
done in a genero.l form nnd y i e l ds :
...
..
I ...
~
. W1 (z -y )dW 1 (y)
W( 7.) •
S'l'A'J' I SrICA:. Vft.R:ABU;s
I+ .
l (
z- x )d ... I ( x )
One obi;ain!" !rc·o (L'S4) -the !"ollowir.g Rir::nn...'ln i-:rteg:rn.J. 8
:or
~i~forcntiable func~io&JJ:
00
00
• _..,J w, (z-y ):, , (;; )cy : J w2 ( ~-x )w 1 (x)dx
W( z)
(50)
- oo
J w,(e - ;; )•,i, (y)dy: r w,(z- xJw , (x)dx
..
w(:1, \ •
00
Denote mo111s 1 1'11.'iances und 1:1CoU1<ll.~ U of tb ir·d order or
Ll1e dist.:ri~ui io!~ func tio11s \..'(z) , 'r.' 1 (x)w1J 111' 2(y)bym , m1 ,
m2 , n 2 , o; , r.r I , UJ , u' ~> !:llltl ,J' ~ 1 • Ec'..lo t.1011 (1111) ;:fields for
g ,(, J
a
'
ar.d g 2 (n) ~ n =
(51)
Equatio1'S (52) . (;£) =d ('•7) ;;ield :
c: •
c:
= Z(t 1 )
-
~f
- E(n') -
• E' C' > - E'(CJ
m~
+ ~(~' ) -
• E[(~+n)'J - 2~ ({ q)
(52)
~ 1 (q)
E'( , ) - E'(n)
-
• E[(C •n >'J - E'(C•n). o'
'!'ho following relation is obtnlnrid in n simllru· way :
(53)
Cont$idec• as
Gnu~~inn
rut
exa.;nple t.wu \n...1.r:·iub!eH ' and Ti baving a
dlBtrl buLion:
(54)
·•,(y,
'.¥ 1 (x}.
•~1
• el'f(~))
;c
\!Co,
ei-I(u) is the t"b:llateC.
~ri'( 11 )
1 r f ( -11 )
2
1
W1 ( •,, 1
1.i'l'Ol'
•
'" [1 + errq:i::o7
'"'>J
li
IWlction:
•
(55)
yr f
•
- orI( u ) ,
erf (co) : 1
'l'ho cnnrn.cte>ristic Iunc~ ion <I> I ( \I ) 0 r wI ( l<)
.follo'•S
..-. 21
from.
197
ADD!'l'lOh OF Vi.111AJ.IJ.J:;$
1 ~Q \:
ifuo (!UbstiLution v
<;> ,( v ) • v?Jna
= (
io: t:n )/\[2o 1 yields
" ;:p(ivm,-+i:v 1 )jexp[ - (x/1{20, - y/]dx .
-oo
I
llsin., (55) and
&J1e
s·~lsti tutioo
(x/'{2o 1 - y /
• ~
yields
The chru:acteristic function <1>( v) of ~- c follo>is from ('17) :
Compariaon of (5'7) and ( c,,J) nho>is Lhat -:hr> .!JWll ' I - :nust
have a Gaussian Oi5tr:.but.ion,. nince tt.ere is n o.:Je:-t.o- ono
rel ationship bet~een dintribution "'u.nction and chornc~t:?r
isi:ic runction. De!!Si i;y !unction "'' X) and die tr! !:>u:;ion
function W(x equal v,(x) end w,,x) oJ' (511J i f ~., i" re pl aced by m = m1 +mz- ana a: oy o 1 c o ~-a~ . Su:r.r.ii ng l .tu<le panden~ Gaussian vsrif!ble~ L'litl1r<r zhuc two , ~gqin .vield!i
e variable with Gau ssitUl di.ncribuc i on hav ing th1> mean
m
=
I
...Lm,
and tbe variance
a' •
I
l:
•••
02•
'
\58)
It can further be Hhown t•tmt th" sum o! L i nd•;r•rnclenL
va 1•iablea approaches a Gm1n!lirm distri bui;ion J'or l<u'@:e
Valuos Ol' 1 i.r the varinlilr:i do not have a Gauasinu distribution . '.i'his i s the centru.l. limi;; t!::eo!'em Of St11.ti'1t i cs . It holds under very goneral assumption-a. l'l~iin and
'' &riance of l:he distribution arl' equal to the sums of the
"-<inns and variances of the variables according to ( 51)
onu (52).
198
' I • S'.IA'!'IST:CAl, If fa_qIABLEs
4. 22 Join! Distributions of Independent Variables
tlce
As " more <:(lmplica.Led exru:1pl<' of
dis ta· ibt~tion
of
the sum of two ralldorn variables conside!' the followin$'
problcffi tf·.at Nill be encoun\;ereO in chupter 6 . A variable
~ hcis a Go.u.;;~iru:i distribution wi~.:i a:ean rr.
= 1 and variance
:-rz , $ second variable Tl has also Gr~u:Jnien distribution
\,·i th :neao OJ = 0 ar~ct variance oz . The Ointr:i but ion function
oJ' {; - ITJI is «anted . Thedensit;y f1.tncbions w,(x) Md .,,,(y)
o!· ;;he variables C a.mi 1'11 are
.,, , (x)
V2Vrro
1
exp [- ( x- 1 ) ' /2cr ' )
=
')
w, (y) = .~exp (-v 2/2o
yc•/Tl'O
'
- ·:.X:: <
2)
(59)
;.:<;.00
O~y<oo
y
< ()
x-y ,;hall yield z . Hence, r.he following rcli:niioo must
l1old i'o1· all values ol' y :
x
=z
(60)
+ y
rL'he C.ensicy lu.ncti()n w(z) and distribution function \\r(z)
C- ITl I ai·e given by th" f o I l o.ii ng ;iqua.tion s :
of toe v"riabl e
w( z) :
'f exp r - ( z +
Tl~ '
y-1 ) ,, /2c 2 J exr ( - y ' /2o ' )dy
( &1)
0
.:
W(z)
1
~
exp [- ( z-1 ) ' /4o 'J ( 1 - erf,rZ - 1)]
: f·. 1(7. ' }uz
20
l
1
r(1
n •
t
=
- CO< z < c;o
'
- erf( u ) ]e - u du ,
U
a
-ro
-i»
z '-1
20
i [1 + er f(z)] + t[1 - erf 2 (z) ]
:.nt u!1 fi:rtlle1· calculs~e Lhe <iensit;y function of ' - ITJ I
~ O :oust be satis!'ietl . w,(:x) is def:i,lled
i~ t:,e eonoitio.!'l '
as
fo:lo~,·s :
1
w, (x)
"/ 2'lr.Ca exp [- (x-1 ) /?o 2 ]
x " 0
w, ( x )
0
x < 0
8.
v;Jr.o
(62)
2
jexp{-(x-1) /2o 7 ]cb: = t[1 .. erf( 1/'{2<1) ]
0
{. canuot: be smaller• cl1an z.er·o for non-llegativo values of
199
4 . 22 JOJN1' IJlSTFWurIOl\n
, _ 1r,1; henca, (61) holds fo=·
z,"
o , \Jut one l:ae tor.iulti -
f~Y by 1/C .
The nmal le:l:: p~ra.issib : c ,_,.a.:_ue of y for z < 0 l!"' r.o-:
~ero but -z ,J·:e to x - O, as ctay be ~een from {60} :
y ~
-z • I z I
fo1·
i
~
0
one obcoin.i int1Lead of (1;1 ) :
w(z ) •
=
~.{ox:p(-(uy-1 )'l2a~] e1<y1( -y'/?al)c1;y
~
o)(Jl[-(z -1 ) ' /4 o'J[ 1 -
ei•rc-t~ 1 )
(G;\)
z
~ o
The diffet·encc betwc<'n (6'1) and (63) is ~ho difr~:-ent aifYl
of z in the nrgumer.t of Lhe error :unction . Tr.is JJakes
it exceedingly difficult to co:1JpGte th" <!i :otributior. fun~
:ion W(z) . As n coz:sequence, tee ;;robnlnlny of C - hi~
c ;, o, being si:io.!le:· ;;nae zero ·•ii l be calcul1oted only .
This requires integratio:. o:: •:(z) free: _.. ~o O. It ,,ufrices to integrate (63) , since (61) holu!? !'or·,; • 0 only :
W( O) =
~
Jexp(-(z- 1)'/411 2 )[1
t:.y
- oo
- arf( - '· - ·1\ld::
II IJO
• 1 t 2erf(1N?o{ - 2erf f 1/o) 2 1 + er:l 1 V2o ) ]
r.0
err' (1/2o)
The integral.ion (6'•) i s very cwnber!Joma . It
ed by KASACK
by pa1'a..'11eter
(6i:)
w111:1
accomp l ish-
intei;,ration . O:>e 0ubstitutes
first W(O) • W(O , s) , s = 1/2-; , t:Oen u • -(~.-1)s . :t. follows the differentiazion dW(O,s)/:ln, tt.e oubsiit•~t:ion
w e f2(u-a) and an integration av~r s .
Consider next the d~stribuc;ior: of l~.e product ; -, o!"'
t:wo stacistically inderende::t cont.i!luous vu.t'lal:l~s !.aviq;
density functions w1 (x) and w1 {yJ . The density ~=ction
or the joint dis\.ri bu \.iou follows rrom '42) :
w(x, y) • w1 (x)w 1 (y)
The probability of a po int with coordinates ' and '1 lyi.ng
in ~he F.1.Coo o l omcnt llxdy e quals
w<x,y)dxdy
w 1 (x)w 1 (y)dxdy .
(i>~)
200
The product ~n wL l
xy equtt la z :
hav~
a certni n valuo
~
if Lh" product
(66)
t. certllin \'nlue of :
anj' be obtnl.neC in t.,.n ·,a:rs due t.o
t ho relacian
xy
(-x){-y) -
c
z,.
(67)
'ru~
Lrnn•foraati ou o<' r.tie di.t:fcron~ial dy oecoitos ambi5U""" . In order ~o o::ilcc i : ~que , one :my distinguish
che t;·,10 ca..ses x ~ 0 U.."ld. :x < 0 . Giv<!'n a. cer1':t"J.ilJ ._,alue of
z , x r.iny assume any value becwef!;n 0 and
or - oo and o ,
providoil y ha" tha vul ue
l')IO
:; =
t
x,
x ~
o:
x < (',
(68)
~~e
diffe1·cctial dy i.s tra::isforli.ed into
dz
d:r ._ dxz , x ~ o;
dy a - -;r, x II 0
(69)
The probabil ity or ,,, lying be Lween z and " 1dz i.i' C lies
between x ~d xf...dx onJ :t Tl lie.a between y • !: o.nd y+dy•
x
z+d.v .
---;::- .l" gi vel! by the followi!lg ~ rod·~cts :
w1 (x)dx w,(:)az
x x
"• (x)<lx "'
i<i>~:
(70)
x " 0
x <
Q
H 8.,C!ll!> l'e><SOnubln 1:0 integr1U;e the firHt product .from
0 to ""'00 and -r;he second fro211 - oo to O, since x m.o.y asswcc
all value a ·~c~wee:.. -oo a.."ld = · Jlo1'<>Vcr , the i11~eg1·als mny
not conve:·ge ai; x • 0 d ue to the fnctors :'. . A oe rt ain interval of widtt 2• C>.L·ound x - O ii; le.rt out nml the limH
of '-"·~7~,, for ~ .... O ic invest~gate<l in each case :
(71)
Eque.t iou ( 71 ) may be replaced .for L>ven r unctlons by
(72)
m,
Let. C nnd Tl !l9V
61 Gaussiau distribution wit:b meanu
:n, • O and v ariance a of and
Equat ion ( 72) yi.elll.S :
=
o: .
201
4 .22 JOINT U!STRIB;Jnor;s
s (
~ ' )"-"T
- '= ) ';
2ii 20,01 ..
exp - x l l<o,
-· x' 2~,l )1-xdJ<
~
~(·
) ~
.. .,
The sub~.itu~ion ~
·•( z)
= .,..1
, .. a,
=x
1
{73)
/~f is ~ad~ :
a .,'I'exp.r -•• ( z 'I--cr,' o,1
z r!loz
•S _•~cs
~
I
'l'be ;Lntegnl
..je:x:p(- .v1. cu'e +
. 11111,
. )
8 )) 1
36n = in
• ,iu
0
is tabulat;ed . il~ll(ill) iz: a Elinkel flrnctton 11111.l K 0(1.1) 1 r:
a modified llnnl<el runctio=i . L quut .ior.n \'/I< i nnd (75) ,yield
for c - 0:
W( ··) • --1.....K ( - "-
no,a 1
,&,,,
( 76)
° :J,01 J
It followa fro:n ( 711) tha'.; w(:z)
i~
"ven :
(77)
w( - z) • w(z)
w(z ) is thus defined for a:l real z .
Tbe distribution l'unc~iou
W( z )
a
1-
-
TTC 1 0 1
fK ( - u-
-oo
0
01az
(78)
)du
cannoi; be rodu cod to tabu l1rce<i functioJJe. .
W(z ) and w(z) if one sub•tii;ut~s
l•'ig . R~
shows
(79)
and
w(o,
-L
) 0'7
01
o1 ·•(x) ,
W(-
o,
2
-
02
)
= W(x)
.
( 80)
The Rayleigh distributio"" is iiaportan; Co:· ;>:·01::.,:ns
involving .fading or narrot-.: bane noi!1e . Dcn~ity f1..;.::ctio.n
!llld distribution function of a vo.rinblo ' >:ieh R;,.:;-leign
distribution at·e defined as f ollow" :
ff exp(-x 1 /6~
WI (X)
)
x
!;
0
I
\.(I (
X)
r
0
W, (x) • 1 - axp(-x2/6 I )
x < 0
x
~
0
( 81 )
202
..J
'1'he "1eru: '?quala
E(CJ =
~
r.i.
°" 1(x}d.x
' 1 orde :·
..J
oncl -che
x
!:iecorj
rr.om<:nt
(82)
ectuetls
;! .
x 1 ·d 1 (x)cx -
•
= o'{n~ ,,
(83)
TL('> vai--iw.H.:•" o ~ !~ollo~·.s .fro:u (29) , (32) &ud (3L>J :
a:
E((.') - E'(') = 6 ~ (1 -
=
tn)
(84)
Let a vnri•ble ~ be indeµemi.e". t of < anrl ho.vi) a Rayle igh disti·i 'ou~iO!.. >1ith d•m:;il.v func;;ioc w , rv) :
y
if
0
(85 )
The Cc:-n~ity !w1ction w(z) v:f tile o:rctluct , ... .;Lttll be cc-tl cul,,ted . l::qut1 Uo n (71) i·a~bei· thnn ( '?2 ) mu st oo u•ed , since
the den"i ty fur.ctior. of ~he Rnyl eigh di2tr i hut ion is not
syn~etrical ntou~ x = 0 . ~$il:6 (91) and (8~) one obtains :
~
L
l
1
l::
6; ;;ind 2a ~
=
l );:.
:;
,
(I'his equntior
2o;
1
. x exp( - :-: /; 1
= E.'b '
1
l
l
1
1
-x
~Jr.>( -z /x ll ) - C.X
h
•
(86 )
identical wi:h ('i'.::) i f one cubst:itutce
= 8: into ( 7;) nnd uuH iplie~1 by
liz
6r ~ :no, cr2
T!.e Ci:!'".!li;;,- !unction o :
o! (76) ~
',., ,, ..
_)
=
/j z
6fli!"
I
Thr
• ..
l.i\Z
2
=
2• )
function
(88)
'
:r.e:i be rc-juc"d
<
(87)
4
6i'"2
1
Y.
froa (86) \'lit~ the help
K o,' ~
I '.} 2
d i~tri bu L io n
)
~-. !'olloK~
2u/6
2z1~1 0,
f
•
to t;.;.bulated funct.ioi:." . The ~ubstitution
J1Cl<1:'1 :
&1
1'h, (x )dx
2, 1c,o,
- t in
f
a
(ix)H~ 1 ( ix )d(i x )
(89)
The inte;rnl
fy[ J,(y) • rn.(;1)Ju..-
=
J\n'~'<:ua::
.. yH',"<:v)
is y_nown . "' 0 (,yj and t: 0 (:r) a--e Beese! fu:1ctions o: fir!>t
end secon~ o:·cJ.er (;;euoann Lu1ction"). Equation (99) ·oe -
comes:
bc F', 11 (lo) ] (':)1 )
Let c oppr·oncl1 zero . Us:;ing tne equat.lou
H 111
(ic) • -2/no
I
0 <
£
(92)
<< 1
one obi:ains
u,,, u',' '< t. > •
·-·
(93)
-2/·•
and
(94j
·rhe tex·m in Lhe l;rackets is non- !°"legotive fot• 1·e~1l po:;itivc
values of 7. .
Let us i nvestigat e whctne:r (94) "qunl>S 0 !'o•· i • o ·,w.:
1 Ior z • oo . With t ho h"l.I' oI (92) one o!.>~"'111.1 ror 1. = O:
lim ..l!L(- 11" '(;>i' )1
t- 0 b1 b?
1
61 61
_,
....!!..L
ti 1 6 2
!:0 hl.z..
•
c:Z
1
The QSyi:lptottc nrproxi:r.ntion
-l!:
11
(ix)
ii
\~nx e-x
holde for large valuer of x . Fne ;;econ<l te1-:t in (co11) ·:anishes thus ror large •m_.ies of z rutd or.•· obLuius 'i(oo) • 1 . Fig . 85 ahows the functions or (87) rud (':I'•) fo1· ;, ~
61
"1:
1.
The dietribution .funct i.on of th" eu.m : »1 o!' two iniellendent Hu;;leigh d i stributed var.iabl~s J.'ollo>Hj J.'1·01a ( 50) :
W(z) •
~(
(1 -
exp[- (z -x ) 2 /-6~]}xexp(-x 2 /6~)clx ~i'OO
('J))
'.ilhe lo\o/or lJ.mJ.~ Of the integral cqueln ZOl'O , Since t;he
4 . STJC lS'l' ll:AL V i\.IHJ..J!LES
1
densii:y runctioL w(xl = x "XJ' -x /;! ) has to be replaced
by w(x) = li !or x < 0 . Tile upper li:ui t i ~ ?, , since the
distribution £w 1ct i on W(z-x) n 1 - exp[ - (7. -X J' /6: ] has to
be replaced by O,l(z- x)
0 !oi· z - x < 0 . Sub:;t!.tution ot
(96)
yields ·,;i th the i,e:p of the integral
n.f ~r:r
len,.~t.t.y
t ransfo=t!ationr :
,,.
ll(z.)
z2
(97)
• - rXJ'.>l-b'°,
l
z' ~· - i[n ll:.1[erf(#J-)
- exp(-~)
+ eri'(~ 51
1<0,
.<6,
}.~ ,
l<'
o:
o roi·
~ <
o;
k'
=
z;,
o
o'I + o•I
A sinple!' formula is obtain<JJ tor 6,
'il(z) = 1 -
)JJ
..-z'
~ 61
•
1:
(98'
Consider ti.e distribution of the quotieut 'l/C or two
independeLL coc.r.inuou.s var1nb1.oo C and ~ 11aving density
fllllctiona "" (x) n.nd w2(y) . ~h" density £unction (68) multiplied by dx iutd dy is used again . The t·elatior:
l
x
=
z
rnt=.st hold i.f -;, j ::- to
:nay be obtisined in two
product ~' :
i
hav~
a cc ....~ain value z . :'!:int value
juct: P.s in the case of the
w~yo ~
(99)
::lt = z
-x
x
Lei; us cor.9i .J or the :;a.see x ~ 0 and x < 0 Geparately •
i:: orecr to 1tW:" t1'e c!iffereutial una.nbiguous . x rr:a:y aosu:oe a:.1 ·yaluc.oa between o ru1<l ,. oo or -oo and O for a certain
Vt'<lue of z , provided y has tl10 following vaJ.uo:
y
~
zx ,
x
~
O;
y =
- zx,
x
<
0
The diffei·entilll i,; transformed im;o
4 . 22 J0lii1' DIST!i IDtlT~O~IS
dY
x ·- ? ;
xdz,
p
2C~
a;; = -:x.1.i.,
x < 0 .
(100)
The probabillt.v o r n/' lyiiw; bet ween, un<l <.+dz , i f ' li cc.
between x nlll1 x+dx lilld i r - lies between , = x z ru:cl :1+d;;
• x(z+dz), i" giver: by t~ll :·nl~owbg ~roducts ;
w1 (it)d=< w,(zx)x dz
•,;
1
(x)d.lC w,(zx)(-x)dz
~ O
x
x < CJ
Thu probabillty w(z )d z of 11/' lyin;; ilat1>eeJJ z ~ml ~+d~
ror arbitrary values of " is obtained h;y intet>:radng f1'0:n
X •
-oo
to
w(z)dz •
.
X
+N :
-J« 1 (x)w,(zx):x
(lXc1z
-~
.
~
1
j w ,(x)w 1 (zx)x dx<l•. (101 )
For syo.t!etric !'ur.c-io::s one! aay w.rite lnr-itead :
w( zJd~
0
I
-•
(102)
2fw,(x;w 2 (zx)x dxcl:
z
2
3
16,li>t-
z61t6 / -
1
~
,
-3 -l
-L
--'--~~,-i'--1•
-l 0
•o;AS,--
Hg.85 (left) Density .functiol'.l and d-<>li'.Lblltion fmction
or the produce of two Rayloigtt distL·iullCO<i variable:'\ .
Fig . 86 (ri ght) Density fullction and dist1·lbution fu 11ctlon
or tho quotient of two Gauss distributed vi.u·iai:>le r. .
As a f'ir,;t exrunple lot '
a.nd 1 have Gaussian distribution with density !unctions w 1 (x) and •,(y) = w1 (x) or
(511 ) a.nci with me-ans m 1 = m2 • 0 . One o'btnins :
20'5
w(z)
..
= ~-~
o,
4 .
jcxp( -x'/2o:Jcxp(-z'x 1;2~:)x dx
1
1,01
0
1
.. 1
I
I
= ~x'C OT 1 ; , )
The sub st H ucion u
eq
w(::1
=
J'+~'"'
J1noz
l
~ .:....
I
~?.:-,
, C-z
srATIS'i'lC':..t 'IARI/JltEs
1
yfolcir :
(103)
J e - utlu
0
o' )l .. 1
:'he <'listd but Jon d.ef:L:c~a 'by r;llis dcn"it.Y .£unction is
Jr.Jlo wn an c.: ..u1:hy dirtributioJO o~ '"" S tudeuL di:itl'i bution
vii th one def~r·r.a ~f fN•adoo. . 'l'he :E "tr ibu:.lcrn funct i on is
fill inverne tb.llgec~ functlou :
. ~
'.>'( z) =
(104)
Fig . Bi:> snowe LIHl f u.acti.oirn or (1 03) and ( 1 04) .
A" a fu:.'tL"''' cxruuple 1.:onr i •~er I.he di::: t,r•ibullon o f t h e
quo Li em; -ri/' of cwo Rayle ii:;:-, <!i !"lc:ributed •:ariu oles ' and
11· Equation~ (811 , (SS) an<l (1•)1) yi~ld:
I
•h.z,
=
L
°:'x exp ( - x 1 /!1 ' ) zx
1
~ ..
'
1 0
( 1 0.c;)
·1·t.e uist:itutlon !'=::,;ion
.i;ti; ne2-p of tr.o i11Legr!Jl
I
II(~)
is
ot~a~ed
ft-o:n (105) with
1
- i xr+'f
(106)
F~g . a7
;;nows
the func-.:ion:; w(:t)
and ii(;,)
of
(105) and
( 1CC J .
Sewn·"l Joir1t distribution.\< oJ' a Hay!oiglo vo.ri able '
and a <Jau:.i~ viu·inble ~ " 1 t l be c alculated . The densi ty
funcL .:.o!l& o.r• n::: folloi;3 :
207
4 .22 JOIN'.r i'ISTRIEUTIONS
'
x
( 10'/)
il' 0
:
,,., (x) • 0
4
w 2 (y) • (2n)"'"c ciq( - y 2 /2c'} , - .. < x <oo
For tile computation of tile dizi;ribui;ion or the quotient Rayleigh ;·c·!!J.l>le/<.iauss variable conai.d<!l' tt:I! :le:isit:y .runct:ion ol' t:oe Go.<.«'S variable i;o oquol :;01·0 for· y < 1:0 .
'l'ha density fu nctiot. of ·Lhe quotient C/11 r.or.ipu l,od for ch.is
truncat:ea diatribuLiou holds fo!" !•11 por.i t;ivr value;; x/y
= z > o . The donsity funllLiou for negativo v~lue~ is its
image about tho orclin•t~ . One oll!;ains:
..,
~
zn;&l J cxp(-x
w(z) -
•
The substitution ;r
_ .L l{:>c
w( z ) - 2
6
~
2
/2~')ix exy ~ -z'x
x•
,,.
zZ
201
T
FT'
111
:.•ielf!s :
'[2a~/;
(c>o'z '16' +
'Ii 1 )x dx
z > ..,
n-:
( 108)
I
:;s-
10s
OD~
"5'"
.0
0
1
-\
7
-3
-1
-1
116,0,,-
0
1
0
I
'
l
l
'
1
3
I
ll'idii>--
:2-~
>g
~
' --1
0
z
•16,0,-
J
-\
-J
-I
~
t'fi.d/6-
fJg
. 87 (lel"t) llenGity funct ion and distl'ibutiou funcciou
0
the quo L.LonL of two Rayleigh dintl'ibutod vnrial, ei:; .
.
1'-.tg.88 (right;) Denulty and distribuoion runction of the
quo~ient Royleigh distributed varioblc/Gnuoc 1li.otributed
Vnriable .
11 .
~'he conr:et~
der.sity
.
(2clzl/f>2 ..1)ll1
n(z) •
+\'::>·
funcl~ou
'{2 lz c ib
..,
W(O) :nust equal t du'!> to
'ti(' l ia defi:ied by :
STA'r!G'IICJtL
v.~JlIABI,ES
is defi:.ed by tte fo=mula:
-:::0<
~
< x
{'109)
~he 8;)'lllllet:-y c!' w(z) . Hence ,
(110)
1 2otz
• 1<6-,-
2
+ 1 ;
-11l
z < 0
:!'ig. 88 nhow:; the f1mci;ions of (1 9J ruid ('10) .
Tllo den~ity =o d.isi:ri!>ution fu:icUou o!' tile quotient
Ca:.ae& v;...riable "Ra:.-lcigh variHLle shown in F:.g . 89 is ob~a1ned i~ a correspondi~g ~ay :
-
d'l~o >
w(z
•
~
fx ex;>(-x'/6 2 )exp( - z'x'/Zc')x
• i» ::. 1 ).,,,
l b ('
2 V2o I + 2 o >
dx
{ 111)
l(
~ z/\i2o
)
\~ ( ~! • 2: 1 + (6 ' -,,1 / 20• ~1 )1/l
C•'o1·
1. L.t!
con:.pul..E:tt i oz::. of
( 112)
the
d<1nsity function of the
pro<luci of a G1;.uss varial;le a.ud 11 Rriyleigli VtU'iable let
tlJ1 d 'n~i ty :-unctio!l o f t!:.~ Ga'.lss variable equa"i z ero for
v < • l'l.<i densii;y function col!lpUL<>d " itll tLis truncated
d.1.&L:· ... but1on a'1d itz ittage sbout the ordino.te yielo i;be
der.Blt;t :·u.-:c~~on for posi~ive and n'lgat1ve •1aluos of the
1·n.11do:n \."nri£itlo:i :
~
,
6
\I
•rtw
<t( 7.)
~a
sub~~i t
~
.'.'
.,l
"'
exp(-/?x 1a2 )x oxp(-x 1/6 I
.I
•
LtLions v =
"'
.
1 +v'
J exp(2ov ) Vv
\o.= ro1 '12E'o/5z
·
nbo
<lv
)~<ix
209
1;.22 JOWT JJJS'!Rilli.ITIONS
·I
I
1
~
.J
·l
~1
3
16111~-
I
1
3
I
1'~
l
t
1f!i6 0-
1t?.i6aFig. 89 (left) Denl5J.t.Y .;-1:.;:ct ::c :__ ru:C d:!'-:;ribut.i n tu... r-]o::i
or the quotient GauEs dis't_r:.bu~ed varinblc/Rb.;;:e.lt-,.~ Cir. i;ribu~ed variable .
Plg.90 (right) Density runci;ion ru:J d!st:-i\Jution Lwdion
of the product of a Gauss d.:.s;;ributcd ""'t u libyleign d~s
tributed vari&blo .
Using the tabulated integ1·al
j exp(-" '+v 2 )
00
2CiV
0
one obtaina
l<{z J = y2~6o-'{2z/6o
:::
~
0
':'he densii;y function holding !"01· ;..o~i i; 1ve ru.;I Ler;a;; iv"
Values of ~ t'ollowa froa the .:..equir·ec~nt of r.y:ni:ic-rj obolf.t
t: =
C:
W(i ) =~e-'f2io l /6a
v ~CJ6
-oo< z
<oo
'!'ho cl.iatribuLion .function i s aetineti by:
(11~)
210
'-
S·l'J<'l'ISTIC..U. VARlilLES
Fig . <)O Ghow~ ":h,. fW1ctionl! of (~1;) nr:d ('1•1) .
The tleusi~:; function of the !iUJ:'I o:· a
llncl a Rayleigh vartnb\P is given az t!:o
variable
P:>:az:iple :
Gu.u::H:
la~~
l( 2
(IO
1
1
'/ ~t1 6 , Joxp(-(x- .:) /2a ']-:< cxp{-x 1/0 )<1.x
( 115)
•
u-2
r. 2
z' 62
z.
2Vz{no oxi\-rr)[1 • exp( - 2o!Cjf',J • Vilei·f(,1 2~p)
pl =-
02 .. 20',
q'·
!»l-,Z-:; l
4.3 Statistical Dependence
4.3 1 Covariance and Correlation
:t bas been assuu;ed so far that tile i·f<Udoo variables
wer" st:atist:ically indeµendem; . Son<' of tue defini tions
of section 4 . 1 >11usi be genoralizod ic oi'<ler t;O be ai>le
to drop the con,1icion of statistical i.mlepeudence .
Gou.sider a din;riOution function W(x,y) of the ~~10 va.t·ia"oles C and T1• 'I'hc rinthematic.al exµectntion of a funo-
....
f .'
~ion g(~ , n) is <lcfinod by the i ntegi·al
E(g(C , ri) J =
(116)
g(x,y)rli'(x , y)
-00-00
2
d W(x :v)
Let W(x , ~,) be <liffer,.,ntiable for all x nnd •J , dXdy ' '
and Jo;t; g( :-: ,:; ) be co:itinuou" cxcert, nt 11.ost, at a fi,.,ii;e
nu:nber of pointn . ;;qua~ion (116) oay cher: be 1·eplaced by
a Rienann integ:-nl :
E[g(, , '1) ]
=
J' J g(x,y)w(x , y)d:xdy
00 -
(117)
-oo-
Let g( ' ' Tl) be the product of integer powers of
~ !llld
n:
(118)
E( 'k '11 J is call od 11 :noaent of order k i I . The moments
E( 'k ,,• ) and E( 'll ) nre ide,;tical with tho 110C1ei:ts o~ thC
one-dittensionnl oarginal <lisi;ribucioc of C and '1 · One defines on analogy co (29) :
c•
4.;
1 COVAHIAHCE AND
comu;J ATION
211
(119)
i!le point with t he coordi.nates' = 21,, Tl • m1 i:- C'al_ed
the !!!Billi of the t·..o- dimei::s::.onal u:. <1tri b•1t ion . 'l'lo<J :Looent.o
about the meo.n a.i·e callc::J
l>Y µ., :
>'>1 •
E((,- m, )" (,.-tt,)
cc!!t:.~a..l
1
]
r
"" ""
J
O.C:!llents and o,rr der.oLeJ
(x-:n ,J'(y-<'
,J' d'•(x , y)(120)
-oo - oa
Exp!lllSiOn Of LU.a factors ( x - at 1 ) ' {y-m 2 ) I lnLo µOWCJ'n oi· x
and y yields with the irnlp or ( 116) , ( 110) wid ( 119) :
0
? and- o ~ ere tbe variances of t!Je rJa.l"l:.!ir..rJ.l
distribu~ior..s
!or ' Wld fl · The i:soment µ 11 ie o!' int.ere:-t. h~re; i!; is
called mixed mo;r;ent or- cov=:'..ance of ' and 'l · It fo:lcwa
!rom the multiplication theore= ( 4&) and (121) that His
zero for statistically independer.c vnr-inblen:
u 11
•
E(,)E(q) - m,m , • o
( 122)
The mathematical cxpcctntion
E[[c, (,-m,)
1
c , (.,-m,)J'J • u 20 cl +2u 11 c 1 c 2 + 1J 0 2 cj
( 1 2;,)
i s the integral of a !unction ;;hnt .ia 11ounew1ti.vo und musL
thuo be nonnegative too . Hence 1 the right h<1nd cide of
(123 ) must be nonnegative . Let at least ono mottenl µ 20 or
µ••
or
be unequal zero . One "'eJ rewri~e the richt h:..JJd ~ide
(123):
~(u,1c 1 ... l..L.,cz >' .-(i;,,1.1o2-i;~ 1 )c: ~
~(u 01c, + µ 11 c 1
~ 01
)
2
.. (1..L,.;.t02-1J,',
( 12- ,
)c:]
and i..Lo 1 a1:e nonnegative .ro r the SllJlle reaoon as (12)) .
The eerma in brackets in ( 124) will be nonneo:ati·~e foi·
f1I'bitral."y vlllLLea of c 1 and c ,. if the following condiLion
holds :
1'10
u,...., -
...
1Jl1 ii 0
(125)
212
•• . STATlSTICAL 'IARL'<BLlls
A cor:-elation coeffic!.et1t
p
is d4"'fjnt:c1 0.1· tlJd i'o.:..: 0 _
'Jfine; equation :
- l
-:;::='
"'=·=·= V1.1,. µ,,,
~
=z
a,
T!.e ralntion:; : ' ;l 1 or - 1 ~ o ' +1 fOl l ow froc (12;.) .
For . ta.c1st:.c.~lly i..lldepend~r..t: VFlri.nl;les 'and n :'ol lows
p • 0 fro:u ( 1 22) snd ( 12•J J . :he inv1~ruf1 re I atio:.. does not
ho U ~--;t:•.tlernl l y; s tatistic al i Hdr-ponconoe 1.!ilJU1ot be infered
fl'OUI p • 0 .
A~·~i.;.me n I 1nca.r reli.Lt:ion.:;~_.:.f b0twr on
'
•
~
+ 3a
\X. 3 -
c ( T'l - rr 7
;
-..
·c
;.u ;.d
n:
p
Cr.\ nltair.:.:
..i10
µ 11
•
E[(,-n, ) 1 ]
•
a1 "' 01
•
l:[(C -i, )( - - :n,
auo1
"E[a'(r,- <1 1 )'•2n(;-:o, )(- - c 1 •-(6 - m,) 1
. t.,,-
* 2o. if - 1,
i:
"': [a
l
1
n u 01
1
(~ -u , )'
l\- ll .
• a.lµ: oz ... (~ -:i ,)Z
,'•\~-:I. 1 l( - - o,)]
nµo1
t- ( p - i:. • )u o1
0. ~....:..:.
01 _ _ _
_ __
S-tt,,'µ.oo
]
~
1
(127)
r' oqu11l~ 1 j'or 6 • r.1 1 • 'l'h~ r"•Bul L may be irivorted . Consic!r>r l' irat the case Lhnt noth µ 02 And µ 20 equul zero . 'l'hiS
a.cnm1 tli~ L Lli" mru:·gimu dist:riliution~ of Lhe variables '
n.nU Ti :..arc coucent-.i~ated in tl:c pointr m , anO rn 1 • Hence 1 'Che-
t.:No - di.cnn:-ional dist.1·il.uti o1J is concl'ntratr.C in the point
.X • !I.1
&nC ''S -=- ~ ; . T}-_e COV61·i:u~Ci" u, , • o/2 must \tanish
us a c ,.._..... quf"':lCc; . Uri the 't!:-,t:;J" hand , t.he 1·el t:a.Vion L. oz •
• u, 0 - ..111
v follo•" for a distribution concentrated
in th< 1 H.•~ x = a:, nn<! :; = "i · ~·tic dcrtni;ior. (126) for
~ Caru:lot be appl.:ec in tti!t cnnc .
I.t le••iOL one <Jf LLe equoL_onH (1;>11) oust hold i l at
l ~J 1 rL oao o.f the mornen Gs µ 01 o.r· µ~ 0 a.t.•e uneq\.la.l zero . Let
\.I.<• nqunlity sign '1old i n (12':>) . '.l'l:e !'ight hand side oi
(1?' l wlll equal zero i ! on(' of tho condi·UouB
(128)
x1 COVAllIANL:E Af;D
4.;,
213
CORRELA'l:o;~
is satisfie<l 11ccordiiw ~o ( 12;;) . I et =uo right i19r.d 'ad~
of (123) equal :.cro . IL rust l old :
( 12'-))
.:ti.nee tho rr.ntt.crtat.ical expecr;al;ior: or n coru1.,gat.:ive 1\tnct'iio n can only be zero if t:he :~tA.....,ction Vr'lr.iehe.!5 *'Ver·y·..f.... e1·e .
:t followr l'ror1 (128' ami (12')) :
' • lilJl-( T)-ln') ~ 111,
(130)
>' 10 • 0
\l II
' • l!J..L{ 11-ID 1)
\102
.._ ID I
':'hese oquatioou a.re i lcncical !"or u 20 I 0 un<.l
to t he relntion
'
\J 11
o.J. 01
-'-
0 1 due
•
F."nce , the lin,.ni· r<'lation (15:;) :io-·•rc:: c nr:a 'l iu;-;ay"
Iollown from oz • 1 . ?:-or:. a linear rel~lion, on ~tc othe:.h.o.nd , rollows ir.. gcn.,ra~ otly p 2 ,i 0 m.tl r.ot c' • 1, according to ( 12'/).
One may in.fer
coef.ficient p is
1'1·0:0 ~his
~·
.liscui;aion t:.~t the ~01·!•e:atio11
meaeure of th<' . ir:Q1u· lHuaper.dence of
two variable:.. . One sayo t wo variableo u.J:e ¢0.t."r·ela~cd f o r
P f. 0 and uncorl'Olatod t"or p = C.
Ae an exwuple consider the densiLy f'u 1 to~ r. io11 of a Lwo dimenaional Gnuas dl~tributioll :
lf(x, y) •
1
2no 1 o1 V1 - ol
-
"(1-p')
On~ obtains for o = O a.nd v 1
w(:x ,)
1
(
' • • V2Vna,
oxp -
1
ex:.i [ ,
r,.' £nll
o; - '" o,
~ ~]
Cl
( 1 ~·1)
o, ~ 0:
'"')
1
v')
2;;~ V2•ha, ex-p( - ~
(
132)
't'he statiotic.'ll independence of the variabl,;n fo11 ow~ i::
thi
; B case from c • O, because of (9) and tha r<'latic:i
d [ll ,(:x)W1 (y))/dxely = w1 (x)w 2 (y) . This roeulc bolds for
011 two- dimenrs.Lonal density functions which i'actor out into a Pl'oduc L ol' t wo one - di.roensional d<moity June Lions for
0 = 0 . l•or instance , of(x , y) could stw>d instead of 2oxy
;:>111
4 . S'l'A'rISTICA:. '/AllIABLts
in (131) c.nd ;:hr. whole expr·ession could te multiplied
1 + :e;(x,;;) .
b~
4.32 Cross· and Autocorrelation Function
T!::e ir.dice!l of the
• ,., , ri,, ... were so
iudividur.d !'esul.ts of
p1·e. senL eon:e ordered
·;ariatl~s ' • ' ' , ' ' , .• .. and n •
far used OJtl;y l:o d i!lo ingui sh the
t;he 1oeasure1:1e.HL1:1 . Thoy did not resequence . l'oi· ir.utnoco , tbe res ult
' ' wiis lloL necessari ly meanured al'tbi· Llie reflult C1 • Let
us now ass=e the ind ice<> indicaLe u s&quenc<0 . Let a meazurtirr.etLL at time t 1 yield C1 , 1 1 i £ a:.eaaur·e:nenc at tioe
t 1 > t 1 shall yield ~»"li;ctc . 'l'l'le sequence :.loesnothave
to be a tioe seque!!ce . C, , ', , .•.. aay be win; er levels
3long tnc course o~ a rivei· or enc tcas;crat"ure at cert:ain
ilacen .
l t 11Akes no diffe=ence for tr.o com1 ute.tion of i;he mean
(') C"f H results '
1
(') • R
A
2:; C1
1 ,
• • •
,
' •,
( 1;;;)
lo I
•~bother
i;he index i indica Lea a Ga1.iue11ce or not , since the
Lai·mli of the sum ma:y bo co:n:nucc<l . '!'lie swne holds for the
moan !'t'}Ua!•e deviation
{134)
Giv~n
;-...o Vb.l'iat:es
~
and fl, o:le may cor:s&ruct the ex-
rr<!s•uon
Ic is ir.irortant 1·or the value of (135) that C; is mult ipli~d wi~li '11 ar:d not "1itl1 111.1 or '11·• •
":quutions ( 1 .53) and t 1 34-) 'll'C idontical "1it;h (35) fol'
R .. , iJ: this limit exists . Lot , 1 equul x in r measureroeut; ouL of a Lolel of R mea~ur·~wenLs . It holds :
u . ;2 COR.'1ELATI011 ?UKCTro:;s
215
lim
p1 •
r1 R-CJO
1.et the pau· C
sllI'ements ;md leL che
P., =
lim
11 , R- co
li~it ~xist
q/R .
one may tllen write ( 1 35) :!.n the for:n of (3;,)
..
l;
(x - :n,)'(y - c,)'
r ,, .
't Js--CIO
I.Pt the time seque nce C1 , C1 , •• • be t·epli•c.cd by u ttmc
!unction f(0;) which assurues tJle ·v alues C1 rit ·t he t i.or·~ e1 ;
j . 0 1 1 , 2 , .. !(9) i s written instead or r(e;) for a continuous sequence . Lee all values of tbi:- ""quer:.ce bt> located in the interval -te '- 9 " ~!! , wi"ern S a.a.y be finite
OI' in.finite . Ono may i·ewrite ( 1 33) antl (1:;;11) as fo l lows
(all i.nt egrala 1·un f rom -tel ~o •t 6 hut Lha limits nro no t
writtan t o simplify t he formula:;) :
.,, • (r(e)). e·•Jne ;da
o: •(t.rce>- q<e>)J')
Rep.acing further the sequence ~ • • ~,,
tion
m,
g(8)
by a t iac fW!c-
one obtains :
•(g(e)). e·• Je;Ce)ue
a; ·
(Cg(9)-(g(a))J')
• e-• f(g(e ) - m,J ' ae
=
a·• fs'<e )da
-
I
111,
o,~ • ( rs< a )- (e;Ce ))J(!(e )-(r(e ))l )
• e .. J[.r(e)- m,J[g(e)-111,)de = a ·' Jt(e)s(e)ae -
,,,m,
T!1e inr.egral in L:..~ numerstol" ut p ±-s calle<l c=-·o:;scorre1 Rtion !w1c1;ion K1g \S vl f o r av • 0 if Gi a 1 f'!'o11c:hes infir.it;v :
Jim s ·
s-~
1
3/2
J
~an
f(e~g(sJ._e"' :1r.e
(1~9)
'Ihe &U ·.oeorre I lt ior. '.° .il.~ ti on • I ' t ~ ~) J'o:lOWS :-01· f( 9)
f{3};
<912
lio
a -~
Th~
a·'
I
!'(a)tte•'J"Jdn
(140)
..917
.;;t:.ort.-t.ime cro.onco::!·elotion or short- time
autoco1..relat ion :f unction are unoC. 1.!' e i s .fin:l Le . To abow
wtat mny be done wirh tlte corrc ~tion fw.icL.Lonn , l et ue
terms
assu.ae thni; f(9) and p:( 5) ai•e cot consi;anL; tho indeterfor:n 0/0 is c:::u~ avoided for o 1n (1~8) . Let ue
.:'""urtller af:sume tl1a.t. at lca5t one of th.:- moans n: 1 and m1
equol~ "ei•o . K1 9 (0) • 0 yields p • 0 and K19 (0) :K 11 (0 )
yieldo Io I : 1 . Hor\C<1 , the cro!l!lCO.L'l'elution function in
a tr.F"anur~ o~ l.f"_"" --:orrela.t.io!l of Lwo fur.ctionti . o ceasurea
t!:i• correlatioiofor the functions !(6) and g(B)otly , but
oinat~
K 19 (~, )
and K• r ~3v ~·itld 1.he coi•rels:&i cn for the runct i o:H:i sJ1l fted •sy an a.rbi 1;r8-""")' wuoUJlt S v . Exwuplee o f cross-
and nutof'.!orr~la.tio11 .functi ouo JJ.J.•c u:io,,•n in 1"iga . ~10 and 71 ·
't'he lilnic" of iaLei;r~tion in those f~gure• o.ro not - - and
-roo, si~cn cal(i,9) nn·:! sP.l\i,U) ore periodic f:.inctions .
5. Application of Orthogonal Functions to
Statistical Problems
5.1 Series Expansion of Stochastic Functions
5.11 Thermal Noise
00n:iidor u root o!' tim..: .ru.iction" g ~ f e} , ~ • 1, 2 , .. ,
which do not !Jave to be ortnoe;oni;l . ::ach ru nctaon sh:;ll
be expanded iuLO a ::icrie:; of the CO!:o]J] oi.c ni·tt:onormal system { f( j 1 9) l iu <oh~ i!leerva.l
- is :;
~
2
Q3
:
00
g, (e) =I;a,(j)~(j ,9 )
(1)
I••
e12
a 1(j) • J. g, (S )r\~ , 9 )d,
-an
'.!'he coef:l:ic t onta n , ( j) hnve ce::.-cein value" l'or a l'ixed
J • j 0 Md vai·iabl e va.lues of >. . ' J'UJ11;Lious !:!>( 9 )
yield l coefficients a, (,I ) . Lee q, o.r t,hom bCl in the interv&J. 0 < A < t.A, q 1 in the interval A/I. < ,; < 2flA , etc .
'!'be fractions q 1 /1 , q 1 /1 , . . . . "hull be plott<1d over the
value of
6A , 6A to 20A, 2tc . fhe rnnult is a ~l;e;o
!unction . Asswne that i& can be aprroxir.intcd .rot· s:u.11
valuea of AA by o continuous densi;~ .::u::.ction . Ttia density !unction can be differen:. t:o:.- each VRluc of i . One
int ervals O
~o
Calls a,(j) equally diutributed w:th t·td·ercncc to j , i f
the density functions are id1mtical ror '11 l vnlu"• o:' .j •
.Furthermore lot tile coeffic ient~ a,(j) and n.(k) b• sta tistically independont for ~ F '< - The •eL o.C ~illle .::Lwc~ions gA(e) is cnll ed n sample o.r whit:e nolae wit!J rer e rettce to 1:110 orthogonal system ( r( j, 0 )] •
>,(Jo) in cnllod Gnussian distribut ed , lf iLs density
!'unction is Ll1e derival;ive of the error fWlction . Tho c~t
~ .
218
s:,','l' fST!CAl
FROBLEfts
of fui:ctiom; g,(e) is c3lled •.i!lite Gaussinn noise or thermal :.oise · , i r :;he '>A ( j are ec;.ually di .;tri buted wicb reference to j , !ltatisticnlly independent u.nd Gaussian dist1'1buced for a ce1:taln j a j 0 .
Fo r the practicul monr.urement of t he coef.Cicients aA(j)
con"idcr a generutor ro1' t he f:mcti.onr. .C(J, e) . 'rhe index
J c~ot run from zcz·o to i n_:'lnity ao in ( 1 ) ; j can only
assume a
fini~e
ntm.bor m of values O... . n-1 . Tioe is di-
vide<! L<t~O con- ~verl!lpping .:.ni:erval~ of du~·ation
funct~or. £~,&) in t~c !'ir~t t !.oe intPrval
s.
T:'le
is denoted by
~ 1 ( 0), thC' fur.ctioc. in <t:e tll!e i nt:t>rvnl ~ b,y g, ( 9) . A
finite O\l.'r.ber L o l in t ervnls is posoible or:ly ; \ runs from
1 to L. Let the m ru uctions f ( j,B ) ue uvoi loble simul.te.neously an(l l et t here be m mul tip liers wtd i ntegrators .
The r.i coeffi<>icnt a 11 1(j) , : = O..... m-1, <:u.c be measured
in Lhe r'i r.s\i in.tervol . Tt·ese coeffic•ents are .represented
by Lhe integrator outpu- voltase>< at the end of the :i.rat
ticie interval o: duration a. Repet: ti or. of' these ceasure1.em;s for all ' ti:ne in~<'r'lals yields the Ill< coefficients
i j = o... . ffi-1 , l • 1 .... • .
Assume t he :.;et of l'unccior.s i:>A ( 0) in thormal noise .
• 1 (J'
I.et us p l ot tile fl'ac L ~on q 1 /
t of meas u1·ementn yielding a
of e 1 (j ) in Lht i nterva! (r- 1 )AA < A < re.A . nie
m stcr t'n.nctions omy be ayprox.ute.Led by continuou!l densit:1 f•.incti.on~ ~·.(~,A) a~ ohown in Fig . 91, if
A.:.. i~ s:.l!':':. cient y scte.11 anC t. sul·ticient y l arge . The
equol rlistribution with reference to j causes the :ollowinIT relation to hold for· a certain A • A 0 :
value
r~n·1lti.ng
(2)
Tb~
cliat ri"uuti o '1 of t'lc coefficients a,1(.1) general l.y c;lepenc.l~ not only Oll the ijet o:!: runctionn e A(a) , but also on
1
Jse of ;;;hese teri>n is not unilorci in the literatu.re .
Tbr!'mal !lOise i:: !'1-eque'-L~y cal led Johnson noise [ 12]. or
re1ns:or noi«e . ·rue noise generated by thermal ngitai;i'?~
of elecvrons iu 8lJ ohlr.ic resistor is thontel noise,_ J.
tho f1leci;rons are descri!>ed by Boltzmann otati"1;ic :-atber
than J'ermi statistic .
219
Fig. 91 DenaHy fu.nctiohn w0 ( j 1 A) o:!' tllermFil noi;;11L J = 0
•.• • m-1; A denotes thtl nor·m11l.Lzed output volcati;en of t he
:n integ1•ators .
the :Jystem (f(j,3)} . llowt'ver , it is ~ndependcnt of the
oystcm (f(j ,9 )) under very g<'nerul assu.'!lptior.s ro1· thermal noise.
~·or a proof of this eL11.tement let
us reµlsco the
completo orthonormal systew ( r( j , a)} "oy ano thcr c;ystem
[h(j , e ) 1 that i s also cornplotc nnd 01·thonomol in the intenal -oe;:; 9 ~ 1e. Tl1e f'unc:tions f(J ,9 ) and h(j , 5) snsll
be bounded . I.et the function.a h(J , 9 ) l:>e m::pnnJed into a
series
h(j , 9)
•~CJ (k)f(k,9) ,
•••
CJ (i<)
612
f
h(j , e)r(~,i)ae
-£/2
'l'he awn L;c J (k) shall. converge abnol •ltely . ".'he ':le.des ( 3)
then converges uni.formly .
SA(9) is eJC])anded into a sorieso:!"the S,YBLeru (h (J , 5)\ :
&,(e ) •
..
L;b,(j)b(j,e),
b,(j) -
I••
Uaing (1) and (3) one ob~&iClS :
612
j g,(9)h(J,3)d9
-en
(4)
220
en
b. (J) •
I
-LC
~ ,(B )
-0/2
(5)
I (k) f (>: , 0)
' •I
- ci (k) en
• l:
J g 1 ~6;f(k , a)cg
,,,
-8/2
•
- c (k)u.(i<:)
2.;
1
k -0
The l 11ut sum r;onvergon absol uLtJ. y , i f all a,1(1<) nre bounded .
Thf ~11c of .:;~ati:iticall:,· iLdepcndent. , Ga::c:-ian distribul;ed vw.·inb-c;; ir- s G~~ssian dist1·ib.:.J'tCd "'"o..:·i•Jhlc . Eence ,
&be \J•l.I) have "G•U!:!lisn clistr ioution , J.J.' Lno a . ( k ) are
!l'C&t i s 1.icolly i nd evendent . '!'he u:eWJ o f t h e> u.1 ( I<) and of
t he l.>A(.I) is zero .
'I'--.e de"sity function •·c(k , A) or LbP a,(k) reaes for
cter~al noise aa follows :
(6 )
Integration o ver A yields 1/ a: ,
w0 (~)
•
T~0 (k,A)dA
--
The variance
a~
in {
•
>/
~
,
ie defi'-o~ by :
(7)
The den• it;; funccior. of c 1 (%)a.(k) equa!s:
w, [k , c 1 (;.)A] = i/2',rJ«'\o
<·:q{ -c~(k)A 1 /20: 0 ]
(6)
The den~icy :'unctiori of the variable o,(j) follows frOl!l
{5) , (?) nnc\ (8) :
7. 11
T!iERJ'IA.L HOJ SJ::
,.~(j,A)
•
221
Vcrilc:ab ex~( -A'/2~ )
(9)
( 10)
'rhe last atep in ( 10J l!Wk~s .tse of t',e initinl !lr.,uortio:i
that tho distrlblltion of 3., (;c) ru:d r.nu:i o~ doce r.ot de -
peJld on k .
I f Parseval ' " Lheor'"lil ( 1 . 11 ) is sat is r i.ed 01· , rurti:::c
it diff et•ertLl:I', ll' t!Je functions h ( ,J, 0 J mo:1 be rcr,r~i;<'r.
ted with a1·bi1'rary nc c;uracy ::.n Lile sense O( a vadnhi:ii;
i:iean squnre aov1 ntion by '.;he s:1stem { f(j , a )1 , one ol' vab~
from (~) :
8/2
r !l I2 (J,9)d9
•
13/Z
• 1 •
4$12
S ( 2;
-sn ...
c (k )f(E.,a)) d9 • 2'.;::~ ( i< ) (';1 )
1
I
•••
!i; :follows fl'Om {10) a!'ld {'l1 ), tnr,;; '.;he cond.!.tion
a~ • o~(1 + c)
( ~2j
is sstisfie<l '\n<I chat c app1-oaches "t!'O tor .; 1f.::ic ie:-.t l ::
large values or ru . '!'ho vnrinble,; o A ( j) ru1d u .< ,: ) Lt.en t:ave
the ssma vu.riauc<: . The density !unctiona or 1 1.g . 91 r nm.~i n
unchanged , if Lhe srunples g, ( 6) of thermal noi!Oe oi·e expanded in a oorioa o:C tr.o ay5toa { r.(J , S)l iusLeud of
11
(f(j, 9)) .
Thermal nolse is usual l y de.f~ned l'l lht1 literature by
a Pourier series rather than ":>y i;he i;eneral orthoGc:on~l
series ( 1) . One may substitute in ( 1; i;he nine an<! co!1ine
JlUlses that vanish outside the inccr•.'al -t8 a 6 a t'> for
tbe system {!(j,9)) . .t.ccordini:; ~o t!«1 rc:1".Jltn of t;hi,;
section, tbero is no difference
whet~er
t=er;i.ul noise is
defined by a .Fourier series or by a series o! fu.r.ctions
{h(j,9)) t1u1t co.nbeexpanded in a Fourier ~eri.;s "" i:;hown
by t>) .
tt hatt been s t ated in section 2 . 2'1 LhoL uudici sie,na J.I'
wa:i-e found to have <Joquency formante , i r" docomposed b;v
WaJ.sh functions, just as they have fr equen cy l'ormnnto , i f
decom.POsod by sine - cosine .functions . Furthornioro , atttlio
222
signuls n: te.!."ec ":>y :;equency fiHen cou!d hn?f.l:; be distinguiahed from si.:;::ia:s 1·i1tered by frequency filters, i t
t..he inJ.'ormncio:: flow ·•as ~he su:r.e . If nu1'!io ~ig::a:s had
tla~ distribG"t;iO!'". o.f l;h.er:nal r..oise, 11.!Jd if ~he ear could
d~co::r.]1oso t~e~e
sittnals ir.to ur.. .:-.~r ini te ::·:r.iber of coapoce11ts ncco:-ding to (1) or c~) o~e uhoul<l exrect rnch re-
eulta . '.!ho e:xp;rir.iental i-esulh sr:ow tlint a'>dio signals
a.re ~ufficieot:y similar to ~Lu11o"l r.oiac and -chut the
eu.x· UacompoBon thorn into su.fficienLly ~1nny cor.iponents to
muke the resuH s a:· ;;ni;; eec~iou t.Lpullc• nc .
The l'e!lul t::: r.mGt also '-'IJPlY to noi!l<' r0r1resented by
elt:iC L!'Ol:tngLJetic r'ldiat:;ior. , sucl. ae l.L~;1d.. . '!hC!r0 is at prer.ent no device k.t:own ttat coi.4ld deCOtllpose light i!lto ftlalsh
f®Ctions nnd produce a seq'Cleccy spectrwn . Devices that
,1cco11pose lig!lt icto sinu~oirtal fU..'1C:..io:.is and froduce !~e
~uency s~ecL::-3 , s~c:i a~ a diffr:ictiou grotir:g or a priSJ:l ,
nrc ;i:ro - 1nva:.·iant; .!ust like the :-:-equ1.::..C.Y filters o.f cocitt.ur.icationa . :lence , a dcvi er: for· lt;Con.}Joeir:.J;: light into
1\'nl nh runcr-iorJs must ~1ave an ox-trn-i1ely fast. t..ia:e depender..CI"! , "'X'!ll sini~1g w!::.,v no p1~nctic:-il ~~ur;t•·0ntio1 .. for such a
device hnn 'become kr~o 1•.,in ~rev .
5. 12 Statistical Independence of the Componon1s of an
Orthogonal Expansron
Lt; hr.u bN>:l as~urned
coeffici;,ri~a
ir. the preceding "ect;ion , that the
n,(h} =<i a,~t) a:-e rtt1tfrtic~lly indeperuienc
It rerr::ii.!lS: to be ..;uown tt:nt thi..: L"1Cependence
olsc; l:o-c!. for tr..:- c:>ef.ficient .. lJ, ( j) nnc! b 1(1) w!J.en j 1 1 .
These t;Oc~f-cicn-::; have a G,u.'1aian distritution and t:heY
are .stati~t:.~all;" ir.dependent, i;" tiHl correlation coefficier.1 i> Ol' t:..e cov:iriance af1 ·:ani ~h . U~ing t!:e 3bS:)lute
convu;leui.!i= oft.he se=ic:;: i::i \'') onn ohtnin!!:
for
1
i( .
:i,,
I
1:0.nr
·-~
I
•
llm
•-""
~
'"'
11~0
k O
; 2: [I; c 1 (b)a,(h) 2:; e 1(k)n 1 (k)]
,t._1
] . 21 LEAST MEAN SQUARE DEVIATIOll
•
.....
L: L: c :1»c 1 <k>
,,o k•O 1
litt
, ~
£ox· auy pair
Denote by c the largent
b,k and ll £ln1tc
223
vnl.ue
( 1lf)
... "' (h)c1 (k)
LLC
1
lie
1uO tr.0
1"hc double num conve1·ges
c 1( h) and c 1(k) converge
....
L L; c 1( h )c1 ( k)
a
tibsolu~ely.
:=i:.co the ..s.u.cu; of
a~solu"ely :
K
(16)
hrO Irr()
Equations (15) nnd (16) yield :
oj1
ii
cK
(17)
c appl'oo.chea zero for lo.rge values of ' b;y cl<>£ inil ion and
the covariance o j1 v-anisbes .
5.2 Additive Disturbances
5.21Least Mean Square Deviation of a Signal from Sample Functions
Lr,t e ti.lne function Fx<a) be compo~nd or th<' :·i:-~~
!unctions of the orthogonal sy:n;e:n ! r( j, 6) I :
Fx (S )
....
• L; llx(.1).t:(J , 9)
l•O
i
( 18)
l!';r( e ) :Ls called chnr ucteL' o.t: an al.phabet . 'i'hE>l'b is only
a finit e nwnbol' o r Guch ctiaracters, if the coefficients
a..-(J ) !lre not arbiti·a.ry but; can asswoe n finite number of
-;.. s·rA'.L'ISTICA:. PROBLE!-1s
valu<"::: only . ?he
~el~tYre .tl1-J1a:iet ,
r.o:c~er~; iL equaln
!i and the
c . e; . 1 cont~i!ls 32 cha-
co~_f~:._ci("r.ts
O.c(t,) :nay- assu.tt.e
two values .
I.et .Fx(0 ) be trbJl:'u;iLoed . A dil'tU!'l.>u.ucc g , (a) is added
(luring t .r•a.nsa:iZlnion ancJ L!1e f=ii gnal
F( 9 J = F x ( a) ~ i;;, (a .l
in receive<l .
n !:crie!'i :
F(5) ...
..L
L~t ua
( 19)
a:-s-.ur.e t!i.at f('3)
~{.i)!"(~,9) •
J :- 0
i.;wL
be
~x:.:.icnC.ed in
~
l_'.
,.o
[a.,(j) - a,(.!)Jr(.,5)
(20)
6 12
•1(J) =
f
-en
F(&)J(j , 8)r.iS ;
,l l'LL"ls rroo 0
~o
iul'inityenc noi:; f.coc1 0 co :i.- 1 . a,(j) i s
de!ir-ed by (1) .
t caet be rtocitfol at i:he recei vei• 1<hich character
o = 1 .... x ... in t•.e one wnich ao:;t rrobnbl7cauced ~he si[11al t"(&) . ·r1:e r-robabili .y of ~ tra..'l!l!omation
of F,(S) :.nto ;.•(a' U'lfe!Ode on i:he ;>l'Obu.bi-Hy ;;hat F~ (e)
·,·Jri.'i tl"!il1$:n.:ttcd . Lei u;:; assu:ne a1l chC::l..ruct.e1·a ar e t i·nnsniti:;ecl with e qual prObHbll ity . ~'he decision dt>pends bhen
i>uly C!l the t.ILoLut•bnnco:· 6 ,(a) . No <Jec.i;lon is possibl e
for 1 ~ing"le cl1!J.t:actr-r , if r:ot:hing is .i'.JJown tfhout -che set
gl(~) . :!o<:e.,,·ei·, it; is K.00 ..m in ma.ny cazes, that ad_isturb11.0ce g._,(e) 1<ith 1·1rii:r. energy is received l~ss o:'ten than
o~e w~~h li;tle en~rl":'' · !~;;i:;ir.£ iL d:.rrerently , the pro' abil:.ty cf rocoi-:i1:i:; a di,;t·.>r!oanco g, (e) with energy bel11Jel'n 1.V and 'r.'+6 1../ tlecr·i:.·Hscff oonotoni cally \,•it.h increasing
'r! . The signal 1''(&) ir· moot lik ely produce<'! bye characte»
F.,(SJ,
1
I"• (0) that may te ~L·tl.l',Sfa r!lled odditivol ;v wich the least
P.ner1-o:y ic.to F(9) . 1l'he r:'lne!~f!':'>' 1 6 ·~·l-~ requi110C. ! '01· t~1is transener~y is u:sed i·or !he defi.u.ite integ!'al of t::e
a function . I~s me~ is the saac ns the ~ne
used i:i electrical engi.:>eering, if the function
r('! re."..:ncs the vol:ase aci-oss or the c11rrent through 8
'lr.ir resisto.r .
1
Tbr ·t"nt
or
ocrJ~re
i;<>ncr~lly
.5. 2
1 LEAST r.EAN SQ.TIAHE JSVIAT 1or.
612
J (F(9) - 1',.(S)}
1
-e12
22.5
!!/2
'
d9 • J[f 1 (a •- 2F(9)F,.(9)1F :(S)]dij,(;:-1)
0
-a12
•rM integral of F 2(6) yielrJr. the en ergy or I. lie received signal , the integr a l of 11J( & ) tbe enerr>;i' o: tlic chnracter ~',.(9) with whicl1 tl:o ~ie;u11 is compareu. !i;e integral of F(9 JF.. (9) b tht correlation integral or tl-c correlation of the signul 1''(9) nnd tke character' l',,(e ) .
The contribution to L',I., by F ~ ) is ;:;!:c :::ane !or all
characters F\1'(9) and may be ig<101·ed . I: , f:.tt"t!1er:11ore, t.!le
energy of all character" i.n the srur.e ,
'<
w~
912
=
S l"J (a )da
=
w,
(?2)
·812
oneme.yignore F;(e) too . 'l'h!'umalle5i; value 6 W~ in dcterl!lined by the correlation i.!ltee;ral a .o:.:e ; ~ trii::: en::<: :
6 •,; • minitr.um for
6/2
f
l-'( & )F,.( 6 )da ~ mexi!:!un:
(25)
~/2
The t rllllsmil;ted character [IK (9) will be detcct;1>d correctly
if AW,, has its minimWll for 9 •x .
Si p;oal det ect i on by moone 01' (21) ano (23) ir called
detection by the criterion or loa~t mean squer" dl'Villtior..
Samples g,(9) of i;hcrmal nOifle :;atisfy tLe CO!'lditious for
Which such n detection is proper . ':here are many LYJ..e" of
additive disturbances for which t.'.le conditions Arc not
satiofied , ouch as pulee type disturCar.ces or nocalled
intel li~ent .interference .
IJeing adde1•s , n ultipliers o no integratori:; , one o·;y :lctenmino in p rinciple t ho moot p1·obably transmitl.Gd cho.racter f rom (21 ) or (23) . The effort required, however , i s
usually too great . Let an alphabet !:.ave n cl.ar~ctcru . n
enorgies AW., or n correlation integrals havo to be co•1PUt ed according to (21) or (2~) . Tl:ese cocputations ~houlc
be done simultaneously. 'lence, n or n/2 ec!ders, tt.ultiPliera and inte5r acors are required .
~26
'.;; . ST/.. 1 Sl'.i.l;AJ, HlOl!L:.1s
Less <'l(pensive ml't!°io<lo
~nto
tint. (18j nnd (20)
"' 1 ,, I
'\'
L.J a (i'
I••
l.\WJ
be obtained by substitu-
(2" :
1n•
- 22: a\;)ia, (,JJ
I 0
~·
I;Ca(ji
!
- ... (.J)]
n •I
+ ); u;(~)
j dl
~
1~0
L'•'~
:oi!'lb.w:i f or
~f [a(J)
- a,(j)]
2
•
oini:n\Ul
•••
O!'
..,
Jniniaiua for 2
..
:Z n( ~hi, ( J)
,. 0
~'he
SWlS
2':
yieltl ;:he
B ~( j
=
L;a'(JJ
ma;v br ignored , !lince bhey
) 01'
I• o
s=e v.._ue for
•'"'
!'Ol" chn.ract.er·s ..-.it
cqua.~
each < . ur.c otot'1~ na f:!'Om (24)
e1.e1·fD" :
,,,.,
6'w\. = oini;t1wn l"o:-
2~ n(jJa,..(j) = caxiu.uo
(25)
I 4
f:q crnti ons (21 ) , (21 1) n:id C-'5) shOw ,hat only the coe.CIic.i.cmtr, n 1 (j), J < :n , of ti':o noUe cnmple f(;(O) affect
LI e deci ion over >:1.ict. c!.1u·acc<'".' f,.( 9) ·..1as the most likely
to ?1-"oduc~ the :-ecHl YC!d .'iib!l •l I··( a J.
oc :nultiplicrs aud .i.ntep·tt Lo i·" rai;lt~1· than n 01· n /2 n.t'll
r~q:.rircd fO!' l;~.e p1•ar:t ical .i.u.nleaenta~.i.or:. of ( 2'•) and ( 25) ·
':'tiis c enn .. ts red".lcti~:.. I"ror 3~ a~ 16 to 5 :.n the case or
the tele-yr ~ :.i:pt.•,1;et .
Tet us nubsbitut<' the "'"" axU) 1 ~,(j) from (20) !or
:; ( j ) ~.!l
p1. j :
6Wv = minirr.ur. ~01· ;>
m•1
m-t
pQ
j :O
>.:; [ax(j)+a 1 (j) Ja,. (j)->.:; a.~(j)~maximlllll
'l'he effect of ta~ <liscu.rbo.nces g,(&) on toe signal decinion i" due rn the su:n 2 ~··
2:;n , (j)a.,,{j) . 'l'be p1·obe.i>ilitY
I •t
EJW1Pl•E~
5. 22
of
227
01• CIRCUITS
wrong deci:;ion depends sol dy on tile
8
~tsti!ltical
912
: g,(e)r(J,9)d& .
-S.'2
Let g,(8) be a soo.ple of <:'1er:t• l noise . J'~.o rtPti,;t~
cal. propertieo of t;i.e coe f ficients a,(j) nr-t! thNL - :mde":
ver"/ general condi~iona - indepen<l<'nt o:· tho ort!-.OfOtlli!
sys tem [ .f(.j ,9 )} uueo . ;!le Lrl.'..nsmici;od s i i;uul ~'x(9) is C'.>mposod o.f ~hose !'uncUuna s ccor a i ng 1;0 (1i') . Honer> , it i"
quite unimportant Ior o'.1e prol.:ab i Uty oJ' n wronc; (lecision
whicll .runcUons f(.j ,B ) a.1:e us~a t o corr.;•o"c 1.l:c "ie;n11: , i i
the di sturbances are :.ud..itive th<:>rn~l r.nl"~ ·
5.22 Examples of Circuits
Iiet u~ diocuss soir.e .... i rcui tf.""
signal detection . Fig . J2 s~o•s
coe!'ficie~ts
a(J)
are obtained .!'rom the received :ir..r- ~l F(; ) by mean:; of
san:ple r unctionn C(j ,a) . '::-lLi.s e : r·cult is iv:ical ly tne
srute as the one of Fig . ;iO , e:<cepc thH th<> clir.turbed co ef.::icienta n(j) ins~ea.d o f tti e imoi!1C'H'b"tl coel'fi ci<iLJ L:;
n~( j) aro obtained .
(1)
l(UI)
Fig . 92 Extraction of vl1c cocf!'icie:...ts
a(j) ::::-oci the received ~ignal ?(b) .
'1 multi~ : it« , I ictegr:itur .
~a(I)
I
1(1,9)
'Jibe sumo of the products a(j )e,( j ) occortli ni;; to (2~)
are produced. b;y ~be cir uui t of l'ie; . 9::S . '.!'hta ctw1·uctu1·s HJ.'"
compose(! or ~ brcc functio no , m = ;; . Hence , throe cooi'fi Qiente a , (O) , 0.,, (1) a nd 0 .,( 2 ) occur that or·e rep1·e,.en t ed
,,.
!l . STl.::ISTICA.l, PROBLEMS
228
by voltn5c~ . The coefficientg n,,(O) , a~(1) &nd ,,.,,(2) , 1 •
• 1 1 2 , ... nre represented by r·csistorn . The operational
n11pl ifi ~r~ A ha•1e ci.:-fere1elial .input~ . The inverting input
lerqinals are denote<! by ( - ), the non- invert illl'" ones by(+) .
Vo(I)) 'lo(ll Ya(!)
•V
Vo(O) V•O) '/eU)
R
.
-~/a.Ol
."· R··
II
-
n
(DI
R
l'ig . 9; (le!'t Sii;nal detecUon by the lar1;1est sum . Al l
characte~s have eqtul2 e~ergy . V0 ·V(n(O)a 0 (0; - a(1)a 0 l1) +
- a(2)a 0 (2)~ ; v,. 'lla(O)a.(0)1a(1Ja,(1)-a(2)a 1 (2)]; V, s
•V(a(~)a 1 (0)+a(1)a 2 (1)-a(2)a 1 (~) • •
Fig . JI> (right) Signal detection by tho """'llest sUJll. TJie
chui-11cte1·~ do r:ot have to :Jave equal energy .
V0 • V[u~ (O)+a~ ( 1 )+a~ (2) -a(O)a 0 (0) 1 a( 1 )a 0 ( 1 )+a(2)a o(2)] i
V,
V( u' (0 )..-E• 1 )+a l ( 2) - a( O)ii 1(0)-a(1 )" , ( 1 ) 1·e.(2)e. , (2) ] i
;,• 1 • V[ni (0 )+a: (1 )+ a; (2)-u(O)u (0)-u(1)u,( 1 )+a.(2)a , (2 )] i
1
:<
R1
•
R/[af(O)+a:(1)+uj(2) ]; j • 0 , 1 , 2 .
_
5 22
EXAJ1PLES OP ClHCUITS
229
!'or the imrleaeutation of ( 2'-') lot un note tho.t; the su;r.
t;',1., . This sw:i mo.y be disregarded, i f the smallest A'i.- o;!:a!l be determined 1dthout
aIJY need to ltllOI< tl:e val1.e of 6W.,. . '!'hn n\l!t of u(j 1 a.,(j)
is produced an before , excep c that the nign must be reversed . Renee , one may u se the c i rcuit of Pig . 93 , but the
inverting ntHl non- inverting i.nput terminals of the operational iunpli!foi·s must be incercha.nged as in Fig . 9'•. 'l'be
s:um of u~( j) J.s produced by un adili tional !ine with t:OJ1 stant volt;age +V Md res i sto r s of Pl'O]lor vnlue .
Circuits are required t o determ; ne wh ich output voltage V0 , V,, v,, .. . in l'ie; . 93 is la1·13est Md which output voltage V0 , V1 , ••• in Fig . 9~ is smallo:1t . One i;ype
of circuit that determine5 tr"c :argesi; or "molleat of n
,·oltae;es uses n rl!.lllp voltage t!.at is co,.,p11r1>d via n co:i:parators with the n voltages . The firsi; coa:pnrntor to fire
determines the c111al~est voli;age i:-: c<1se of iw incrcnsi:>e;
ramp voltage; the largest voltage in cle t.e!·111ined 'by t!le
!irst comparator to fire in case of o d.ecl·ean\ng ramf voltage. An advantage of this T.YJln o f circuic is ehaL cne
ramp voltage doe a aot ba'le to .,ary linearly with time and
t'he.i; voltage fluctuations a r e i'.nirly unimponarnt . The drawbnck io tho non-inst;antaneo us operacion .
An insto.nta.neous co;nparator is nnowr. in Fif~ · '}' . 'Jlhc
vol t ag(l at ~he coilllllon point of each group of " diodes
equal o the largeat applied positive voltar;e . Let: v, be
the largest voltage . ibe vol~agc at the nou-;.'eversing L:Jpu; teminal (-) o! amplifier A, is laru;er ::hnn at; the
non-reveroing input te=inal (.) . ,;ssumir.g surric:.ent
Pl.lticntion the output •toltage B1 will be nt negative
sat;urntl.on , which shall be indicated by B1 •-1 . 'l'he runi:;lifiers A, and A, receive a la.:"ger volt.age a~ the non- revers'
l.llg input terminal (..) than at th~ rov~rcinc one ( - ) .
80 •h are drivan to positive saturat.ion, <lenote<.l by B =
2
a, • ~1 . The output voltages ll 1 , B, !lnd ll 1 lndicote the
largest voltage·V 1 , j =0 . .. 7 , by representing j ao binary
lllJ.;uber . 'l'lte diode characteristics must be very similar
oi a'C.1) is the n=e for all
=-
230
for good results . r at.tpli!'.!.ers are rcqui l'tld tor comparison
01· 2' voltages . Vurietious o~ tbE circ11i:. cw1 deteci; .,,hic.h
of ~"veral vo:t.ucon l111s c;he larf_jest or t.h· ~u.allost mag.ni_
tud <• .
t•'ig . Y6 shoNr: ~111otbe1· cir·cul t f or <lGLt.:!'!tlinntion of l;be
largest vol ,;agi.-- . 'l'llc thr!':'e a..rr.plifie1·.; ;.., , A 1 and A. 1 a.re
ctriver: to poEitive or negntive satun1LJ.on by the oii!fei-cnce~ betf'..-ecn th.~ t.tu-e.P volcages Vot 1.', 1'1.J'!<l vl . The 3 !a
G possible :;ieruuLn~ion" of &lo.<> out;n:t voltage" fil•e s_0 .,,11
i:: tl':I'.! tab~e of !:i:o l'il!.UI'<: ca,..t:.on . 'fuey denote no;; only
thr. l argest b;.it al ro che sec on<'! and Ll.ird largest - that
io the !'lmalle"t - voltit,;e . 'H:e voltab"" v,, V , and v, may
be posicive or nee.~)civ<-~ . Tl1i s circuit. ia uiuch more sensitive than the onn of Jo'it,: . 95 , sincr> t 1e voltages are fed
directly to tile rui:pliJ'iers raLher than tnroui:;h diodes . I ts
d.r·awbqck i:; tt.e l~z:gc number o! aa.plifie""s 1•equire-d . A
co:r..paris:on of n voltng~s req,'.li:!'es .ii.eos\.i.rn:r.en~ of (!1-1 .._
... •1 • in(:.-1) vo:!.~age dil'f<:rcmccs . E:ence, a
CO::al Of ~ ll( n-1 ) di ff H!'elloial ru:ipJi ri8l"S '1.l'e "ceded. The
c!.rcuit oJ Fi[ . C.:',, ontl1r:ot.he!' !;j=lll·i, rQquire~ for .n =- 2r
voltages tg, n <li.Cferom: ; a I ai:.:;ili.fiers only .
+ (n- 2) •
5. 23 Matched Filters
(t
l:ae boen a,.aWtl'd so r'n~, ohat the coefficients a(j )
n.re obcaiued by mul tiplicaciou of the eignal l'(S) with
!'( J, a ) and ir.tec;ntion o: tLe product . .A 11athematicslly
equi•rnlcm; but Lecl!nically very diffe.-er:t 1?1etb.0C. uses
rentcl:ec! :iltei·a . Hi" cu:;tocary to usu tbc pulse response
l'nLher tha:r: i;he fi·equi>ncy r<>sponse of r.ttenuucion and phase
aJ1lft to c har.. c t ei·i~·· matched filters . Consider a nar row
block pulze P(0) huving tho amplit ude ·J/c inside the inLervnl -h ,; e :: h and Lhe a:nplitudc O outside . This pulse
:ipproc,cne,; ;;lle del La func;;ion ~ ( e) for vanisitlng values
of < . Coz:sidcr ~urther s bank of filte<'S . Let cbe pul" 8
6(a •§ } ct api:lied at time 6 = -i to tbe input of &he fil ~cr j . The output function H.j,e ) , -i a e a+!, of (18)
twd (20) shul! be produced . ~(j,S) is tho pulse response
251
.5 •2:; J1AT0HED FlL'l'J,iiS
:ig . y lictoctio:i of
the la:--
gent; poritive voltnse Y0 to
R
:Le lb.rgest "tol t6t;e is
u.~ val:~en of
3 1 , E 2 an:l s, r.ho·..•a .
-,:7 •
der.c:-c:incd by
,. ' .
v. ".'
,,
Vz.
'I v-.,
¥1
+1
B, - 1 -1 -1 -1 11 -1 +1
I-
••
B, -1 -1 •1 11 -1 -1 +1 +1
B,
-1 '1 -1 ..1 -1 +1 - 1 +1
~.
}l;Y'
BJ
81
e,
l ig . 9i:> D~tecti ou o! ::hr!: rcl·ltive vgtues
of voltages .
'lo v, 'i 2 Vo v, v,
largesr. vol wage
Qccood largest vol. cage v, llo
'lz vi v,
third lru·gest vol-cage
B,
).;
Va V1 Vz
'
5,
v, v, "'v,• v,
I/'
I -1 •1 +1 -'i
•1 11 - 1 +1 -1
,1 +1 - 1 -·1 • 1
I
lfo
-1
-1
-1
or fil ter j .
'l'h<:> ti.Ile function Fx(9) of: (18) can be produced oy arp!y.ing the pulsos ax(j •6(9+~ ) -i;o 11 fil"<'rs •i~h pul~e response f(j,6) and suinming the outputs . These filcerr are
denoted as transmitter filters .
The receiver filters inve::-t the procc::::: . ·rhc !uocLious
Fk(Q ) or 1'(9) arc applied to their inputo tl11rlng ".It<' time
lntervn l -1 ;1 e ~ 1, and the coefficiento o.~ ( J) or n( j)
in (20) are ob·tiU.He<l at the output of l'il te!' J at ~he time
9 • .,. . Let the functions f( j , 9 ) be reprollontod by the
orthonoroal system of pulses D(0-kc); k • 0, ±1 , ±2, ... :
a(.j)i(j,9) • n(j; L:;d 1 (k)ll(&-kc)
Ill
d1 (") =
'
J J'(j,8)D(9 - kt)d8
(26)
lff;oC/]
f
f(j,O)d8 • f(j , kc)c
lo C• Cf2
.112
k = 0 ' *1' . . . . . ±1/2£
T::e fu.c.etionn _f{ j, a) are g,.nera.11;; r.ot represented
exaci;ly by th<' "Ysteru [D(9-kcl1, sir:co tile
in (26)
rnprenents r st,;p funcLioa . However, if c becomes sufficiently sc1ull the mewi m111are de,1iation bot ween f(J 1 a)
and t!le sLi>p function becocic~ a.r·b.i t:-ari I y em all for i;hose
f:inction:; thist can oe i-;euernted .
Let the receiver filte1· j rro:iuce the ouo:;i;ut h(j ,9 J ,
-i ~ 9 "
i f ~he input 6, 9 • ! J b applied . The input functicn D(9-kc)
ll(~ ' +t)
produces the output h(j ,9' ) =
h(j , e -~-kc) i ..f e is sufficiently •mall . Hence , i;he func~
Lion a(j )f( ,J, a) applied to che i:lput produces Lhe following output s1&nnl :
=
t,
a(j)~ d1{k)h(j ,0_,;-,:c), k • O, =1, .... ±1/2c
•
a(j)
L: d;(<t)h(j ,-kd
(28)
•
nc •im~ e • +t . r.ec u~ S".Jt r ci -:>J~e
c 1 (i<) in f28 . The S'.llll ;;ie.d:'l che
e andh~ie-l1t f(,j , J<t)h(j,-k<) . l'hio
Wl intogrnl, iC < is sufl'icieutly
"; j l
"'
J r( j , a Jh ( J , -a }da ,
8 • 1 io ke.
.in
'!'his intec;rnl equals 1 for
h(j,a I • f(j,-9) .
(27)
·--
·-·
.:'.'( j, kelc frol'! ( 26) ror
area of s1.ripes of widcll
SUJ:I mu;; be x·eplaced by
s mAll:
·-·
d6 = li111 c
(29)
(~O)
The coefficie nt a(j) i~ obtuined ot the output of the
receiver filcer at the time 8 • ~ . 'rhe output 0 is obta.Uled
on the othe:· hnnd, i!" the function a(l)f(l,8), 1 '/- j, is
233
5 . 2'• COl'IFAJfDOJlS
npplied to the input. or receiver fi1. car j :
vi
Ill
n(l)fr<1,a)1>.(j,-e)ue •a(,)5f(l,a) r(J ,9 )d9 = O
.'t/1
Ji l
0 1)
-112:
:'be pulse response of t!>e rece~ver filter j :nu~t ":le f(j ,-a)
if the pulse responno of the ~r=.s.Utto>;' fil10er j is!( J , a) .
Transmitter and receiver filters :..re ident ical for even
functions f(j , 9)
f"(j ,-8 ) , .snc! rot• odd functions f(j , 8)
-f( j ,-9) .
Matched filt<>rs do HoL ne~d multiplieril to deta1'minf!
tl:e coefficient a(j) in (20) • . Hu ia frequently !ill a<.l v8Jltage over corrElBtor circuits . I.LI ~ene~al , one cannot
0~y whether correlntor- or :natch.-d fllt.ers are su;•criol'.' .
Mult ipliers for Wal~h functio~•, " ·•· ·, are ver,Y accur'n~e .
Matched filter s , on t he other hC1nd, I.lo not have to ho conntructed fro m coils and ca1oaciL o!'n , but may be d.rcuito
like the one sho"m in i"ig . 36 on 1 "go 90 .
0
5.24 Compandors for Sequency Signals
It is well known tbot ins-;Rntn.neou~i co:r.;ir-ession o! s
fr11quency limited Gignal producor. ~ si1:;nal , tb.a~ L<' not
frequenc;y limi ted an;ymo re . The renaon for i;JU. s is that
compre osion of sino £unctions al·1vuytt i;;er.crates burrno11ics .
fnin ia not so ~or ::ir.qu<!nc;y limi t"d fiwci:;ions componod of
Val ah runctions . Fig . ~7a sho~s ns an exn1pl~ ~~o char~c
ters F,(e) and l"x(&) .
P•(e ) • wo.1(0 ,9 )+
~
j
1
I
~ (e ) = -wal(O,e ) +
·x
(-1); aal (i ,0)-
-L:
±
,
cnl(i,9 h
j •I
.1
( - 1) sal.(i,9) ,.
L: cal(i , 9)
' -s
l:, cal(i,a)
"I
Son.ding these choraceers through a co~p=esso~ t..i.sv int
tho Ch&.1'acteristic abown by Fig . ~7b produces the Rignalc
•.:Ce ) and F;Ce) of Fie; . 97c . These s.i.e;nalscontain cxnctly
tho Bllllle Walsh functions as trhe chnractors F•(9 ) and 1>..,(e) ,
tbe;y 're only multiplied by different coerficionto .
"-o'.sider a compressor characteristic ,, = Eerf(,/1{2o) .
5. S:ATIS?fCAI.
234
PROBL~s
Let 'ti 1 ( x) = \.1 1 ( - <>:>< 'l!:x) be the ampll Ln:;lo <1ist1•l hut l on :rune.
t;ioa of a .•. ignal bc1'01·e compression . The ftrnction W2(y ) •
w,(-o::<rr~:1) follo..:s truo (L . 11):
CondJer fu:t hcr a sigi:al courosed or tt.e 16 ',1ulsh func.:iou"' oi' f•'ig . 2 . A11 1G 1·u,,ctioJJs c11ual 1-1 in t;he inte1·val n < ~ < 1 /16 . Aii:ong the 2 14 binar;; charac Lers that
can b<' produced !'::-om LLe 16 !.u•ction:; there ; p 1 ~
cha:i c:er 'o'lit!: ...11~'11tude 16\ .. 1) • 10 in t!:riz interval; 11
16
~ (
hnve Lhe ~:r.plitude 1 '(11)11( -1 ) = 14 ;
0 ) chnracteru
0
) Charactcrr h<tVe tho arr.plituile 14( 11 )+2( - 1 ) = 12 ;
120 =
etc . The oame result holds fo-:- tin.v othr>r ~lmc interval .
<;}
(-g-
Her::ce, bir:n.ry +;l1r-u:~c Cf\ rs componeC of ~,'a.;. sh 1'\lr.cLions have
a .:>ernoullia..."l aap:ituje distr·ibutiOll . Let ~ cbarneter be
cooF-o:;c-d of r.. ··~alsi.. !'"unctions 'fl'ith rur.plitude +~or - a . T'ne
probabilit;v p 0 [ (:r.- 2b)n) of swop~lng; an =rlitude (m-~)a
equaln:
The dintril'!Ution func:iv:_ i!: W9 (:x),
w,. ( x)
«hc::-e [ x) denotes ~he !n~e~t i.l.ltee;er s'1al-er or equal x ·
1
17 ( X} C't!l be SfiJrOxin:ntc!! -fOl" lRJ."~e -v aluon Of tr. by the
6
e1·ror ~·nucl..io n:
\I (
'a x. ,'• "•
'
~
(1 +
X
1
er 1'l;' c£
c.
1l 1 (x) ;
1
:-ne dc::-ivai;ive w, (x) .i.~ sho~ in Pig . q8b . Cocpressor cnsra.ctet•ii;it;ics Tl= Eer!(CA{2o 1 a1·c shown ioro • 0 . 5E, E
n.na ,'!:: iu Fig . 980 . '!'he corr·esponding; denai ~;y functioll
235
'i . 24 COMPANDORS
f
F~JJI--
·E
a
}
·f.
«
Fig .97 Compresoion of sequer:.cy
..Utiplex eignalc . a) origi nal
signal b) compressor charact erist:1.c , c) compressed signal.
Fig .98 (rigbt) Coutpression 0£
sequenoy multipl~x signals . a)
compressor cho.rncteristice , b)
density !unction of tho otatisticsJ. va.riablo
c) density
functions a.rter compression .
I
11
·I
:1
;I
;I
11[
'
b
I
It
-.,_
t
/"
•
,,
I
.,,
lit
" .'
I
I ,......I?.'::. ' \
'•
I:
•'
,,
• I :
c,
~2(y) lll"e
v z(:r) •
t
shown in fig . 98c :
z
exp((1-o 2 )(er!'-1
~2 (y) • i(1 • er!'(c er.(1
t>' J
f>J
Note thnt tbe Gaussian distribu~ion 0£ Fig .98b 1s trans!ormed into an equal diGt:r·ibution £0!' o • E .
Figo .98u and c also show linos denoted by "13%" . They
bold f or It non-reversible compressor with tho characterist ic T1 • c tor ICI ;!; 1 . 5E and 11 = ±1 . 5:E: fO!' lcl ,; 1 . 5E .
'!'his coutpressor clips e.11 amplitudes ubaolutoly larger
23&
;, . STATISTICAL PROBLEMS
tha'l 1 . 5E , which are 13% of the runpli-.;udcs in t l:ce case of
a Gaussiar. distribU'~ion . T!:is clippe1· will be discussed
in more detail l~ s~ction 6 . 21.
5.3 Multiplicative Disturbances
5. 31 Interference Fading
[,et a radio signal be trb.llsmitted via several :paths .
'l 'he samples of the sarJc signal intel"fere with one another
at the receiver . Consider as example a sine- v:a,re transmitted
via two paths . The samples A1 coe 2n'V 0 6 and .4, co~ 2n\1 0 (S - a,)
wii;h a delay di~i'erence ~. are received . The sum o! the
two s.am;:.les may be written in two fo r :ns :
A, cos 2nv 0 0 + A?cos 2rrv 0 ( B-Sv )-=
(?2}
t
AtCOS2itv0 Sv )cos2nv0 0 + A 2 sin 2n\1 0 tl s i n2nv 0 9v
+
2P.,A 1 cos2n v 0 9v + A,)
2
llZ
co s (2n\1 0 6-a. )
A phase sensitive receiver rccc:ives one o.f tbe two terms
of the second l i ne . The ampli tude of the s i g na l received
varies between A1+A1 and A1 - A 2 or between A 2 and - Ai. . A
phase insensi tive receiver det ermines the amplit ude of the
o scillation in the third lin(> of (32) . n varies between
A 1 +A 1 and 0 .
The mathemat ical reason for this variation of the ampli1,udea is evidently that a t i ne shift Sv o rr of an oscil1atio?1 co::> 2n v 0 8 has the same eff ect us n..n ari:plitude reve-rsal , cos 2nv 0 ( 6 - 11) = - cos 2n"o e. It appears reasonable to
use other f\lnotions for which Lhe equivalence bet~1een oime
ah..U"t and amplitude reve r'sal does not ho ld 01' does bold
for large values 01~ s . . only . A general theoret ical investigation of useful functions is mathematically very cOlllpl i c ated . It- is, however , obvious that a s uperposition o.f
time shifted , differentiated \-/al sh functions according to
Fig . 77 would not cancel by interference .
A sia:ple:r application of orthogonal functions for the
transmissi on t;brot1gh c.n interference- rading medium follows
5
. }1 IZV"'l'ElU'E!EJICE r'ADlllG
n•olll the Carrow bundWidth of' tl:.P 6j':;telJ! diSCUSSCC in Sec-
tion 2 . 15 .ror telot.;vr.o tr:111smis~ion . 'J'he cor!Cept. in ns
follow a : Fr equoncy di vcrzity is a wol t lcno>m method for•
improving t r anRmission To:i ability mder the inJ'luonce of
interference fadiug . Signals are "'10dul ated onto n"v<'rn:
carriers rather t~an one . .::-ore t!:o.n 2 or 5 carrier:: can
genorall;y not be US< d cue ~O ba."\dwi:!t~ li.nitationn . The
na1:row bandwidth !'&quired by Lhe ~y::tcm discussc<l in section 2 . 15 makes iC poonible Lo ut'.~ 1: nnd L1ore cnr1·i~n1
inst ead of 2 or 3 wi~i1out excessive ::cquii-eir.entn t'or ll11ndwidth . The question is, w11etl:er it. 111 worthwhi le to !'!'read
8 r ixed tr:mstsitter power over so r.tan:; Cll'.i~ier!"" . A st.ort
digression into k:lot•U re:;ults o! divcrsit-.; :;ran!'tmission
,-.,u.
is necessary before an nns,ver can b" !,Ci
L et a harmonic osci 1 lnti on with !'reque::lc,y v 0 b<' 1·udinted . Using t he Rayle1.gh :fadinB modol, one ootaic!l u.~ ~he
recaivor input a
vo\to~e
e(9) :
v(9)cos(?nv0 9 • o.(3})
e (e
(;:;:)
v( 6 ) is a slowly fluctuating env~lope, •·h~ch is µracticn.J.l.y constant duri.!li:; an i.ntcrvul 0 -l!- B, ~ S !! 90
nnd which has a llayleigh dist,-ibution wit:t tl'. e fo LI owing
denui.ty £unction :
e
w( v)
2v
(jT exp( -
P.>
I
v
• 0
1
6
?:
0
-.; <
iJ
.;e, ,
( ~4)
equals the expectation E(C 2 ) according ~o (4 . 53):
6' •(v') ~(v' (G>)
05)
The phase o.ngle o.(9) also flucturi~t>s slowly. 1L ~l\a!l
ha.ve
constant density functio1~ :
- r.: -=
0
a.
.=:: + r
(~o)
n < -n , a > +n
An 11opvovement in transmission l'olinbilH,y rcqui1•00 , that
two Oto inore statistlcully ind epcnrlent ' copies ' of tho niannl re receivec! . Hence , t>he rleni;ity function oi' the
5 • S'l'ATiSTI c;.r. ?ROBLf:!1s
,;oL'le di&t:r1 but..iOL of tt.e 8Jl.;>l:'..t..udes shall be a product
of den•it;; fnnceions (31+) .
r, number of i11et..hotis are known for tho rocepL.Lon of atatisti cally lotl,.pendcnt copies of a oigna1 . Space (liversity
U!:;es several '1.!tt<'r..na!: GpaceC. ~t.l.f ficiently i'tu' n.pnr-c . Angle
Cive:.~sity obtoins copic5 bj" :r.eans of Cirect10.ca! antennas
with. narrow Le~ . Two ;>olw:i~ed nr.tennas disc:-iminating
bet;1eea i·ii:;t.t ·ma left cirr.:ulnrly rolarized
provide
!'Airl:t inde11emlcmt "O!'iee in the short wave !'cgion . ih'e'lllency divcr·,,Hy l' "C'> :>eve1•r;l ~inunoiaal C()L'l'iar::r 8.l!d time
di;."ersity trar.srel ts -the ni.gnal .re~e~tcdlJ' ·
Sa~.~ing obt ~ined se·v~ral tnUepeuden: copies of a. signal
the ?=ble:n of making br.st unc of t:hec nrine" · There are
basically throe n"thod~ available . a) Th•' copy is used
which 11.:is tho larges"t ave1:ne;o rower dUl"iue; a tj me in-terv~l ak (o~Um11l nc l cctio.u) . b) >Lll copies w.•a added (~ual
ga i.n swr.:n-.tion) . c) All cop.Le$ are :rulti;ilied co.fore swn-
"'"""$
c.ntion t:1 f.:..c !..o=~ tt'i.a7- dep
··10 C!l the~r nit~t·f..t.~C
pol'l·er d.u-
nte:·v"l ek { naxinal t'!ll~o sm:mntiou) .
For n ·..i:i.ptiriso,,; a:: - he tl ree oe'ohods I et q statistica.Ly ir:dopenclent copies of the signal F(e) 'be available .
Let fading tra.nsfor:n copy 1 rrom F( 9 ) into G1(a) . A sample
f',(9) or thcrn:~l noise is acldod co G1(e) . Hence , the f ol=~ng an i
lowing
i~
r"ccivecl as co;y L:
(3~)
11,(a) = G (e) • g,(a)
G, ~e) i~ :-cpre:ier.eeJ. du:·il15 B short ti:nc a , hy the following equation nccording Lo ('3) :
1; 1
v
(&) = v, (9 0 )cos[;:>nv0 9+a ,(S)J, 0 0 - !0• ::! 0 lll 90
o..r::d n 1
&0 -~ s,
::;
a"re
:J
+t e,
(:;8)
.c;.ssun:i.ed t:o br. constant in the lnterval
a •;0 +te, .
of v 1 be::ig smaller than B
th!'P.~holJ v0 o:· , ;>ut:;i D£ it diffC':-ontly , tho fraction of
ti:ne whicl· v 1 iG srr.aller ~}HU. v 0 .fol lo>.:s ft·om (;\4) :
T!:e ;>robnhi!it;· r(Y1 <v 9 )
.v
v2 2v exp ( - rr
v' )d v
0
= 1 -
exp ( -v 'I~ 2
)
(
"9)
;/
5
.;1
INTEllFEliENCE FJ~il;G
23')
Le1' q stai;isiically indepeu<lent copiC$ oo received , all
h!IVing the &11Dl8 :li::tri.bci~ion . p• ( v 1 <'r, 1 le tue probability
=11at i;he pplitulle~ v, of all q CO>·i~-- "re &tmllor th= v 0 :
l'lle average po,,-er o~ ~nc copy t~ 1 ( 9 ) "in<i tine ir.terval of
duration 0• 1 tltut in an integer rrml tip lo OJ' 1/v 0 , fol Lows
rrou. 08):
·vr
0.-e,12
A.•Q
1e,
/2
G ~ {e)dS· tvf <e 0 J ~P 1 (e q )
tv/~1 ·,
(41)
Let P, denote the average noise po·.-. er l~~t1 :'>
with copy l. The 3ignal - to- noi:;o powl'r ratio ,
1
l"Pri.>ived
(42)
is a quantity lhat llu~;;ua::;e>= due ;;o i;>,e :a~ing only . The
probabil ity of P, be~nt belo• a tnr1>!'.1'0~C r. foll01<S fr'Oa:
(3'l) and (42):
.: I
v I /2P, < Pg /P,
'.F,
)
v ,'/2F,
p(P 1/P,< F 9 /P, ) = p( ·;1 <v 0
W(P,)
W(P0
=
W(·1• )
=
)
= 1 - exp(-v, / 61 ) ~ 1 - •3>."Jl( - 2F 0 /6 1 )
Let tho copy wHh the largest vnl ue J' 1 /P , ""' s,.lected
!rem the q available copies. ~'Le pi·o lrnbilH,y ~I.a~ P 1 IP,
is small er thnn P 0 /l', rov al:. cople-s follows !'ro:i; (1rn),
i~ all copies a.re stati5tically inUe;.iendeut. :
\l, (P,) • (1 - exp( - 21\/6 2
2P )
<1"I
•
ooJ
;>~
Tf-
dW.(P 1
)
0
1
=
('•"i
);•
J
(IO
0
qy\'l - e
•
t
•·•
)
~
o'' dy • t_,
I
"'
2P,/62
1•et ue denote tha overage signd - t;o -noiae pO>HH" :·at io
or
~acb copy by I\ /P, :
~· . Si' Al' fSTICfiL FROBLR16
The :iveragr; signal- to- noise power L 11Lio o!' oh<' best copy
' 2
is obt&ined "'it!:.. t!ie he~p o: tne rdetion (P1 ) = t6
= 1\:
0
P,q!Fr • (f ,/!', )
c
1
2:; :
(%)
I •I
'J'ne ratio (? sq /? r )/(Ps /P, l io r.hown i~..r'ig . 99 by the
point a dtmoted b;y ' b '. One tul:f rMdil y ""o that the avertigl'l signal - to- noise ;iowe" rnUo iocx·cnsoo inoignificently
if moro thnn th ree or foLt:· ucipie~ Clf'f· 1rncd for optimal
so'Iccci.on.
•
9
8
~7
-
•
o<
••
•
•
I
!:"
... 5
... . ••
-~·' •
:£ '
•
1
Fig . 99 .tncren~e or the average
s i gnal-Lo-noise po,,.er ratio by
dive::uity roccption accor ding to
PRt;11:11>.r: . q nu.:nber of received
copies of L!:.e signal ; (F 5q /P, )/
(?5 /P, • (average signal - to!'!oiso rower r9tio of q copies)/
(average signal- to-noise po;,oer
ratio fol' 1 copy) . a) opti.J:ial
sel~ction , b) equal gain sUOllation, c) cn:xime>l ratio Gui:u:iation.
t. 5 6 1 8 g 10
·---~---
J
q-
ltop l aci ng optimal selecUou b;y cqLtul e;aitt swnmation o.r
q copies yi<'lds , <1ccording to 11RENNAN , t he followi ng rela~ion :
(4-7)
P1 q/P, • (P 5 /P 1 ) [1 + trr(q-1 )]
Psq/P, etnr:.ds r:ow f or 1'hc ,,.,.,rage :;ignel - to-noise pow"r
ratio of i:l:e sttn of all q copies of the sisnal .
1'h~ ratio (F sq /:,)/{?, /P,) ia showc in Fig. 99 by the
:poim;s denoi:ed by ' b ' . Ovti:nal selection and equal gain
"umm9tion differ on:y slightly if 2 copieo are used (g •
2) . However, equal i:;sin sunmiation yields
i.J:lpr overnent
of ~ . 5 dh over optimal selection ii' q • 10 copies are
"°
uoed .
for maximal ratio summation th A ampl i.tudes of copy 1
in A t:imf' interval o f dU.l'D ~iOll 8 k il:I :nul tip lied by a
weighting factor which iE propc.,...~ionn~ to -;;::e rmn-vn!ue
of copy l und .:.nv~··st!'J.;J prt)po~cionnl ~o the .r•.ns-val11~ n!·
tbe noise of ~nat coiw . llR3t:Kft11 der·.Lved Lhe fo ll owing expression replacinr; (tlf>J and (47) :
Psq /Pr • (P,/P,)q
Psq /Pr now
denot~e
(48)
the nverage signal-to-noise poW"e!' ratio
of the weighted surr 01· q coµiel' nf ~h,. ~.Le;r:al .
The ratio (Psq/l ,)/(F,IP,) is rhown in Fip; . '.."J by tlle
points denoted by ' c '. l'.n.v..imu.l ~'l:ltio mm:maLion is l"lOr.1ewnat bett"r than equal e;:i.ill SUJDJ:li;tio,•. The di.f.::erenco is
les!"l t.-han 1 dB for tt.e l"ange v::....
vrclu~a
of q shown i'lr.d DF-
proacbes 1 . "5 d.5 for inr :uiLe ·.raluA.o of <i .
The increaoe of enc o.verage ~igr.n.1-tc -uoise po·..rr:-t· .i:aLio
provides a
good menns for cornpariniz. Lhe variouo m<"tllods
fol' utilizat i on of copies of the oig1w l . The J'rtic~lou or
ti:illl dUl:'ing which tr'll.!JSCJission i" possible i!l, ho.,iover ,
a botter measure foi· rne r<>:'..in\.:lii;:y or a link . Equot.lons
(39) and (43) yield nuch n zeas.Jre . The r£rst g.ives tnc
fraction of time during wticl. 3 \'Oltnge v, ia ~olow n
thveahol d v 9 , tho uecond t!>e fraci;ion o.C Lice d uring wai<.;h
the average signal- co-noi se power ?"ntio P 1 /P r ir. bolow u.
thresbolcl P9 /P, . .Lot us now i·owrit,e ("3) by int!'oi.lucin~
the 1tadl.an Pg
; ,, for which ~,ilP ) equuls t :
9
i
#(PM ) •
= 1 - exp(-.<?.,/6 1
(4J)
)
It; follows :
ln 2 '> 0 . 69}
1
2P,/6
•
(P 0 /P,..)ln 2
("O)
+ 0 . 693
P0 / J,..
Equation (43) may be rewritten :
ll(P, ) ~ 1 - exp(-O . f.93r./P., )
'!'he j)t'Obnbility o! 1-, /!' r being larf:et" ehnn Pg IF,
P(F1 fP1 >F 9 /P1
)
=
1 - W(P )
9
* e:xp( - 0 . b93P,/?,.. )
(;>1)
t•CCl:l"C! :
('.;.2)
P(F1 /P,>Pg/P, ) is shown in Fig . 100 by tbe CUJ.'Ve Cl • 1 .
e Ol'dinate O~ i;hat figure shows ehc: pcrcentaF:e of the
6
: ~ '~4Utt!Dolirlclom.atlQll't
Tb
?42
5 . STA'US1'.ICAL PROllLEl'la
time duri ng wh i ch F1 is la:'ger than a ch.r·eshola Pg . Here
P 1 and P g are dHided by the r.tedian P,.. fo r· normalization,
l i q copies are received one ob ~ ains l'rom ("4) the probe'bilit:r that P1 /P, ls l arg<H' chan P 9 /P, for at least one
copy :
p.(P , /P, >1'9 /P,) ,; 1 - [ 1 - exp( -2?9 /0'
)J"
Orie may t ·ewrite this equation using {SO) :
(.53)
p• (P , /P, >? 0 /1',) is shown in .l"ie; .1 00 bythe solid l i nes.for
q = 2 , L ana 8 . 'l'hese Cu.rves give the percentage of the
time llurin~ to;Liclt divcrsit:•t' traJ1smission is possible Lr1
optimal selection i s used with 2' , 4 or 8 copies and i f a
ratio P 1 /P, lai'e;er tl:an P g /P,.. is J'equired .
'!'he C.aslled line;; in ?ie; . 1 00 sho·..; the percentage of the
time <luring i,..1h i.ch P 1 ::>f a sua of 2, 4 or 8 co1>ies is lar;~ei- tltun the thI'es!10ld P,. Hence, cney give the Sr-action
of cizr.e duri n g whicn diversity transmission i s possibl e
i_f equal gai rr stw::nation i s used with 2 , 4 or 8 copies "(l,.Dd
i.f a i-at i o P ,/P, larger titan Pg / ? " is required. These cux~
ves may b~ computed «ith the help o.r (4. 97) for q; 2 'while
nume!'ical methods have to be usf'd r or J arge;r. values o! g .
5.32 Diversity Transmission Using Many Copies
'11be r.iechod.s discussed i r1 r:.he pL·eviouc section for obtaining statistically indepen(\ent copies oi a s i gnal usu -
ally proviC.o only a few co-pies . Polarization ciiversi izy
cannot yield more than two copies . Space and u.ngle diveJ;•,...
si ty could theoretically yiold many copies . HoweveI' , co.:nsidei·ation of co~t and the space required limit this nUllll>er i:i practice . Foi· instance , antennas have to be spaced
several hundred l!leters apart for space di versity in t he
shoTt wave rggion . Frequency and time diversity- ~e the
only pract ical methods that can provide many copies of the
sit!;naJ. .
In order t>o apply tb.e curves ot: Fig . 100 i;o frequ.ene;t
!>·-'2
D!vi;:RSITY r~..U:S.'11SSION
H~ . 100 Relative time 69 dunng
wh1c:i ehe noraali&ed signal po wer of a diversicy l;ransminsion
excflcdu a threshold P. IP. (tleri ·:od from f i gures due to BRE;lNAfl ) . q num·oer o!" !"eceivod copies ; $Olid lines : optimal se!oc·Uon ; dashed lin e!l : equal
1~u.in sua:mation;
d.ci:;hod-dotted
1 iue : recept io11 wi tho1it <Ji ver-
"it:r .
end ti.me diversi10y , one '1lU.\lt keep in mind tl:·1L q ' Oqtrnl '
signal" are radiated '1hilo ocly one n1gnal is raa111Led Jor
space nnd llllgle dh•ersity . Given a cer:;ain !lve:-ege Lr9n!>::tittcr power, the average power t·adie.oed pe:- sit;na l is
S1llall er by a factor 1/q for f «equency and Lime <llvcrsicy
than for space and angle dive1•1Jic;r. 'J:·l'.is d~·awbe.ck of fi·equoncy and time diversity mo.y , o.r course , ·u e compousate<i.
Instead o.r using q antennas !'or l'eception an in r:puce and
englo diversity , one may u~n ono an~e.u.ua with q-~imoc the
gain; this would just compenoote ~he reduced si.rnal powei•
or each copy .
Lot the transmitter power a.nd the receiver antcn:m be
!ilced . Replacing ord.innry transmission 'q = 1) by q-fold
frequency diversity will br.i.nt. e.r. impr·o vemen-; only iI ~he
average signal-to-noise power ratio a t the receiver i nput
is :!.ncroaaed , despite tho dccroaBo of t he aigMl-to-uoirc
JlOWcr t•etio of each copy by 1/q. Reduction of the 4Veri;ge
signal power per copy by 1/q reduces ,;be media11 P,. in
Fig. 1ou to P,../q . Given a ce1·tnin threshold P• the ratio
P, ll'., l>ecomea q?, !F.. . Using q- fold frequency di vLr~ity ,
the traction of t;~e during which the average signal po\fer
exceeds qP• IP.. oust be larger than the fraction of
..
,
5 . S1'1..'. LS'PIGAL PaOBLD!s
~ilte
durii::;; 'Ahicl. thn avercge si!;Jla l ;>O>fer· exceeds p IP,.
0
for ordinary trar:s11i~"ion. Consider an eX'llLple : Tl:.e curve
q 1 iL Fig . 1CV yi,.lds .18: 9'>'!& fo:::- 111logf0 /P~ ~ - 11 d.B ,
while the cur,•es fol· q : 2 yield for 10 log 21-9 /P,.. = - 8 d!J
the values 68 • 9';J% And 99 . ;>%. 3enc() , l;woi'o_d diversity
increases tho frac~lon of til!ie durin~ whlch 'the average
::ignal µower ir lari:;ec· t:.. an P, !'rem 'l~;N to -·):!:>or 99 . 3%.
On•-1 eta~-- reaCi l:; see thnt nuch a!l i-:x:prove:ncnt is possible
ii" the cu.rves q • 1 nnd q " 2 a=e separuLed horizontally
by
~t lea~t
10
lo~
q = 10 10£
2
~
5 CB . Thr.
poi.z:.ts denoted L,j' :-1 en ' Bbow <ncre this separa~iO~ i!:; just ' dB .
:Svi1en1.il,y rwo:·otd di\ler~ity wiLh equal g3Lu nu!lltl&tion is
worthw1Ule i.!' uati"ft1c tor;:,.,.. O)'e!'alion occu 1·~ Jo·r more than
1
1
...
'•0% ot
i-:hil~
l.h~
the ticio, wh~le opttmel selocU011 will be wor&hor.l,y .:r !iati~r1.1c~or·y ope!-·aiian is µot1t:5ihle for more
;:~~of the"ti:r1e .
BaseC. on thr nax:&E> con~idera-;iou.s thf!' oepara:;io= between
tt.e CUT':e" q = 1 a:>d q " 'I must be at lenst 10 log 4 ~ 6 dE
o.t:d h<'tweeu q = 1 an~ q = 8 at least 10 log 8 ~ 9 d3 in
orde:· to :i:ake I •u,.h l 1 or eif!!'lt>fol :l di v~1·sity wori;hwhile .
The points in fii:; . 1cl'' denoted by ' b dll ' Md ' 9 dB ' inaicate 1·1here t heGe sepnratio ns fn'e junt b and 9 dB . Optimal
snloction i• worthwi il<> unly i.f ea~l1H' actory operation is
pos.:;il~le at. lenc- /lJ'fe- or SOJ\, of ~he t.;.i.mn . ~··oi., equal gs.in
S".llll!!.ntion o~: Lh..rr.~ points '3 dE. 1 , 'o dB ' and '9 dB ' arc
locutc<I on tbe lin<' ll.2 = '-::;!(,. F~equenc:1 diversity using
equ~
gain :;mni>sLio:i i,.. worthwlll.J.e if
~nti~fnct:ory
tion i~ ;>osd ble a~ !east 40~ of the time
l!o~t '.>Jltli l..be large:-j, Jillltiber of coricn .
!i; has been ctincur;sec in section 2 . 1~ t.Lat
operaand is then
a cert~d.Jl
fr·oquency bo.ndwldth may be well utillzod by sine and coz .We fU l ses . Six t"1ctype circuits 1·equiro 11bout 120 Hz
bn..110wtdtl.L acco.rtlin; tQ 1'able 4, twelve circuits a.bout
.:40 Hz . ;, ,;oi;al of 2400 E!:: bandwidth are t·equi.red for tenfold !requc;ncy diversity tra!um:issior. Of theM twelve ci.rCU it" . -,.,. spacing or tte ten copies by n;ul tiplos of 24-0 !!:
i~ suf.t:icieut in the nhort wave rei::;ion .
6. Signal Design for Improved Reliability
6.1 Transmission Capacity
6, II Meas ures o l Bandwidth
Ii; we.a recoe;ni~ed 'tcry e<rrly during Lhc <luvelopmeut of
commun;i.cntione that the possi ble tra!lS!lliSuion rate O! Gy:Dbols o! 8 COOllllunication clrnnnel deµemleo 011 its frequcr.cy
response of attenuation and phase S!Oift . J>'or innt:nncc 1 the
!s:ious theorem by NYQUIST [ 1) and r::Jn :·lt:LLE!'! ;: <>, 31 utat.es
that one independent symbol may be t:-an~:oilte<! per ti.me
ineerval or duration - throi:gh an :.~c'lli=c<! ::':·c~uenc:1 lowpass f ilter or band1<idth 6:f . ,,·here
T :
1/26f .
(1)
'!'he tre.n$miasion i·ete of "ymbo1':: is Je!i11~,1 in tc.iu
caseQy ~he nwnbeL' 1/T o!: independent s:1rcbolfi tran:;;n:itted
por unit Lime [ 11, 51 . SH.,1.}fNO!\ t ook i nto ucoo\.lJlt tllat vhc
po~eible branswiaaion race or i.nf or:r.ation rl'l i ondna on statistical disturb~ces as well as on i;l !e r 1·equt11cy r r:;ponse
or attonuation and phase shif;; [6 , 7J . H~ oM:nir,..,d the ce lebrated formula for the JXlSnihlc trqn,,miudo::. rate of
information through an idealized .:-requency !o~..:pa.ss ri: ter
~der the influence of additive t~erc~ ~oise,
C • 6 f lg(1 + P/P61 ),
(2)
\there C is the transmission capacity stated, e . g ., in bits
per seoond . 6! is the .f requency bandwidth ol' t:ho ideal 1zect
10
'-'Pas s filter and P/l?61 is the quotier:.t (ave<'oe;e sie;oal
Power )/(aversgo nobe po1<er i n the band. O lli r :; o.r) .
It i s important .ror the presen L purpose that (1) ""
"ell e.e (2) contain the frequency bandwidth or . A consistent t heory of communic11ti on b aaed on complete sya~ems of
6 . S!GH.U. DESIGlj
246
orthogonal functions require~ a defini.t:ion of tra.'ls:nission
capacity chat does not need the cor.ceft of frequency .
First, frequency is eliainated from the average noise
power P., in the frequency band 0 :Ii f ~ Af . To do so let
the noise , represented by e vol~ap;e , bt app:'..ied through
o.n input resistance R 0 to an integrator t!:1H; integrates
tbilS voltage over 11 ~ime i.m,ei·val At . A totnl or , integrritiono i " per.formed. Tho int~e;rotor• output voltage q11
the end cl' the ~-th intee;rabion wlll be dona Led by v,. ll'or
<v,)
thormnl noise ·the mean vol tae;c
is oqunl to zero . '!!he
i:ienn square deviation from ;,ero, multiplied by 1/;10 , is
an o.verage po•11er and may cha.:.•acte.c·ize the noise just like
P4r
does:
(v}.!Ro) •
(3)
Ueinf: section 5 .11 tllis result :nay be generalized . i.et
g,(9) of (5 . 1) be a '-'Oltai:;e across a resistor Ra which is
caused by ~he::-cral noise . -;'h" no~atior. v,(t) will be "Used
1.nscesd of e; , (a) and the Iunccions f(j,9) in the samo
cqu&tion a:re replaced by tllo normalit.cd voltages ~.
whcro Vis def'in(?-0 as follows :
®p
1
= 'i.'V'
Jf(J,9)f(k , 9)d9
011
T'
I 'l l
J V(j,t)V(k,L)<lt
=1; T
=
e
- I'll
'rt1e coefficiem;s of (,? .1 ) are repr'o<lented by normalized
volt~i;es using the notation 'r'V,(j) . :Equatioi: (5 . 1) theA
ansUBen che .!"ol lo7:ing .:orm:
g,(0).
•r'
v,(i:) " f;a,(j)f(j,e). v"'
j:O
a,(j).
•r'
en
v,(j) =
~
v,(j)V(j , t)
1
,.,,
J g,(e)r(J,e)ae ·w I
- e12
(5)
joO
v,(t)v(j,t)cit
.. 1·12
Lot Lhe voltages V· • 1/, ( t )V( j, t) be applied to an integ:ruco:r and integrated from -tT ' to +t,T' . The output ·?oltage at the time t.'r ' equals -'I ,(j) i.C the time constant
6.1
1 H£ASURES Ol' llAHDWID'IH
of the integrato:' ir chorer: equal to tne ur.it of
The quantity Vi(j )/Tio , witr. dimen~ion or power ,
derived f rom the 01 rh put volt~p;e . Lot V; ( t) if ( 5)
red, divided by !R0 , and then i.nr;egrr.Led froo _,T '
m
1 f V~(t)R~ di; = /'V;
T .f.11
time T .
may be
be :<qur;to i'l" :
00
..
,
,.1
.
• ;\J
)R
(6)
Since che left b~nd side is the $.V<'I'nge ;iower of the
noise sample e:, (e), Che riglch hand oido musL have the suwe
moaning. A certain t~rm
j )/H 0 lr: the sun represents the
ave.rage power of tbe coa:ponent j, or f( j , 9), of cl1e noise
eB!:lple g,(e ) . Ave:'at'"inb the terci Y!CD/R 0•1er 1 Slllllples
of noise g,(e),
Vi<
lim
,_..,
'.l! <-.
L V'(·j)l'r'
"' , a'
,\:I
(7)
yieldo the average power FJ of tho compo11ant j o~ tbe noine
s11111ple s or of "tl'l' noise" . The dislri'lJution of V,(j) is
the same for any J in the case of t~.em,l noise . Henc<' it
makes no difference ~·hi.ch com_pone~t i r aveL':?.ged . ln this
case one inuy repluce 1 the avera5e ovo1: ~ by 1;be nv..,rai:;e
over j . Furthermo1'e, the average oJ Ir. componeni;n eqltals
:n tiltieo tho avoragc or one componen L:
m-1
1
(l:v'lj)R)
l•O .l
c
=
m(v'(j)lr')
A
a
=
(8)
The value o::- P 1 ie quite indepcndeuL of tbe orthoc;onlil
syetom [ f( j, 9) l . Multiplication of tho noi"e sOll!ples g, ( o)
by the ru.nctiona ot cm or·thogonal $,ystom (h(j , a)], which
have the same orthogonality intervalo a• Lhe functiom;
f(~, 9) and can be expanded into a :>cries according to
(· · 5J, yield voltages v;(j) in,,tead of v,(j) . Ii; i'ollo•·i;,
however, ~fro!!! (S . 4) to ( 5 . 12):
(V!(d)R~) • (vl'(j)R~)
(9)
~·rhig exchange of time and ensemblo average requiree that
ha 01·godic hypothesis is sa;;isfied .
>.> . $1GNAL DES!G!{
Tt.:!! fil:.i;:hcs Yl•e in\•e5t~gntio!l -qbo,:t the 1·e;:.lace~ent orf 61 :..n (2, . ~ct uo r.o"N turn to t;.ile ret lnceoent of !lf in
l.h~it. ~quation b;i a J.1U!'1Jn:eter th~t i:"" i11\!r•penCent of sino
and cosine _fWlctiona .
;:t:+.1 or·c-bogon.!:1.l .fWlct;ion~ l' (V , 9) , fc(1 &)
'
:rmy bo tronnmitted through a
commtmication c!.:<ur.!'1 during tt-.e oL'L:10-onr.oliLy interval
-~ ~ e ~ i . Coi.;.sidcr a~ =-~eci:;i.l C0.8(? t:... ~ :~1Lncti0!1S Of the
Fourie!.· ser.:..es:
Ass1une t:hei:1
r ,(1 , '3) ' .. .
!{O ,a J = 1, ~,d,&,
-i
~
a
,
c
,r, (l ,e)' r ,(1 , 0)
~ ?, l
=
• '[;:,coo2:-1S,
fs(J.,0J
1 ... 1 , 9 : t/T .
(10 )
':lleae sin~ an<l oo:'=!ii..c £!£~£!!E2 are O!'t,J1onorr~al in the i.nteJ'Val -! ;,; ~ :l! f nnd widefinec our.mdo . J,et tnoro be stret ched by che sub:cLic1Jti.on 9 ' = 0/~ aG in . ecLion 1 . 21 :
!(0,3'): fi0/~,9)
•f? cos 2ri(0(~
r,(i,g') =
-t
~
e'
f2:-inZ':'li(61~' =
~ ~,
-~s <
s .,
t
,·2sir.::>n{11ns •
r$(i/~,3;
~
The <\urcotiOJJ ot th<' orthogonal.icy ictcr.vtt l bas been
nc!·eased .fro1t 1 •;o ~ . Tit" numbe:r o.f functlorn; transmitl;ei.l
1 er· u:rit of -im£. sl.1ull rCJ:nnin co:'l.Gt.w:.tt . ~(f:l ... 1) functions
r.n.!:1.t b~ tra!lsrr.it:t(I<! int.he i.::rt:erv-al ~ - t.in.e ... as large. The
i:ulex i r-.ms fro:n 1 :.o k, .:l:ere k in defir.ed by the equatio:.i
j
(2-+1 ;.; - 2k.. ~'
k
0
t(l
T
!-
(12)
+ •J /2~ ) •
Let ; app:roacl: j nfinity . The tice limi tcd sine and co31.Yle elements becor.rn tl~i:; periodic slnc o.nd cosine fUllCti ons wi~h tte frl'<!Uonciec i/s ~ v a !'T . 'l'he frequency V
r11r.e f1'om v 1 = 1/~ ro v. ~ k/s Gince i runs from 1 to k.
Ttl<' difference v, -v, , <ienoted as tlie frequency bandwidth
Av,
~~
given by
~v = l~T =
:im
1--
(v--v 1
)
:i=:
!-«>
~/~.
6(21+1).
(1})
MEASURES OF 3._o_!ffi'•'IllTll
•
6 11
A!•
21+1
-pr•
I
0
2 'i
24-9
(14)
lll/T is the nwtber of o!"thoe;onv.l func~ior.s tr·nn51tit~ed per
unit time T. The bandwidth t:.v or or is a mca<;urc of t;he
!1Ul11ber k of orthogona. n:lno 01· cosine element" t1·011emit ted during the interval of ot'thogonali t.Y , if the UUtUber
of eletnonts a11d ;;heir orthogoa11-ity inter'1aJ -H · a :i ts
approaches infinity . Accorcing to ( ·1~) one r.iqy use a./"'
instead of t:.f wbicl: :_,, tl:e nUDber of s~e !llld co::ine ele-
oents tranS11itted per U!llt tllt!' : •
The freque11cy bandw~tlth Ar tr. only e measurn of the
numbei· o! sine and cosine !'unctions tllat can be t:.;1·w.t1tlit 1;e11. Oil the other lu'.U1d 1 m/' 1 mn;v be int;erpretnd ne. u 1:1eaauro or tlie number of or;;hoi;onnl fwictionn t;1nt CUll be
transmitted per =i~ .of ti:n" , wHhout rercrenco to Ginc
and cosine funct i ons . r.encc 1 m/T is a i;encrr,li:uclon or
the concept of !"reque:::cy bll!l<!"•idtt .
The difference b81'WCe0 uf W-.d 1>/T goes beyond t~.C 6=eater generality of u.;': . It is o:·ten cu.aber,;on~ for theoretical investii:;ations thai; cvN·y funci;ior: occu:;iie" e.n ir:!inite section of the time-fi:•equency- c'"o:nnin . ':'!Je hntched
section in J!ig .1 01a snows th" S<Jction of the time -froquerto;ydomoin occupied by a i'w1c~1on l;hat diffe1's from ::el'O in
tho interval t 1 " i; ~ t, onl ;y . Fie; . 101b sh owe t!Jc ~ection
occupied by a i'requency- limi1,e>d f;.uici;ion that is uox:-zero
in tho interval f, ,; f j r, . The batcted area.!! can be Dade
finite only by truncating Lhea 11.l'bitrari:y at so:&e value
or r or t since ther" ai·., no tim" and 1-reque1tcy-liir.ited
tunctions .
It has been shown in section 1 . 33 that Lhe!''' i.t1 n clu"s
ol' ~ime a.nd. sequency-limitod. t'Ltnctions . Tllis makon it tempting to replace the tilnr - l'requency - domaiu by u tin.enequency- domain. But this woul.d unnecessarily dit;tint;uisil
the Byl'tem of Walsh l'unctionl . ~t is better o i•·t. roducr.
8
'time-function-domain' . Consider a syste~ of functions
(1'{J, 9 )] , which are orthogonal in the fillite interval i;,
§ ~ ~ t, and zero outside; j
• 0 1 1 1 • • • j,. ... j,. ... Let
b . SIGtfftJ, DESIGN
sigm1l,; be cooroscd of .l"U;lctions ;;it:-. the index j rlllUliug
Croo j· to j 1 • According to Fig . 101c, tlu: cin:e isplottea
along the "bscisoa ond the indices j or ,j/! = j/( t 1 -t 1 )
along the ordinate of <> cartes ian coor<linatE.> sys tern . 'l!he
signnls considei·ed occupy i;be hate hod sec ~ion of this timerunction-domatn. These s ignals at"e "xactly ~ime <>nd "function" lin:ited .
;.et us investigate the connection betweei: sequency
bandwidth t.:p and :r./? . The system of functions
(f(0,9),fc(i,9),r 5 (i,&)J , i
= " ••• 1;
-1
a
a !;
(15)
shall be orthoc;onnl w1d ?i ,;!ca.J.1 equal ~ho nwnbcr or zero
C!'ossings in the orthogonality interval . The same co nside!'ations apply as l'or ~he sine and cosit1n olements . :Equations ( 13) and ('ou) are obtained agali1 , buc the normalized
sequency u has to be "uos~ itutcd for the normalized :'.'requency v . Purtber:to!"e, 6v is rerlaced by Au:
li.Jt
i~
= D.~
y
=
(u.-u I) •
.
la
i~
2~+r1 • .Ll 'T
"'
k/s = H21+1)
(16)
(17)
Comparison of (13) and ( 11;) with ( 16) and (17) shows:
n) The norma-izcrl Crt!quency bandwidLll ov ir1 a measure of
(>{2 cos 2nv0 ,
the nu..mber of functions of the sysLem
'(2 sin 2nv9) ~ha~ cen be crru:slli tted ln a normall-zed time
int.erva1 of duratoion 1 .
b) ~e nor:nalized sequcncy bandwidth du ill a !!lcasure o!
the nw:ibe!" of ~unctiono of i;he more general "yatem { r c<u, 9) •
f 5 (u,9)l that can be transmitted in a normalized time in~erval
or du.ration 1 .
c) The .frequency bnnllwidth Af = Av/T i~ a opocial case of
the scquency bandwidth Ore
=
6µ/T, but m/2'.l' ia an even more
p;eneral measure of bandwidth since it applies to all complete sys~ems of or~hogonal :unctions inclu<ii_ng those to
whic!: the concept of aequency in its prenent definition
is not applicable . m/2~ equals "one hslf the average nUlllber or functions tro.n~mitted per unit tiloo T" .
• 2 <rRJ,.tiSt:ISSIOi< CAPACITY
6 1
i'i .101 Time-!'.:-equ~ncy-domain and timP- fUI.ction- domain .
a )gsection of t!oe timc-frequency-domai:: occupied by a lil:ie
limi ted signal; b) r.cction of th~ title- f:-,.queocy- doauin
occupied by a frequency liaited nie;rtal ; c) nr.ction of the
time- funct ion-domain occupied by a time ur.d runction lirr.i-
tod signe.1 . !",-f,
=
Af ; t , - t , •
·r ; J ,-(J , - 1 )
=
m.
6.12 Transmission Capacity of Communication Channels
Consider signals l'x(B) i;bet ni·" composed of t;hc ~ya~em
or !unctions (1'(0 , 0 , fc(l , s) ,f ,(i ,B ) ] or'thogonal in the
interval -1
x.
.e
11 2 ,
:r 9
li ' •
...
(18)
l.~t Fx(9 ) be tran""'i tt'>d t!lrou...-h a co=unica~ioc channel .
Then
for the ti.me be:lng thnt the fUr.ctiOOS f(0,9 ) ,
an~ <1elayed by ~he
time 9(0) durin g transmission . Using i;be at~enuat ion co ••f!lcients K(O), KcCi ~and K,(i) of section 1 . 32 , one obhi:l~
!~r the signal at the receiver:
!
SSSU!l'e
cCi, 9 ) lllld 1'$ ( i, 9) at·c only attenuated
Fx.<e) = hx(O)I [O , e - a{O)) +
f
(hex (i)f, (i , 9- 9(0) ] 1
(19)
•••
bsx (i1~s [ i,3-9l0,] l
b~(O )~K{O)ax(O) , bc x(i)=K,(i)acxCi} , b 5 x(•) K$(i)u 5x(i)
!Ilhe recoiver shall determine which one of the noositle
Cbe.rac•ers F.,(e) was cransmitted . The lea3t-mea.D~ squ&rP
c!eviation criterion sho.11 be used for the decision . Sanplc
6'
•sna1s F .,M( e) ll!Usi; be produced at the rec6 iv>'r •hic'" e.re
11 8 similar as possible to the rece ived signals Fx.{9) . Jt
18 t hen necessary to decide which intesral T( ~, x) io c:nal•t!st:
''· 5lt.WAL DESIGN
1 .. 112
J°
I(Y >Y,J •
[Fxr(9) - :; , .. <e>:'r15
(20 )
1... 111
tt
=1 , 2, ... x, . . ....
j
x.•1
2 , ...
1
Le-"t u:- ~e:sucu tr...:: sa:.111p_e- :·u.~c;.io :_ s : XM\0; could be mnde
e:xactly
~•
i;o the
r~ceived
:ii.1n1al1> :
.,(e )
Tlic~,
eq·~al
(21)
i"i~gra•
• x
.r~ ,.1J is tl:cnzei·o . '!'he iJ1 Lc,g::-.tl I(9 , x) tor
Liust djffr!:.· from ~.e r o bJI tiL lr:ar:.1.. AI~ 'l'he minimum
~= (..:~.u:o- ct '1....t·bi~r&1~ily
!:.inc~
i& is only
;.•oss.:Yle to Cctern..ir:c a fir..ii:-r:> d fferr:n<"'o . :i; ftjllow·s .rrotn
( 19) trnd (20) due to i-he o:rtho"o,-111alit;1 of the aynt"in
{-" O ,:l },fc( i,& ) ,r,L,Sll:
di.frert:i<.:c;;
so:J,ll
""
l(~ , x) ~ [L.,, (OJ-b, ( 0 ) ) 1 •;8{[bH
(i)- b 0 .(i) ] '1
(22 )
"I
- (bs . i)- bsx iJ]
2
I
l
61
Cc::.:.1<ler thor..~ si;rr:::.lr. Fn(d) that d- f'fer ! ' t·orr. F~,(9)
·In only one cl' t1he coel'ficieuLs q .. (Q\ , cxCiJ oI a,x{il ;
or:c of t-" foll >wir:g ... ondi~iou.i::> cus- I o:d :
(2;1)
fb, .. (.l-'t>cx\i)]
1
~~I,
:u,,.(i'- t sx (lJ)' ;;; 61
"'l,e iir. .. c~l valu•~s 6n(O), oa 0 \.i and Oas (i) by wliicll the
coeffi cl e1.>ts of twc signals mu,; t' d ii'fer 11 t tee LrnnsmiL Loi•
foll cw fro~ ( 1 ) ~ !l!ld \ ~ 1 :
'"·r(O)-n x (O)J ~ ,c,a{O) • (61)
I nc~(iJ-ncxCU I ~
I "s .(
l
- k
sx
oi I ~
oac( 1)
LI a ( i '
112
/K(O)
(td)
<
(.:, J
111
111
>
(211)
/lc(i)
/Ks { i )
1nt a,(O 1 , "'cx(i) a :od <>.$,.(i) be r c,,"tricLed to value s be~""en •A nr.d - A- 'rhe r.umbe!' of :>ossiole coel'ricients in
tr.~u
riv(•t' by
l ..o'
:-, ,
ra " .eAl\(0)/(11 )'" +1
teld
r51 :
Ol5)
253
The ' ones' on i;h.e
r1L~Lt
hu?;.tl aides ua::e i=to account C!lc
pos s i bility that t!.: coef!tcicnts coy l:ov··· the ,·al.ue zero .
TM l argest integer s tloett Mt i:::fy the .iLeq;ial t ci.,. (?~)
must be ca.Icon fo::- r 0 , i·" an~ r . , . rh.c perntissi ble vnlue s
of Lhll coeffi cient a.,. (o) 1.U'e 0 , ±t: a( OJ , ±2Au (O} , •• . iJ' ru
is odd ; for even r 0 ~hey are ± tAa(O) , ± i i> a(O) , . ..
Le t Kc(i} , fo r i.,. le , und K5 \ i ) , :·or l > 1, , 1,n flO
siall thot the fol I owirw re 1 etior.:; !:old :
(2&)
No information can be trttnr.mi :;te:..l wi t. !i s :~ins:-~ 1'u1tction
r«i, 9), for i > l e, rr f5(i , 6) , for i. > ls . ?oI' t1l:ropli f icat ion let us put
(?7)
1,
w!:lere 1 is calle d tlle hmdlilf.J.t .
ai:;t ed beyond the :>anclir.i~c but
The coefficients 00 ai:; leas~ I.WO
! 5(i ,S) , i > 1 1 must be ehw...bed
111.rormat :.on cru: l.:!! trwcic tiJe yroce~s is ~lrr ..:·ent .
;unc;ior:.:; f c( '.. , 6) and/o::to obtai116.di.f.fOrl'nc :i;isnal. Thi~ type or trar:><rr. 1,,.si.on i.; irlf·OSSicle if ti .... f1Ltt' nuetiora increases so l'apidly t.r.yvr..tl ttie ·oandlir:11t t.t~t• t. tee
cond'i t i on
..,
J.: ((b, ., (i)- bcx ( i )j1 +( b$ ,., (J.) - bsx(i)] 2 l
i.1 . 1
<~ I
(?8,1
is s atisfied for ar:y pair x ru;d ·i .
The number of disti nguil; 1able sigr:a~!: t=t:t. c&.:J l'e LransUtted during e. tiloe int:el'Val of dura-;;io!! '! i: t!.~t. ,,z;i """
't :: ·;he (Jl'Oduct
I
ro
Tfrci r ,1
1.1
.
Tb~ i nroi•mntion transciH~ed per uni-;; o: i;ime , or the trll.l1< miesion capacity of ,;ho cho.nud , is t!.s logaritlt1:1 o: tb1s
Product divided by ~ :
c -
I
;r ig
..2:, (lg .... ,
+lgr 5 ; )l
(2'1)
254
G. srnr;AL DES!Gtj
Lee;· us cons ider a special caae of ( ~· ) . Ii; .=:-ol1o"s fro111
(25) for r 0 = r ,
; r5
i· :
K(O) • Kc(i' = Ks(i), i
- 1 ..... l
All functions f\O,S), ~c(i,0) w.d 1' 5 (i , 9 ', i ~ 1, areatLN>uatod equally . It follows i'roa. {?.9) :
c
21
l 1 1,,. £
11
0
( 30)
Lt Lb e system of funct ions u sed
t ho peri odic sine
and coaine functions one n;a,y subsLitu l;o t.f f1•om ( 1 £~) an,d
obtainn :
C • 26 f
1 g r • 11 I
At';>
lg r 2
( ,1)
'!hii< fora.uln has the otructm·e of SHAl:?iO!! • s formula (2)
alt;housh it was deri\red untle1• dif.fcr~n!: assumptions . It
will be show!l in uhe rocsinder of ~his s~ction tha~ r in
(30) end (;,1) i s :-eplace<i by (1 • ?/'f61
i f t h e Same
3Ssu.mptio!ls are a:ade as in tho dorivetion of (2) .
Conoi<ler signals F,.( 6 ) composed of t' :·unctions .f( j ,a) .
>'"
'Tile orthogonality i nterval is -~ lS O ::
.. ,
l•x (O) •
i
or
-aT ;;
..
,L; ax(j)f(j,6)
t
::
tT .
(}2)
(L'(j,u)} = (f (O , a ) , fc(i , S),fs(i,9)}; r • 2k 11 .
'fh" iutcgrnl c!' Pi ( &) yields Lhe ave''"!>" power of the sii::nul :
•II
J r;Cs)•B
1
1/2
- J
· Ill
Fi(e/T)dt
=
Px
(33)
- 1/l
1
l~l
1' ,
-1 /1
, .•
L
a~CJf 1 (j,t/T)dt •
1~0
Inotoed o!' representing a signal by 9 time function
1"x(0 ) , ono n:ay 1·eproscnt i t by a point in a r-dimensional
c~1 r L esi<Ut signal space , accordi 11g ~o aection 2 . 11 . Let
~ho UllH vecLora e i , J = 0 ... t·-1, point in ·~ho direction
o I' 1,110 ,. coo1•d i oete axes . 1'he aqu1u·e of the le ngth of obese
1.UJ.it vcctor·s equw.s the integr al o! tno aqLLnre or f ( j ,s ).
6.
12
TRANS~Uss:o:1
1n
,
J fl (j,9Jdo
ri
255
CAPACITY
•
- ei
•
I
s r '< .i,
"l
,
L;T 1a L = ~· •
· Tfl
.111
·rer
04 >
A ,igna.l is reprenented by t llc r ol 1 owlue; su:n :
(35)
F,,.
F~ r ather than F.., (e) is w:-itten i n vecLO<' represcnLnLion ,
and Fx represent s El certaiu point in t.lie r - dim1'nsio1tul
spaae . Its diotancl" rr•om tlle origiu ls Dx :
t•t
D~ • (':'
"6
-~
a:( j )e,,
l/1
1•
<· I
•
[ T L.
"' a..,'
j
i.'
I>
-·
I 1
(l'Px >'
( ;;6)
1=0
A sample of therffial noise ,
00
g,(e) .
L: a,(j)i'(J,0 ) ,
O'/)
jaO
may also be reprcsenLed by a
II; •
..
2;
j•O
~ec~or:
a,(j) e 1
( :;d)
According to ( 5 . 24) and ( 5 . 25) only tho r· cociponents £(.i , a)
or e 1 that occur in the signal nre J.O•portant in t37) and
(38J . Hence , g , (a) ia divided i.J1~0 two parts f' 4(il) ano
S.l( 6 ) ; the part g'.(( 9) 11ay be ignor,.d:
e;l(e ) •
•••
L:
a,<jJr cj ,e J
f•O
•••
2: e;(j) e ,
9l
00
g~(e ) •
2: a, (j )f(j ,e )
00
g ~I
ju
The distance of the poinr
D.i•[T~a, (j)] 11 '
09)
I••
9l
=
l;o.,(JJe ;
;11
.fro:i: the origin
equal~
u·.
A •
(40)
i•O
Tha nvorage power of many noise samples 9A i s denoLed by
P._ 1 ; the indices r and T indicate the nw:iber of orthogona.l.
256
r, . SIGWJ.. DESrcr.
coinponer:'t:z of
ort~ogonalit;f
!,be !'lOi;;f? 99..1.t:'le and tbe
~u!·atio!J
of
the
intPrval:
It has Ueer. sLowr. i.n ~tccion.s ; . 11 ancl 5 .1 c und.er vet'~
general assunrtiotie, t~.e"t Cl.c di.:;trtbuLion or che coe:f!icients :!,t(j; it! tt:c Gtrm.:- for.· all j , if 1..h~ gA{S) ue sw:ples of tb;raa! noi:;e . t:qu:;!;i:>r. (41 )
as
Db;t
thus l:e re•..-rit-ten
£01:~;.:s:
(42)
'!'Ile a•1e rnge over X for 1'iJ<ed j
A.Verage over
~
may bo !'1>plnoed by tho
:·ol' .Clxc(! k. :
r,,r
(43)
The subs ti tuvio:..
]j.m
·~
.
P,
•
::"
yiela.s :
.. "'
liu.
,_
:
1
t
'
?:n_i(j)
Comparison of (110) nn<l ( 44) shc.-s tho t ~he disl:o.nce
Dl
of
ell ;:>oinbs g', rroc. the origin app1·oache" ('rt·, 1 '"'for large
valuts <>f r . rte point n repreo;enting ther:nal noise are
loca~ed in signal spnce arbi~rarily closo to the sUTface
1
Of a r-<lin,,nsion9J Sphere aith radius l rP .. 1 )V •
'11".e ave~·agc po,·1!'.'r of ' si~nalr- Fx follows from (3}) :
p •
lin
l - W
1
-
l
L:rx
\X~l
1
liw
\--4>0 t
±..,
I
°"
L.J "'X (j)
('+5)
.('"1 j tO
This eque.tion m'ly h" ret·:ritLen , if Lhe coorfici cnts e.x{j)
have cc.<> ;;ame tli~L•·ibuHm: for all j aud if they ore stA;i~tica:lJ ~nCop~nder.~:
I •
li:n
·~
'
7 2;a;(j)
• )(: I
•
lim
,_,.
=l
I
°"a'(j)
,.,
L.J
x
(4&1
6·
1 2 TRAli@!:SSlO!l CAFA~ITY
The substitution
..
liJ!l p =
,_
i;oi;parisor:
1 •
:.
25'/
:i'ieljs :
('17)
l,_..,
im
;:0.1-il C~1
i;r.o;;~ tr.n :
)
all J ofr.to f x
b.l'C
locaLetl
arbi •rarilY clo,;e to LI:" cu.!'~ace of s «-:E!l:ez:sion'll s~l.e:·e
l(it'l rndius (TP/' ro:r ~".:-ge val',"" o! r .
A signal •1:itn rut uUrlitive ar;i!1C ... r.ua~: ... e g ~ zuperl-:r.i.1ooed
is
J.~opresenteC by l.. J11)
point.
,_ I
Fx + gl
=
Z [ ax(jJ
•
a , U •] e ,
1=-•
Tile points F,,+ 9l , x
1 , 2 , ... ~J'e located artitraril;.112
.
close t o Lhe sur1a.co
01 a spt.e!'"f' \\J.L h racI i;.i:: ''~('
.., - i - !',,, )
for ll't'ge value~ r.•f « :
..
lim
,_ {T
'_J.~'
. '''!('r1.r,•
-.
)Vl
( "-~)
' -~
F.\' cau!;ed t;-t-~e
u~oturbed ai gnal
Fx + 9A i .:" t he rliHtt:u ..cc be t wee,.... Any Lwu
Bif>1111l point s iv at least. 2( .:.f',, 1 / 11 • fhe r·ossililo nw;i'Lt!r
oJ llOints ha'\ring thJ.::f :r.i.rrtm.m di:: t. i l"') 1-"'rom or.c '1J)ol.J1er
!.. equd to the posaiblc nu::iber o! ci;:r.als . -O fot·:~'IJ.i::ri
this number cocsider the . .;olw:;e '.' or n r- J.J..ir.en.;:;i anul ~ptere
b11•1ing the r adiua R (>.>, 7] :
Ono may deci de uJ'vunbiguou!:il,;· whirJ
v
n''l
• rtir +1)
fti~~o~l
r
.R
The volume Vc between two concentric sp:1r·res wi l' rr1dius
R llnd R- c app:ronches for l=·se nWllbors r tJ.e vr-ltmc #:
V.
r. fll
,
'
• • r<11r+1)(R - (R- c)
J.
rr r"
r("!"l.'- 1 ) R'( 1
-< 1 -fr" J
t ., ('\()
Henuo, mo i;t o f t:he volume 01· the :r- dimonsi on a.,. a1,hor"' ic
ctoart to i ts surf a co . A good estima co or the po,.,,,ib Ir· m.unbe" o
s i gnal points is obtained by dividi.os J,J., vol1rne
rl ..... ~ fl"lflW"'iibCWI of lrda.11\11,ot\
t> • SIG~~A:. OES!G!I
~he
o!·
s;.i!...e.rc 111tl1
airr 1·&ciius
..
l
'I r11 p_,._p, 1/'1
' ·f-1
112
...
1·adiu~
t>y thl'lt or a sphere
)">.
P
'·'
(1 _.. l'/l ' ' r 'I '"
(51)
Encr. sigr,111 fx(e) hlLS t he duration l . 'i'he limit of the
Lr.o..;uaa.L!.!''lOn rste th•ts Ot:coa:c~, :
or~or- free
c
li!L
, _.., I11g\.r~ _.. r11 '·'
&
\l/1
J
-
-
~i:n
~ le;( 1-J /P, • 1
r - oo CJ.
)
(52)
Or;.~
mny so<: th&t.: .;· 1 which is c!lt'" r.11uL{..1~r of orthogonal
in R n1,.i;1.al Fx(e ) . :nust approQCi! in,'inity . The
int~~val of arthotr.or.1.1lit~· -l'!1 6 1.. ~ i'P u~u.y bo finite or
i.r1tinite . rt.'lle .r·nl;iri J /P,, t o:: ~he avt11:ngc r;ip.;11al to the
uverqge noise roi<.-" meiy a:.so be rinit" rir inf.init~ . £quation ('I'•) !:ihowa t.h:1L tnc a:verage noinc pONe!' P,,t is in.rin1 te for fini to T; ncccrrdi. "lE t.o ( 4? J triu au.:ne holds t rur
.!'or the a\·erage: :;tg11al po'°'nr ? . The -rn:-.seti!:i.sion ca:pacity
g::-ow~ beyond all l>oundn i .. ~ is finite and P/P,. 1 is not;
fur.crioc~
Zt'l"O .
Co=i~1der t ;-.~
tJf•'ri &l cases o~ ( )2) !'02· whicf. the orl.ihogon~lity interval npproe.cl 1e;; infinH:1 . For tt•o first example le L us use 8 ey.it;l'lll 0 f m o=~hogor.nl f unctions ( f( j I e))
l.h~LL \runisr ouc1:;1idc t;he i ute!"V9-1 -~k'1' 1 a to~ -!k'111 + 1' 1 ,
where k =- 1.r /'l' ' le 1n intege1' . Conri<lf'r , .furt;hermore , o
"Y"'e:i: o:: er orthogo<::;,l func;i.ons (!'(.i ,<1-1 )] thin; are shilt.~rl C:: - 1 an·j wh1 ch cay !:.nvr: i:he same shape as
he .runc"ic~.: f< ,j, a) . Thee~ !'1!lct:.on;, ''anislt outside the interval
-;k'l'i ':? t; ~ -tk'1''•2T' . Continue this w"y tmtil vte
1!,Yl!tr:n (fl ;] , 9-k+1)1 i.· reoclled i<hicn vn11irl:c" outside t;he
lr-torval ~kT ' - 'r' .i: t ~ l k'P' . rhe totnl number of or t.hogonal
fu.nctior.~ produoed i" then :
(5J)
!'he :actor r/.. r ir. (<;;:>) heccmen rt1</2kT' • m/2T. Tl;e
aver<::ee coii::e ro"'·~r P,,r beco!les FS\, 1 • bei:&'.lse m functions
or:ly E•re r.on -~«l'O i:. any one of ;;he ): time ini;orveJ.s encl
tecause : ' ~n thn ,,urat<ion of th;: orthogonality interval.
t ·
25')
12 TRAJiSMISSlON CAJ ACl'lY
one obtuin" f ro:n l5?) :
0
0
~~'! 2~~ I
l !'; (1 ;f'/Pmk, XI' ) •
~I c;
( 1+P/Pn:, i·J
(~11)
The dC;rivation of 1..h1n: for:nula e:how3 tt.at: one doen not
nave to wait inJ' ir.it;ely :ong to obl...iin t::ie in.fortt:ilion in
tbe signal F~(a) . !'~rl of tile in:·o=aLion is avnil:ible a;;
¥. 'im• interva t" .
Ar. a second ex,,mple considar m
?l ;1 si.ne nnd cosine
el ements in the i ntorvnl -~'I' ~ t ~ ~'r . rh;,se 1?lomo11Ls r>T'e
stret ched by a factor ~ > 1 . Ac cor.1ini;; to ( 12) ono L'.JS to
substitute k = ; (l+t ...1 /2;; ) f or l i:: oi·<!l':' ;o '<C<'F lllo number of .runctior:~ trru~s:li t;t;ed pe!" ~.ui t of tin,. co:1s :.ant .
•be number of or:.hngonal ;-11n-.:t:io:HJ in "!;he :.r1~e-rva... -:~T
li b :; i~T is given h,v '
the ends or the
(
t: r.:
,,1 _,•
•
J
fror.i (1i,) tha• ;;hn f aclo1· 1·/2·r i.z: (~2) i~ r<'placed by :i:s/2~T ~ a;/<.'I' - ~r . T'Le &vorage noise powe1· f',,,
i~ replaced by P. ~ 1 1 . -t :·01::.o><s tl:a: InLI' = ?., "ince the
!b follows
•J.gnala occupy the frequency b'lnd 0 ~ f ~ Af and :..11 rlin.and cosine co:rnponontu of t he noisr 1.1W'Jpl ez: witt. J'r~ q_i..;cncieo in this b=d ul'e received . SITANNO Jf ' s f oi'mula iti chu.o
obtainnd from (~2) :
O•
lb
t-oo
;fJr lg ( hl /lm1 . ,
":>
•)-
)
.H l;:;( 11P/f•. )
Some Care l!lUGt bo f!XerCiS.etl in illt(ll"f. rr>t.:.ztg
l;ltrJ r n!•ru.t...tU::l
(30), (::;1), (52) , (S11) Wld ( 56) . 'L'lll'Y hold fo r tll0 l!'l UIS •bsion of ort;hogooul functions wHiJ the one in<l •'l"'"J~n.
Variable time . This corrcsronds to Ll:.e tran~a:i~!lio11 of
Signe.la repr!'sented by voltages or currents . !..r. e!eclrollll!gnetic wave trnvollinr; in free. space in the z-direcr;ior.
alao has the independent variable t oul;y , but him t'IO o <'t hogono1 positions rot' the polari?.ation vector, and the
tormui a.11 apply to eacb o.r tbern . Rowovei·, in a wavo trnv11l hng ...n., wave guide in direction z , the variables x li!ld y
17.
2(;0
b . S Gt:1.1, DESIG!I
i:.c:y appear in adriition to t. as indr·rendunc Va.l·iablea .
These nddi t; on al degrP.i:;P of Jr·et::loo nt.ow up Ufl rr.odes a.no.
t!:.e formu all apply to CHtcb oJ: kerr. . :ieace, SlfJLllNCN 's fora:ula should noL onl;; be vhw~d ns the liuJ.L or what elci.sting colt=L.w1icution cna"lrlels cnr_ t:rans.:oit., =>·~t !iS a guide
to '.letter cb.aunels . =:qua-;ic:.s t 301 and ;:2) sho'A tilat the
nunbcr of ~rwea:itratJe O"Ll1oronnl f•;.nctious i:; the pririciptl f~ctor doi;el'IL' • 'ng t,.9n:;:ui1rnior. capnc .Cy . A possible
way tc incroaae cLis !!Wllbor "lo Lo use channels chat transmit zii:;unls wnich ar•e voriablti~ of time 9nd opacc coordinac-es . Opticnl telescopes nre itsed in thi.t: -...·o.y .
6.13 Signal Delay and Signal Dis!ortions
Several siupl i:':ting as~wnpcions !Jave boon rllide for t;ho
Jeri vat ion of Lhe transmi ~a Lon cnpacii;y ( 29) . '.::be eliminabon of cl1ese assumptiona wU L be investie;oted in this
sect.ion .
:.et the t'llr.ctiocs rc<i,ij) and f,{i,a) ln (1\))bedel.aycd
by Sc(i) :i.n<! a,{i rath~r ;hiu:. by a co1J11on delay time
3(0) . Tlle funci;:ionsii. i19) are then'"' 101:gll1' ori;bogonal
and 9(0) ie no longer Ll:e ce>lny time of tne signal. ?or a
more generul derinition or o signal delay t;ima let f(O , 9) ,
r c< i ,6) !illd f s ( i ,a ) be t1•iwemlt to(i indiviu1.wlly . 'l.'he .funci;ions K(O l.!'[C,&-9(0\], KcUJfc[l,9-9,(i)] and K5 (i)f5 [i ,
S-e 5(i)] ~~·" ,he:. received . t.et thelL be croascorrela~ed
wit:!! sa:oplo functions i·ro,9). fc(i,9/ and f 5 (i,8) . Ttc
time tlif.!'crcncc bt::d.. Wt!e11 P • <• a..Y!ci tl:..e ab:.solute ciaxima of
i;nG crosscorrdni;ionftu1cUorw yiold the delays e(O), ec(i)
and a s( i) . 'foe value~ of the m1.1.xinm yield tno sL tenuotion
cocfr1ci.en;,e K(OJ, Kc(iJ anJ Ks(i) . Using those coe.fficients one nay derive a srunfil<' function f ;M<A fro:n Fx(a)
in (18) :
q,,(8) = K(O)llx(O}l'(0,9) •
..
:2:;
"'
[KcCi)a 0 (i).Cc(i,8) + (57)
• l,(i)n 5 x(i)f 5 (i,a)J
'Lile received ::ig!lnl F xc<a ) baa the sain<> shape, but 8 must
:261
be replaced b;;
9-Ho>,
nend side . :.et the
e - ec<il o!·
e-e.(i)
on tnc right
cro.:;sco rrela~lou function or FX• (e)
9I1d llxe(e),
r ~
J r XE
.....
(a)"'(~-a')da-rra•)
,
• X"'
yield an aOsolJtc o·.uc.i1rJJ :or 'l ce!"tatn •1alu-:- o:· 0 1 • 3x .
Thia value is defined"" the dc-ay time or ttte pro~'lt';t•t:ion
tim<! o! the sigr?11l J;'x(0) . Since it ia :~ot imo•,rn o L Che rroe;l.vcr which signnl ls l'Oing to 11rr.i.ve , it is advant-u~;eous
to define a pi-opogatio11 t irr.e indepez.tient of
x.
O!le may ,
!or instance, a"1ernP"::e che i:alues -Ox , ;;..f 'tl:e=e a.re H ::!if-
!eront signals ?,,.(e) Lo defi"-'C " Fl'Oplgai;,on tille 9,:
'l'he propagat ion timo of a sign~l ~ui·rying inl'C1rma~ion is
a statistical variable which can 1)e identified ir1 firlit
approximation only •'l rh the conc~1 t~ o f g::-our ,J.,lay or
"ignal delay origin11l.;,· defined in op~ics r2J .
Th~ received sign11l F n (9 J ta< Lh< shap<! a:: (!>'/) i f a
is i•eplaced by 9-9(0), 8 - 9c (i) or a-9 5 (:.) on ~ho rit;h L
han<! aide . The sample ru.ncti o:t Fx M{G) h~s this nhnpc too ,
buL 9 must be replaced by 9-BL on tho d r.ht hand ~ide . Lei;
~hn f unctions r[O,e-e (O)] , fc(i ,9-9c li)) and:, [i , e-~ 5 (1 ) ]
be expanded in a ~eries cf the systcL [f(O,S-il, J , fc(i,0 S.J,!5(i,9-Sl )) . Oae obtains it! anolop;y;;o(2 . 26)tl!e:ol lowi~ equations , iu which v = e - &, , fa = f[0 , 9-o(O)] ,
r, • tc[i;e-sc(i)J an<! r 5 = ~'s[i,e -e,(i) J is w1·ir~cr !'or
abbreviation :
00
fo • K(O,O)f(O,v)• ~ [K(O,c:: )fc (k , v)+l.(O , ~k)f 5 (1c,v)J(1:.0)
• I
00
r, • K(ci,O)!(O, v J•
l; [K(ci, ck)f cCk , v )tA( ci , :;k1:5 ( k , v )]
•••
!s • K(al,O)f(O,v)>
21: [ K(si , ck)fc(k , v)+K(si , sk)I 5 (1< , v)]
,,,
J,o b these series bo oubstitutcd il'lto the !ormulu for
P~t(9 ) . The firs t torm
Fx , (a) hss the following form :
or
6 . SIGH AL DES!GN
(Y.(0)&,.(C;K(O,O) •
+
!:••• [Kc(iH•cxCi)Y.(ci,O)
•
(61)
K5 (i)a 5 x (i)K(si,O)))f(0,6-&LJ
Tt-.err is r.sutual ir:.terfe:.· ence or cro.l"'n:&l:t let~e-:en t!:e coeffici.-r.t" . Ii; is possi ble in princir.l r. to devise distortion cor:-cci;ing circuits thot co:r.pcn!lbltt: t;lie c ro sstalk
"'Hl,iu the nccm'ac:; of ruea.sur<"r.ient, re that K( O)a,. (O)x
.C(0 , 9-al i~ ob;;ainedinpl ace ol' (n1) . J•'n(9) ond Fx"(e)
tu·c th<'n iclont i cal .
Lrtus ru:rther asswne that f (O , e) , fo(i ,9 ) nnd fs(i,9)
1u·e not only attenuat e d nnd delayed dttl'illg transmission
but ''IJ.80 e.u.ff-cr a :ir.ea..1', tlco invqriant.. distortion .
f(O , e J, f'c(i , e) acd fs(i,e) a:-" ~ra:-:nrorrued ir.to g(O ,B),
11:c<i,9) "r.d g5(i ,6 ) acco,-tlillb to !'Ccttou 2 . 22 . ~et the
f\llic~ion"' r(O,E), fo(i , a) acd fs(i,a l><' tran!ill!.itted individually . T:,e co:-rela-.:.on f"1l!ction~ oft:':<> received functior.:- g(0,3), gc<:. , 3) and r;s(i,9 ,11tt· a=rle functions
;'(0,9), fc(i , 6) ru:d r,(i,e .a producf'd . The " imo shiSt
l>~cwo~n i:heir absol:ite mr.xi:oA and 0 • C yields the deleys
9(0) , 3c(1) anti. s,li) . :'ho valuonofthe maxima. yield t he
uLLenuntion coefficiem.s K(O) • K(O , O) , KcCi) = K(c i,ci)
ru1d K 5(i) "K(si , s i ) . Sampl e f u.oct;ion:i li',f., ( O) of (57) me;r
'be constnict ed witn -.:iese coo!'ficlents . Equation (58)
yielll!l ~x und (59) deSi n esa.rrops1>eLlo11 time 9t- !'low le t
chn dincorted .functions e;(O ,e ), gcCi,9) and g , (i ,e) be
"xrnnllrll in a series of t!.e systein (l'(0 , 9 - 6 1 ) , .fc(i , 9-9L ) ,
!\(i , S-9, l} . 1'1e resu1tU!g expressions =e .formally the
:-a1:1~ at tho"" in ("' l and th" nair.o conclu~ions apply .
6.2 Error Probability of Signals
6.21 Error Probability of Simple Signals due 10 Thermal Noise
t.;on:ittle1· the trans3•ission of Lelotypo cllflractors in th•
prr1ncnco ol' the1'ma L noise . ·rhe pro'oal>i 1 ity o! qrror shall
bti c:ottnuccd for :;cvc-r•al methorln or tra.nnmisslon a.nC. deleCL lo~ . The ge11erul .form of such choraclers represerJ;ed
t; .
21 ERRORS DUE
·ro
TllERJ1AL
:io J SE
263
bY t ime functions in :
F (e) ~
,
2:' ax<.J)f(j,G ) ,
x
=
1 ... 3?, 0 = t;/T
'l'J!~ [Unctions f(j ,9) 'U'e Ol'ohonom~l
"
(62)
J10
~
i.
T is the duraUon of a
i:: the 0.nLerV!tl -S :!
cha:acttr ·~hi ch i"
i:elc~ype
usually 100 , 150 or 1o/ no . TtP. con.t'ficiem;s a,(j) '.ave
i;hc vnlueG + 1 and -1 , 01· •a and -o , foi· a ·uai.ar.c10d ey~1tecr. ;
thooi are +1 an(! 0 for· an o n- o l' :t sys Lem . A samplo g,(e) o.r
additive thermal noise tnn d'or oe Lbe character Fr(9) ln~o
tlle signal F( e):
F(8 ) • Fx(9) ~ g,(~)
s, ce) •
P(6 ) •
... a,(j)l'(,j , 0),
2::
l•O
~
(63)
Ill
a,(j) =
!.,;,(a )f(j , 8 )dS
·II?
a(j).f(j,6),
a(J) • "x (.1) • u .4 ( i)
1•0
The energy of all chnracter:> is the ss:ne in n !>111:.nced
system . Using the ea:;t-:r.ea:i-~qllce-dcvi<roion criterion
!rom sample functions F.,,(B) ,
Pv(9 ) •
±
•••
(64)
a,,(j).f(j , 0) ,
or:e may decide, according to {5. 25),
give
~·~ch
value o:· 1 wi::.l
!; n(j)ny(j)
I•~
it s maximum valuo . All coef .ficients a(J) :nust i1P.''O L1'.e
aqe sign as the coetcicients ax(.1) i f tl::e o<D:imUJll is -o
occ..ir Cort
X· ~e suit (65) then hus the Iollo•d.ng ..-alJe:
x•
Ir, tor example ,
$
o(O) had tbe opposite sign of ax(O), I.be
8IU'l (65) ><oUld be larger for the characte= F,(a) "Aith the
264
G. SIGH.o\L DEsrou
coeffic ients
for f'x(9) :
a,, (0)
a
- a x(O) , a , (k)
= a x ( k), !r
= 1 ... 1.1. tb,!Ul
·'
., (+ l aCO) l +l<1( 'l )l+la(2)l < ia(3)i+ ia(JJ.~ , ~"'1
1 ~
-0 2_,a(JJl'"p(J;~
I••
- la(O) I + ia{ 1)1 ~la.(2)1"1"(3)1 + !a(4)1, i'"X
The following two conditions must be sati s fi ed , according
to (6j) , in order to have d i ffere nt signs for a(j) Bll<I
a x( j ) :
a) sig ax ( j ) I. sig a , ( j )
( 66)
b) l• x(j) l<!a,(j)! ,
equivalent 1 ~;f;B 1
> 1 or
~< -1
Big a x (j) meazui 'sign of s.r(j) ' .
ln the case of therma l noise the pr·obability of a , (j)
being posii;ive is : a nd tho probabilii;y or bei ng n egative
is a l so t. Hence , the p.robubi litj• of condition (a) being
satisfied equa l s ! , indepeudenL of the s i gn o f a , (,j) .
of
Tlie distribution
x = a, (j)/ lnx (j) I i s needed for
the computation of the prnbability o.r condiLion {b) being
setin.fied . Since ax ~j) cnn ~e +a or - a only 1 Ja.x (j) I is a
constant . 'fherefor<? , x !:'.as the sa.'lle d:.stributioll as a, (j) .
The densii;y functio11 w0 (k ,x ) is obtained from (5 . 6) bY
subst.i t u ting :x- fOT' P. . E'rom We (k 1 x:) !'ollows the conditional
densii;y funct i on w( x ) .t'or the condition k = j . The probabi.lii;y that k eqttnl s onr of the m = 5 values of j i s "f/m,
since i;he coefficient a a, ( j) have the same distribution
for all j i :l case of thcrn:al noise . Thun the density ;t;unction i<(x) follows from (5 . 6) and (q . 5) :
w(x)
w, (k , x) _ .~~1__
r.i-1
-
x = a , (jJ/lnx (j)I
V 2>/ rro.
a
0"11
{
-x
'/
_,)
2v.
(6'i?)
a,(j )/a
Each coeffic.i~nt ax(j) in (62) is transmitted 1<itb equal
er1ergy . Hence, tbe av ·~rage sif!,nal µot·.. er P equals:
p
= 'i'1
'"s
- 11:
'
L;a~(j )
j: Ii
".""'
~
fJ!RORS DL'E 'Z:O :JIERJ1AL liOISi.
moy be generaliz'.'rl and solved !'or a 2
1'hiS ,•esulL
~
8
1 •
:
P/m
Squetion
1 4~'
yields
co~
1:
r • ;
(GlJ)
The mean squnrr..
devia~io.n
l)ecccics
('10>
~ ::il'1 1 IP
•
where Pm,r in the aver age powe.t· o! m ortl:.o~or..al conponr ntn
o'
•
of thermal noiGC in an orthogonality i!'H.. erval of o ~tl'~tt:ion
T.
Using (56) onr- may i·ewrH"
o'•
P6 , IP,
<>!:
( 71)
6 r • ci/2'1' ,
where P61 is the average powr>r· of Gheroal noise i:. u !ro quency band or width 6 f.
The probability p(:x>1) + p(x<- 1) thut x b l arge,. thau
•1 or smaller toan - 1 fo:lows fro:n (<'7) by L"li;cgr1• .1on :
p(X>1 )+p(x<-1)
?
~
• ifWno-a f• >Jl<f'(-:<: 2/2cr!
• 1 - erf(1/'{20 0
)
=
(/2)
)(lX
1 - ei•f(VP/21'., )
The probability p., tha~ condit i ons (SJ ns well
ot (~)are satisfied becLneP:
•~ ( b)
(73)
The probabili ty thnt the oon~it ions of (66) ar•e not llati" fiP.d is 1- p, ; tbe pr•obabiliLy that th"Y "re .,oe sa~irfied
for any of tho m • 5 coefficients a.,(.1) is ( 1-p, )"'; the
Pl'Ob11bilicy cbat they are satisfied !'01· at !ea~:; 0?:" of'
the m coefficients equals Pm '
])Ill • 1-(1- p , )"' • 1 - ( ~ )'"[ 1 + er:( VP/2P 6 , )J'"
('/4)
'!'he pi:obability of error p., does not depeud on th" ::ystem
or functions {£( j , e) used , provided these functions sati nry
th~ Conditions of sections 5 . 11 and 5 . 12 .
The numerlcnl valuen m • 5 and ll f
*•
0 ~ 3 ~ 16 . (.,
ttz
266
c . SlG!IAJ, DES!GU
app l.y to the much used tolotyoe st1mdt1l·J of 150 c1::i pe!' cha.
rac l.<' t' . Curve ' • • of Fi - . 1r12 s:.owr Jl,. = ?~ or ( 7" J as .rune.
tior. of Pf:G' .for tl:esc J&::.ies of
i::d ~! . 1'h& aeasu.ret!
poin1s 'a• """" ott,i nnd w.'..th ar ou.rly ver>1ior. of the
cquipmeLt ~11own ln Fig . "',O wl 1:11 t he sy:iccn [ l' (~ ,e )] consiatin~ of sine ~"ltl cos~ne riul!!es ac~o=•: l ing to Fit' . 1 .
•'
'
L\"
.
I
\ -- ;...__
't
\
~·
~
.
\
-
\
\
\
_
.\_
\
n
I
'
---
:::rro~ l'J'Obobil i.1;y r i"or I.ht! r.·r.ceptlon of teletype
<>i1>na.l.o !:luperiu1por;od by addir.ive thenr.al r10loo . l'/P~ 1 •
ave1·ogr signul rowe:·/avet•&ge noi"c power· in ~ 1 6 . 67 f[z
widr band . a) balanced ny>1tem, detect.ion by c1·osscorreletion; b) ballillced system, !'il•eri:lg by a 120 iiz ·~ide idea~
lo·~p5S5 Cilte1·, :lei;ecLio11 by aJ:Jfliti;de samplir.g; c) Sllll'O
as ( b) but; or.-o!! sy,,tcm ; 11) same as ( c) but sta1't- stop
synchronization ais• urbed by the 11ol>Jo .
Fig . 102
Let ti::c syste:=. [ f(~ ,e )] eonsisl.
.f(j,S) _ si;in(m9 - .i)
-
j
• 0,
n(oe-j)
1,
2,
-
"' =
-.r
1;he functiocs
drn(~·-.11
(75)
,,(a ' -j)
5,
a -
i;/T,
e•
= t/('r/m) .
Eqaalion l /'l) o;>Jll ies to this syst.em too . lhe energy of
tr.eae funci;iowi is concentr81:ed in the froquency oand
-li :! v = f'!'/m a 2 with t!:e bandwidth 6f = a /2T • 16 . G ftz .
According to sectior 2 . 13 the san:e i:alues n.re obtained
l'or tha eocf!i cienta a(~) w!:e;;her F( 9 J ic lllUl eiplied by
tia• J'uJ1ctions (75; and t.;;e product is integ!'eled , or whe the1• F'( e) is po~sed th POIJgb an id,,al frequency lowpaell
i'ilto1· 16 . 6 Hz wide and the 11t1plitudea nre sampLnd . Hence ,
( 7q) nlso hold A for filtering =d 811plii;ude sampling o !
€ . 21 El!P.ORS OU<: TO
'rH~"'li·:~.L
2v7
110 SE
thO pulses ( ?'i) . A lowpnss ri.U e r 120 f17, wide incrensea
tbe average nolt11' power in (711) by 120/1b .6 ; 7 . 2 . '!'his
::ellllB 8 shift; of ;;he curve ' A' in Pig . 102 ·uy 10 log 7. 2 ?
a.58 dB; tl:c !!hlfted curv~ i" Jeno;;ed by ' b '.
consider ar, oic- off syscem . The eoe rricient; ax 1 j) aa•y
ass ume the v nlues ~ll or O i nsc<:?ad of +a (')r - u . The fo l lowing condi tiono mu s t be satiofl ed i.n ord<'r foi' a co~ffi
cient nx(j) to be detecto'd as + b icsread of 0 , O!' !JS 0
instead of -b:
a) s i g (a,.{j) -1 bl f.
s i r; n , (JI
~~
l< Ja , (j) I, equivn l ent ~~"_..4,..~·- ....
b) l a., ( J·)-~b
~
,
l a x\J - /21
'
1
I >< -1-1
(76j
sx(j) - ib :u.y be •l b or -~b , since ax(~) cay be +b 01• 0 .
Tue conditionn ((b' and 167) are thu" ;;t.e nai::e, but •3
and - a have to be replaced by . , b :u:d -th . The av,,rnge
power o.r the m eoeffici.c::t:; wit;h valt1e:i +b or 0 cqu«l"
p • i mb2 and it rollows :
(ib )' • P/2n
(77)
Colllparison of (68) and ( 77) show,-; thuL P has to be 1·ep!aoad by t P in the equations holding for a balartcetl eystom t o g et t he equati on>; .for an o:i- off s ,y ntero . Thin mea11 u
dB .
a shii't of curves •a ' and ' b ' l.n b'i;; . 102 by 10 l og ?.
Tbo t1b.i.!ted c\ll"ve ' b ' is denoted ·uy ' c • . Tho :i:ea.surnd
points ' c ' were obtained by .addi:ig ~beraO.: noise to the
blo ck pulses of teletype cht1racters at"te1· •hich t:.e diu tu1·bed signa la wo r e fi l t ereCI b:y u 120 f.z w.Ltle lowpass fil ter nnd then fed to the receiving magnet of a t<>let:n:io recei1fe1 . The measured ;ioints agree Sqil"l:; ·•ell wit!' curve
*»
'c' > although the block ;'·ulSeS did not hllV<> t!le Sh8!'" Of
the 1>ulses in ( 79) , the lowpuss r i l ter: wa~ not ideA! , 1u1<i
tho magnet o .C a t elotY!'e receiver n orks onl:r very rougi1ly
118 Oil amplitudo enmplei'. 'l'llo lllea.sured point" ' d' hold for
the same 1'eletype transl!li.ssion, but starc - otop pulser wnre
trllllSl:litted throush the noisy channel for syncbronizaeior; .
•he r oint s 'c ' and ' d ' depend strongly on the care token
Ln nd~usting t h e teletype reeei ver.
208
6. 22 Peak Power Limited Signals
IL has been tt•WU!l'ed so hr th"t ~he average signa! poncr is the dee riin.ing f1<Ct<>r in U.e e,-ror probability .
However, power amplifiers e;eni;r ally limit t!Je peak powe.1,1
rat 11 e1• t hnn tl1e a ve r a ge po we r . Conslde r an 8.tlpl i fi er that
clips = pli t udes '" :< £ IUlC dclivel'8 o peak powet· f , . Average power I and pea>: po1<er PE of a signal consisting or
bin~l'Y bloci< rulses , having po5it1vt or r.egative al!IJ>lit udon , l.il'e t ho sruue . The e1·ro!' probab ility p 1 oJ' ( 7.3) for
one ui1i:iL isr l otted i n~'ig . 1 03as f uML .ioi: Qf PIP. , = PE/PAI
1111d Jenoted "fheoretica licit ' .
1.et t!:.ese pul~esbe amplito..de 0:0<11.Uated onto a carrier .
Thr cu1·ve ' 'I 'hco1·et i cal limit ' would still apply if L)l,
ca1·1·ier is a '"''nlst· carri~1~ . The p~a l~ power ol' u sinusoidal
c·1rrier ·~oulJ :iave to be 3 dB larger to yield the s u.e
averni;e :;;01<er; •be c..rvo ~eno:;ed b;Y ri./n ~ 1 in Pig .1 03
hold~ :'or "
dnusoi da. cnrrie L", ar.i)'>liLude> modulated b;Y
b i 11a1•y b l ook pulses .
Only a·~out one '}Uarttt1· or tl1e chnnnels in n teleyhony
ilul ti;>lex ,;ynte:n :u-a busy du.:-i.ng ;>eak traH"ic . U:;i'1g lL.oclt
pul !les fox I'Cl'i trtmsoj nsiou , -che &JDpli!"iers are u s ed 1/4
o.t" the t i mo o nl ,v 1 while no s i goal3 1 or at leE1St not very
uaef11 I signalo , are tra.nmr.l teed 3/4 of the ti.me . The peak
power must b<' increased by 1C log 4 ~ 6 dB to obtain ·ne
same nverage :,ignal power l;llat i;he 1u:ipl Lfier would d eliver i f useful '1ignal s woutd b e amplifi ed all t ho time .
Tne i·esu ltini,; curve i s deno t e d by m/u = 0 . 25 in Fig . 103 ·
The t'atio o/n 1 s tl:e 'let l vi ty factor , 111 bein1> the !lW:lber
of busy chaonols and " the nun:ber of available chan.nel 0 •
Very l ow activity f actor& occur in ~he gro und s t ations of
multi~ le access satell i te syst oms , since the aunt of t)le
activicy :actot·s ol' ;;he ground stations is equal to lrh•
actl vity factoi· o.O r;he satellite trM:iponder . A repre s entative curve for :n/n = 0 . 05 is shown in Fi g . 103.
L:ousi d e r now t h e traruimii.mi on of: bin ary digits by sincconir.<' or Wul"h. pulses . The r esulting s i gnals k'(0) have
ve~·:; la.rge peek~ although most amplitude" are auch Sl>Bller
• 2 p.E..itt }'()'N3R I H:l'l'E:l SIGNJ\LS
6 2
~·
...
~!
\
l~
•
\
\-
\
n.. """-\
....
' \ 1lh
•'f
"':;<
\
'O'\ •
m
l•' i'!i l& J;I
J>if.10; (left) Error rrobabiL.ty r an function of Pr1: 0 1 =
= peak eigna.l power/overage noise power in ~ ~anC or width
I..!• m/2T . Solid lines: time di.visio:t, sine .:-a=:"'ier , ncti -
vhy factor~ 1, O. 25 nnd O. 05; dn!lhed lines : 11 sine nnd
cosino pulses , pel'centage of clipped ruaplii;udeo shoim .
Pig . 1 011 (right ) Propnbility p(F(B )) of the arnrlicuue:i Ol'
the 512 signals Fx(0) being in in~e1•v11l" cf t<i<lth 0 .'1.
Gnuaaia.n density func tion with equnl mean and u:ean. quore
deviation shown for cocpal'ison . qx(O), ax(i) , ·ox(i; • :1;
l'x(9 ) • ax(O) + (2l;:cnx(iJcos2nie • bx(iJsi::2.,i&] .
1•
than the peaks . Fig . 1()1• shows the rrobrl>ili~;v p[F(a)J of
GUCh a Sif:l'llal hnving Wl u.mplituclo '.1lLl:in an intervnl 0 . 1
Wide. Superimposed is a Gau ssi an density functioH I 11viJtg
tho same mean and .,eM nquar e dcviacion . AccordillP; ~o tbc
X'eSlll ts or sec;;ion ; . 24 this de=ity function aHr->x mates
very accurately tho probability funct:ion of the rur.plitudes
or Signals consisting ore z= of Walsl pulses . The plot"
or ~'ig 104 are symmeerical for nogot:ive values o.r .F( e) .
'l'he nvorage power or the signnln \<o uld be very n111a·11
ir 1lb~ large but raro peaka would be h'ansmittea . 'l'ho large
Peaks oust be limi.te<.I. ~o increase the average signal power .
0
270
6 . S:G'.IAL DEs!G!l
T::ie aasile<! lin,.s in Fir. . 10' !lhow l;ne result.~ of wnplitu46
clipping !'or eiue-cosinc J. ulses lr. -h e 1n·e.,.1nce of additive the1·mol noise . Tho para.meters 0 .01:% , 11 . u%, 13% !Uld
52% indicacP Lhe percentat"<: of an;plitudl!~ ol ipped . The
curves hold for DC transtt.i.csion 01.. for trans~ission by a
Wa.:.sh c1u·1·ier . They also hold a):>proxi>l.a~cly for single
sidebaud codulai;ion of sin'3 c~.rriers . Wal nh pulses yield
•1ery simil"r curves [1) . Little ene:::f); i::: t.l'ansferred to
adjaceJJt f1·oq11 ency bands ·~y clipping oJ: r:um~ of sine alld
cosine pulcco (2] , while no energy is t1·ai1s.rerred to adjacent sequcncy ':lands by c1ippine· o:: sun:s of Wahh pulses .
T'ne .fol-o>:ing conclusion. rte:1 be a.rmm fl"O!l Fig . 103.
Seria: transmission of b ... n11ry block pulsos produces trhe
lo;·;esi; error rates, ii' tlu activity .'.'actor is close to 1.
Parallel tl'!WSmlssion by sine-co nine or Walsh pulses jdelds
l o wer error rutes , if t he uclivity :facto= ls 0 . 5 or less .
The ~x~ct percentage of clipped u:r:plituu~a ls not critical. The clipper charnct<>,.i3tic and the l<>nsity functrion
or a clcipped Gaussian =Flit:ude distribution is shown in
Fig . 98 for 1;,11\ of ;;he em~liLC1dr.01 clipped . 11otc that a decrease or l.lle activity fuCL01· kccpn the energy of apuJ.so
unchanged. in t;he case of noris.l trar1.smiasion, while in tho
case or !'<ll'Ol Lel traus:ai~nion the ave1-r1r.;e po•<er will be
xopt con.ilLllilt anil the enet•gy of a pulse "'ill be increased
if an auto:ootic gain cont1·ol n:c;.!ifie;- i3 ..tsed .
:!J . ROTll of ':r.ct-~"'lische lloctschulc Aacr.en t &!l shown uhat
~D"'J.f<lldors u .. ing tho e1•ror .function charoctoristi.c discussed :ln socClo11 7 . ?.4 yiold betteL' resuHs thun clipping
fol' signn In composed of Wa o il fllllctions , Jll'Ovidcd the error pr:-obnbiliLy is below 10 s .
An increase of the required peak pO'der is needed ror
equa_ error !"a~es if the block u~scs are reolaced by
other puls~ ~hP.pes used i.I1 aerial- tra.oscissioii. - Tabla 12
shows the increase llPE required. for some typical pul 5•
nllapes . 'rtte oolid curve s in l" Lg . 103 have ~o be !:lhi.fted to
the right by ~P, to apply to those pulses . The 'raised cosine pulBe in frequency domain' i" defl.ned. by the equation
271
T ble 12. I.r:c:r·tas~ A t'e: of jtea.k .:oic:;!".~l
p~wer of s block pulse Ioi· equnl
pulse
nr1·01·
10-.,·er
over r.hr. jlCD.k
pra·oabilic;v .
t.P,
st:.ape
[ cHlj
1 DC block pulnc , E for 0 < L < 'l'/n, 0 ot!ler~~iSi?
cos~~~ pul~~ in frequenc.)' t.lo11ai:.; roll oft fac~or r
,,.
3 aW11e, r : 0 . 7~
2 raised
4 SGJJ18 , r = 0 · 5
5 rnioed cosine pulec in &iae domoin,
Fi~ . 3~
6 triangular pulse , £i1•nt/'I) , -1'/11 < ~ < 0
E 1- nt/T) , 0 < L < 'f/n
! ( t /T)
~
0
1 .8
j ,2. 7
1 .. 1....
1 .8
E s:i,n n!lt;/T coo nrm;/r
n "t/I
1- <2rr"i;;'f;l
r i e the socal led roll-of.!" facto1· of the lo'.·1pass fi l ~.,r
used !or pulse shit;>'lng [3} , a in ;nc :o•.J.IDbei• of chonnr.11 ,
end ~·/n is t h e cturi<tloo of e block pul$e i f n of tllr.m hu.ve
to be transmiliiied du.::ill1:5 the t:ime T .
6.23 Pulse-Type Disturbances
The error probabilit;y of digiLal Slgna~s is i ndci;enden~
of the pari;icular nyncem of or thogonal func~ion:; •Jned for
t!)eir transm.ission if the disturba nc"D a L'e caunod by additive thermal noiee . l'his i<i noi; so 1'01• pu~sc-type li1nturbancea which are ll10re icportant LLau ~w~rILal uoille on
te l~phone
lines .
Let us assU11e 1'l'ac the ru::p~it;ude of a d~st=l:in,, p•Jlsc
is much larger than t;he large at woµl Huctc of the u.n<l is-
tu:rbed signal . Thon le~ these pul son pnas L L!'ou~;h ""' UlllPlitudo limiter . U ~he t•ise an<l foll times of the pu I.nos
ar~ ~U.Cticiently shore , block :pulse" or nu·io'.ls length but
equal amplituda will be obtai?:cd at Ha output . Let t~,e~e
Jlul oes be observed durint; R ti.Le int<H'V91" o! duration T;
there shall be one pulse in r interv&lc . r/R io the probabiU ty tor i;he occurrence or a P••lB'-' d-.II'ing an 'Lnt"rval
or rlu1°Qtion T, i£ r on<I R a.re very lo.rgo . W,(T) is •.vrHLen
fol:' l'/R as r and R a1,v1·oech inf inity; w, (T) is i;lle ll.intribut ioo function for the occurrence or ~ puJ.se.
2'12
,, . SHillAL DES!GJ;
Let chc dur;n:ion 6<, of c:ie ;>ul~c,. be observed and let
q oui o: Q have a dU!·ntio;.. A-., ~ ·, . 'rhe l:..oit q/Q !'or
ir.finite value~ c; ~ and Q is denot"d by '•' 2 (T, ), the distrib1.:tio1. funct:ion :or tl.e lcui:,-th o;' :he vu~~es .
I.et tl:~ cccurrence and ti.e leng;;ll of i;he pul<es be statiJUcully .:.noepen<ieci; . The dist;·ibiition fwcction ·,/(T,-,)
of tlie ,jo lct cist:ribution i~ th~n dcCir.ec by the product
W('l',r,)
~
W1 ('I)W 2 ( ; ,
) .
<?al
W(T , ~ 1 ) can not be deten1iJJed l;ly ;JOp1;n1 t e 1:1e1;L1.1urement or
W1(T) and W2(T,) if stntlstical i!lOOpeudence does not hold.
A tota!. of RQ rather th= R ' Q m~ti:illl'e:rie:,t,s would then
Oe !'eGui.l'Cd .
'fhe dlzt!"ibution fucctioc. W(r ,-, ) s:Hl ies •<hen only
one pul~e occur!" in an int.e.;:-vnl of duratior_ ~ . :.:t aore
pulsee occ".l.r , coa:.;_)Ut&.tior..s get vrry involved . Hence , it
is &.Se~od -:-!.nt; a.ore than one rulnn occur!' very ~re
quc nll,y .
!Je:iot.e by p t::e protabili;:1 ti.at n sigctl of durat_on
·r i .' cbulp:~J beyo!ld rocognitlOJI by ~ pul,;;e of duration
b-r • • T, . 'l'te cor!dic.ional ;.1robabi l i -::y of s.n error eq_uals
;i, , undor the condi;;io1. that a pu I :1C of duration 67, l!. 1,
.J.$ J.'fJCe-1.YAd :
Pn • r\t;i'r 1 6r, ~~, J
(79)
Tt.« condi <:ional prob9.bilit;r Po moy be ca:culated .for
vs1·ious rul ,;e st:apes anc detecLion met!lods . p can be cotnJ•i:t<d if W('!', r,) i~ l:no..-n from meaAcire:nent!: . The knowledge
of r. ~uf~icc~ :or a comparison 0£ th~ ausceptibili~y o!
va:r1011.~ pulse sha;;,es and detection methods to disi;urbances .
Let the transa.inad cl',aracter consist of m block pulses
ns •r.own in Fip: . 3 for :u = 5 . Euell p\ilse hai> the duration
•r;., , A posi-t;ive or negative amplitudo shall ·oe detected
b:V nmp.Hucle sar.i1>ling . A dist1.trbing puloe with duration
61, > ·~;m a au sos an orror with p1·obubil H :y P• ~ lt since
t:.nlf of tho dioturbing puli;oo cho.n1>a the Si6J1 of at ieasb
one of Lile :n aampled lllllpli t;udee . The probability Pb ui~
27;,
u . l!~ pU1S£ Tl'TE DISTIJRBAJ;:ZS
,,.
0(11
•"t •U ·S · 4
OO
02
0.4
06
,.,.
(J
lo i
I.
111
rnhi.t'a--
10
6r,1cr1m f--
Pig . 105 (le:'t ) lro·oability p • of"" en·or c~used by a 6i."tiirbing pul"e of dw:•tion 1>;, /(T/r1) . 'I . b lock J•Ulno •. of
p 1 g. ~, upl itude ~lll>pli:ig; 2 . cnme bloci< puses, corrcla:ion; ; . '..lalsh pul .10s • cor.c elau io!l .
Fig . 106 (rigln) FNbahlity r.{~) ot" tt.e 11.11plitu:les b , or
disturbing pulse• ,;£~el' ar.ipl itud~ l i.initici;: ; signl'llG cot•sieting o! a:. = 8 Wnlnh pulses .
cz•euson linea.cly •...•i t.h 6 -s i:i 'the into rvn : 0 ~ fi'fs ; T/ru ,
aa abown by curve 1 in Fi5 . 1oc. .
Lot tho s i gc of t.hc a:r.pliwdos o!" ~!:e bloc<. pultoa be
dctel'l:U.ned by crosscort"clatior. . ~'hi" ae!ll:s =hat the in=c gral o! the pulro" is c>atl~led . Tr.e LU11pl l. -ucies of tho t""ceived aisnal can ta tl.11i ted. ar, +n ::u1d - a ii' Lhe un<liijwrbod signa l hno the a:np l it udo 1 « or - a . i\ d iAtu rbirtti;
J•ul ue with posi tive !J,rnpJ itudc !:LJJ t!'!'ir.1rosetl on n bi1 ~uul
pulo o with amplitude .,. will !o.s- ~uppre>· <ed compl~tnl;v . Or
the other hand , t
a;r.p!itttde o:~ a ntagtttive Ci_;tu1~t.:ing
Pulse would be li11ii:od to - 2a sir.cc •n-2;, is - a, tr.e P:nolleot amplirude the lll:1.He,r \;0 1.tl rl nl low . On :;he 1.>verni;r. ,
onu hali' of the distur1>ing i">Ul !lo~ have an on:? li t ndr ;2u
oi~ - 2a , the otter huve ar. a.nplit.u(le .. r!"o . No oi. . l'Or will
occu 1· U the duratioi: AT, of ~ile ~1Gturl:ing pt...~ .. ~:- ..__ SC•
•!:ort that the .fo~lowizlt; re1.a"io~. l1oltls :
2861, < aT/ro
61,
'
(80)
T/2m
'l'he couditional probabili •-y Pb depentle !'or
·• c!all7 5
'!1 2aT/m
or
T/2:n " A1,
li
T/o
(81)
2'/'-
6 . SWNAL l>ESIGll
on ~he position in ::Llle of the disi;ur bini:; pulse . p• Jlllllpa
fr·om O to 1 at Ar, = ·r/?1:1 'llld increase~ for lu:rgel' values
of 1'. ~, linearly to ~ '"' n:iown b,y cw:·,•c 2 i n l"ig .1 05 . There
is a ntrong c!Jresho~d effnct aL 6-r ~ T/2:r. .
Conside1· ~he transmission of cnaracters co1tposed of It
liaJ.sh fl,;,!lctio:lf! . lei: eacL l'u.cc;ior. have Lhe llllJ'."litude ... 8;.,
~lie !l1:1alles; amp ... itudes of a sum
of :n ouch fwlctior.,; is 1 a nod - a . A.u wnpli.tude limitel'
or - a/a. . Tue lnrgcst and
may thus cl i f at H• and -a >ii thou;; changios the undisturbed signal.
Let :n b~ o power of 2 . At a certain :noment a charactel'
has the empliLude n, if m-k ~nlsh ~i;nctions have the BJ::p~itude
•a/q UL.d k hav·e 1.:
a, = (1 - ;:'k/m)a ,
~ =
o,
P
s.:nplitude - a/m :
1, ... , m
(82)
'.foe proi;>abill~;y o.r a , ocun·ing i s denot od by i·(I<.):
(83)
Th,. 1111plitud~ 1' , of n di::t~rbi:::;; pulse superimposed on
the a.n.pli'Cud11 a }> of t!le uignttl :OCl..,-Y have ooe of -che ~wo
fot~owit.g valu.-o af;;er• 11.tlJ litud<J lir.liLing a~ ±a:
a-( ·1-2k/m )u
-a-{1-2k/m)a :
(64)
::>ka /m
(8.5 )
-2(1-k/~)a
Tl!e probability r , 0:) of bo.vin:-; an ""'!' itude b , beto'een
-?a and +2a !"ol!.ows from (6~1) :
'
l'~\ k I
=
J...,.,,,
'If ""
( 86)
(m)
If
iu. ex=pln of r.,(k) is >ibowo ror
ti>:e as
,...,u '"' nega<:ive
ciisr~ibut..~oD nf~er
ct:
8 in Pig .106 . ?osi-
dieLurbini:; palseu hove a Bernoulli
at!!pli.:'.ldtJ limiting .
:!:e ci1 oreicorrcla~ion of 9 binary sifinal 1''x (a) co!D.posed
of 11ialsl1 pul,,es wol\j,e),
....
F ,(a)
fr ~>x(j)w«l(j ,0 },
i•O
a,,.( j )
:<1 '
275
yields
J111 F,,(e)wal(l,9 /dt
•
arr
m
, "....
=
L/T ,
1 • 0 ... m-1 .
(l'l7)
.111
According to (84) !llld (85), i;he al.isolute value of th Cl amplitude of t;'1e disturbing palse canno: b<" larger th:ui 2a
and no error c&n occ·1r i f iLs duration OT, is so m>nll
tbet the relation 28.0T, < aT/m hold:; . Hence , p. is ?.rro for
Afs < 'C/2m .
(81.l)
This iG the sa!te value as for bloc.1~ pulses .
Tbc calculation of tile conditioual erro::- i::rob.,1.:111 ty
p, is very tediou~ for longer distur3ing puJ.:es [(;) . Th,.
result of the calculu.ion iz sbow!l by curv,; ~ of ''ig . 10>.
'!'he error prooab il Hy iG some•1ha~ lower t!'lan for hlock
pulses and has Beveral t hresholds .
Batter results a.re obt•ined if tho zi 191al componed or
llalsb or sine- co .. ine pu-ses is !llllplitude l:Uni-ed ~. ~l.e
t:-anamitter as cE~cussed ir. section . 2? , s-'.:lce the acplitude limiter a:; the receiver ci"y tt.•·n ?:>e set to lo.,,.,,r
levels .
6.3 Coding
8.31 Coding with Binary Elements
lt has been discu.ssed in section 2 . 11 tha:; a sig.'lul may
be represented by a cime :·unction Fx(O) , a vr-cloi· Fx or
8 set or coei'!iciento ax(j) . A set of U di:·rer·eot $1!;nnls
ia Called an alphabet . A certain fu nction Fx(9) is n cha racter or the alpbabot . Some problCl:IB of designing tile
ctuacters will be discussed h!?re for wtuclJ ori:hogono.li~y
Loy be used :;o advantage .
A disturbance of a character 1:1ay cause it to be "·1 ntnken for a different cbo.racter. at the receiver . A aui t.11ble
Choice or tbe R cha.ractors of an alphabot may reduce t h e
):>:t'Obability of this happening 1'or certain types o r diatlll'bances . Some methods for making a suitable choice will
...
276
6 . SIG;JAL DESlGN
be investigated . Let the
o coe~ficlents :
R
"x(O), lix (1) , .. .. , ax(o-1);
Suen alphabets ...re
.fl.UlCt; ions { f ( j
)a j) ~
call~d
chnra~ters
x
be represented by
' = 1 ..... R
block codes . Using a system o£
orthogonal in the intorva.I
-i
~
or:c o'btainf'i the representatior. U;y tiue functicns:
9 ~
i
1
.,.,
Fx(S) =
2: a,.{j)f(j,6)
J ••
(90)
Ocn eral..:.y 1 'tine signal av cl:.e ln_put of a receiver may
b~ a time dependent electric or magnetic field s t rength,
in other inst:ances a time dependent voltage or current .
It appears reasonable to use the 1'epresenlatian by t ime
fu.rictions when looY..in.r; for al.phabecs t·.. ir;h low error probability. However , i t has been showr in section 5 . 12 tbat
the functions f( j ,e ) ar e unimportant and the coefficients
ax(j) alone deciC.e tbe probability o~· error in case of
additive thermal noi se _ Dil'.fer'ent syste:ns or i"unctions
{f( j , 9 ) ) require different f:r-equency bandwidths for t~·ans
missi on and che p:ract1cal difficull:ies for their gene:vation eu:d detect io~ are d ifferent , but t~ey do not ;i.nf11.lence the erL'Ol: t·a t e . One may represent t he characte.ra by
the cocfficienls ax( j) in thin <1pccia1 case .
_t_ furthe.r silaplif:.cation is nchic\rcd by restricting
tho coefficients a..,( j) to t;wo values which a.re usually
denoted by +1 and - 1 , or by 1 and O. One often makes the
additional nssll!llption th<>t a disturbance leaves a coef.(icient unchanged or changes it co the other permitted valv.e·
Th is means a cilru·aeter v1itil coefficients ax(O) : ,.1 8;1ld
ax( 1) = +1 , WL'itceu in sh<H't notation as the character +1'1'1,
can be changecl by a disturbance into one of the tour
forms ~1 +1 , +1 -1, - 1+1 or - i-1 only and not , e . g ., i.ntO
+t+t . It has bcon shown in section 5 . 21 chat the coe:tficienl a x( j) is changed by a disturbance into a(j) w1ti"b
1na,y have any value even chough a..,( j) can be only +1 or -1 ·
::'ho1"'c are n ntunber of reasons why only the values •1 1Jlld
277
_1 et•e ofLen pe!'lll· "•oC for a( j ) . A;; th~ bog:.n.niug of t'lovelopmcut of codini,; thnoL'y ii, •as unun I. Ly assumed cl\at t:ie
ru.nctions f( j, a ) •el'e blo c k pul~er ~nd we1·e decodod by
,.mpli tude ;,amy.Ene; . A positive lill.J lit\ldf"' was iute1'protcd
BP +1 and a negative one an - 1 . Thl quantization cn!lngc:;
c;be sums in ( =i . 21.) a..~<! (5 . 25) anc g'°n"'1·a11:r i::cread~S the
error rate .
Di aturbar.ces from ~OUL'Ce!1 othr-!.· t.hur. aOditive t:lo:lrmnl
noioo require, i11 pi·.i.Hciplo 1 nn illV(;tGt1gat iou of 1.hoir
effects on i;he Lime L'Wldion~ l·x(9 l f ~ x( j }f (j , 9) l'llthcr
:l!an on the coefficients ax(jj . It l.:.a been ~ho•nn ln t;he
previous section tJ . ?;) tb.a- i;I:!' effr.ct o!' i:ul.>e- ;;yre disturbances depend.•o:. the Ghape of the flmctiorcs f(j , S) =<!
on the clipping r.oplitllde . :;esµito these re"''t" , it in
ous~omaL'Y- t o considl:l'r• O:i.iJ tne coeJ'1'.iu.LE!ut.s of code ulphabe~.s
that are deaignod f o l' resi~cnnct> t:;o 9ulse-t.,ype disturbances, and i;o distin5u.ish on_y b"twocm posii;iv" and
r::.egative values o! t..Le coefficient!" . 'l'he :"e8sons for '-ui s
are the requireccent.: of :-imp:e imple:nentat..ion a::d comreti.bility with exiscin1o •1qu:i.pa.en- .
l'he theory of coding by ·~inary el~l.+"1-G is based ('JM I.he
aasWJlptioo chat tho 1mdist.urbe-.1 cocl'!'lc1enta ax (J) ar. 1·1 011
•c tbe disturbed coef.ricionts a(,J) c1;1n be ~1 ana -1 , 01'
1 and O, only . The coef!'i<'ient!l ax<.1) sr.il .~(,1 l ere uJ-uull;;
called eleoteni:s in thia ca.:e . A:yliabet:B coni;ist~ngo.!' ci1aracters with equaJ TIWDber of ele:>entc. are cc.l c:! Cinar:;
l:lock-e.l.phal:>ets or binory biocE.-cndo" . Since U.e .llldi<'tu,.betl as well ne I.he disturtecl clir1N.1ct.er.; cofft.ll~U only
thn l'le1neuts +1 nnd -1, one way co ndi.ler· t hem to be bina.i·y
~Utnbors . t;u.mber theory i.ppliod to binary mw:°oers rr.ny ;;heG
bo used in the inve:;tigal:ion of co4l.in5 rrobleo.> . P.inocy
COdine; has been crcnted in a large nWDber of publ~cntione
st:il'ti."!l< «ith liAMl'!ING ("'-5] . An excellent sumnoary wns gi ven in 11 book by PE'l'ERSO!' (6, 7 ] . Non-binnry alphabet:; have
also boon investigated llaing numbor theory (8 , 9) .
'l'be value of a code alphabet .for coO'.munications der~nds
OQ
the "rror rate i;llat
can be achi1wed . Cooputation of
278
& • SIGH AL DESIGll
this error rutc is otten very cii.::-icult. Hence, it is com.
mon to use Llie ' Hammin{I, disLn.'lcc' for ,judging che qualit;y
or an ..1pllabeL la t h0 tileo1•;v of coding b:y biliary elements .
It denotet< thr number of binory elea:enLi. in •.<nich two charac"t.eL:S dif!'"r . ~·or i::.i:sta.nce t the cha..t·u.c-t;ers .:..1+-1+1•1+1
and ..1+1+1•1-1 or 111"1 o.nd 11110 O.alie the 3r-l!llling distance 1 . ~IH' probability of decodii:g 11 11 n;;ll.l'bcd charac~er x into the wrong C!laroctflr t often d~cr-euscs with i ncreasinG lh1mming dist,rnco bocweer. the two charncters . Consider , Jor inatance , characters co.nsistin;i; or a sequence
oi· block pu ses . The larger t!.e number OI pulses in which
che charoctern ditfec-, tho larger may be ~be nUlllber of
disturbed pul~es without; nn er••or occurring . The Hamming
diS'C!iliCe J.U pAri;icu:arl;; u::r>ful E the reak power rat;her
than the ene>rgy of cl1e LrOJw:nici.ed sle;nnl iG limi ted .
The l'ollowing example sbowG i;l:t"t a l'll'{;O t!arumi.ng diseance uoe!> noe r:ccE'"sa.rel:r meon a low ;.irobubility of error.
?.'" charactcz·!l ca?: be cou.st1'Uctcd .!"!'om io elemf'nts +1 or - 1.
Tl:e sILallest !Jrut=.i::g C:i&tance c be&wecn two characters is
1 . O!le :r.ay increaze d by coustructing tllO 2m characters
frorr. m' ~ m elements . Tho oncrgy of each ~ranumii;ted characte:· ia lnc!'cnsed by the i'ncto!' m'/10 if ~be e nergy pe:r
elecaent ia kept constant . The decrease or the error probability ia pa.t·t:y derivocl from eb.e con£truction of t >-,e
charac;;crn and pru·tly by their larger energy . It is o!ten
re ..socuble to base ~!le comparison of t"o alphabets on egual
enert:>.r 01' tho characteru or on equal average energy . A
characLeL· with m' > 01 elewenca must thon contai n m/m'
i; i mes t:lle ullorgy per eluo1011t . Hen ce , the Hammlng dist ance
is increased, but the probabUiL,v of eL'rOl' for one element
is also increased, e . g . , if the e1~.rot_•s w:e caused by s ddii;~•;e LLer:tnl noise . J.t csnnoc be decided without calcU_ation •hicJ1 effect doitlnotc::: .
AlybabeLa witb one parity ch<lck aigit ore a.a exalllpJ.e
of " redu1H ion of the en·or r·u Le under LltO iJ1J.' luence of
11ddit1vc chnt'mal noise by inc1•easing the &mFll lost Ha"1llline:
di si;ance . Consider t!:.e 2 m • 32 chnractera of the teletype
279
8 1pllebet:
1 1 1 1 1
1 1 1 1- 1
1 1 1-1 1
1 1 1- 1-1
( 91 )
etc .
'!'be smallest Eaaming dista."lce equnls 1 . Lee a pa:-ity check
"-igit ;1 be added to oJ.l cl:la:-ac•er·~ having an odd nuu.ber
of ele1:1ent;s 1 and n check digit - 1 t;o nll characte1·" with
llll oven nUlllber of olemeJ.J ~s 1 :
1.
2.
3.
"
1
1
1
1
1
1
1
1
1 1 1 1
1 1 - 1- 1
1 - 1 ~-1
1-1- 1 1
(92)
etc .
'r ho smallest !Iucur,J,ng oistance h11~ tnus been increa~ed
to 2 . The ener gy JlOr clell1ent atu nt bo i·etluce<l t o 5/6 or ge nerdly to m/(m+1) . The ;'act"or m/(111•1) approacher 1 for
l arg" values o~ ''" •hile -;;he H=ing distance i<:i r.till
doubled .
Tno smallesc Sw:ur.ing distance between the charoc~ers
o!' an a.lphabet may bo JLade 5, 4 , . . . or ge::cre lly J , by
addi ng s uff i ciently runny chec~ cligit;s . 1'bese alphabets
arc; called system~tic nlp!:abet" . Mnl<ing d ~ 21 •1 one mny
decode all characters correct Ly, if no more than 1 ~l<'mcnLs
have been reversed by dist:>rbance> . <! - 2:!. terai ts the
correction of 1- 1 reversals and the detection wit!:out corr ection of l reversals . Eence, one dlst.ingil.i~hes bt=twecn
l errors-co~·rectlng (IJld l error a-de Lecti.ni; alpnabeto . Thi~
di stinction i s necenoa1'y only :if the <l i nturbed cocffici~n L o
a (,j) are limiced i;o the values +1 oi· - 1 . Accordir.i:; to
(5 . 2~) and (5 . 25) the relation 6'•'x • 6W, would have to
hold in order to make an error de1'ect<iou without correctio"
Possible . The probability that t:.Wx and 6W.,. are equ11l is
zero if the disturbances are due to thermal no is" . Tltere
are , howeve'!' , cllsturbances for which this probabili·cy is
not zel:'o .
•'he 2m character" const'!'ucted from m binary elomcnts
280
b . S!GNA.L DES!Gf(
1 an<;! o forn: a g r oup under addii:5on moc.u!o 2 . l.Jote that,
the Walsh functions have the sarr.e Seal,u:-e ["11] . Ao a.l.phab<)t i.s called a ·oL'>ury group alphabet o:- a o:i.nur;y grou:p
code if i-cs cha.rtlctcrs are t:1 ~ubgroup of this group . A
sysc:e1tatic g1'oup codo :i.s a s.,yst..erentic code whose chara·c-
ters form a group .
A special class of binary group cc<ies BL'e the Reedl'luller codes [5, 10] . 'l':'leir churacte:-s contaJL m elements ,
:n being a power of 2 . The nwr!Qer of en eek eleJJen{;S is m-k
and tile numbel' of ci'.nr~cter~ is 2• , k luis clie value
k =
Z' <i) ,
r < m .
... o
(%)
The s:nallest HaJ:irming distance is rl =- 2""'· 1 •
Conside.r· an example wbere m = 21. i:. 1b and r -=-1 (Reed'l'luller alpha.be~ of first order) . It fol I ows d = 2 ,_, = 6
=d k = 1 +"- = ? . This alphabet contair:s 2 s • 32 chs.racte""
constructed from 16 elementn, 1(1- ) = 11 of lihich are checit
elemen~s . This a.lphabec is denoted as ( 1 6 , 5)-<ilphabet a)!
genere,ll:y as a (m , k)- alphabet . Table 1 3 shows the cha.J.'acter~ of &his (16 , 5) - alphabet with the elemencs ~epresen
ted by +1 and - 1 . Compare the signs o.i' the elements oi' the
fi!'st 16 charact"rs with the Walsh functions of Fie; . 2 .
The signs correspond to the posicive and negative amplitudes o! the Walsh :'unctions . The signs of the elem.,nt,;
of characters 1'? chrottgh ?2 arc obi;ained by reversing the
signs o:f the characters 16 thro1lgh 1. 011e me.y thus conotruct a Reed-Null er alpha·oet with m cnaraco;ers as follows:
The
'lalsh .functions wal( j, 9) , j = 0 .. . m;;/, rep1·esented
by im plus 1md minus signs y:i.elc\ one half or the cha.1'8Cters; bhe other half of the cbara.ccers are represented by
rho fwictio11s -c;:il( j , e) . Thu::; tile Reed- Muller alphabets
rm
belong
to the class of orthoeonal alphabets .
6.32 Orthogonal, Transorthogonal and Biorthogonal Alphabets
To snve space let us consider a("- , 3)- alp!:iabet ins bead
of a (16 , 5)-a! phabet . It contains 2' = 8 characters . The
.,. 32 OR'l'HOGOl;.1,1, JGF1!1G::STS
281
Tsble 13 . The coe:'oicier.ts a,(j J o:: ':!le c_aract<-r1' of ~
(16, 5)-u.lpt:abct 'iccordir..g to Jl::;"D-.iULLE!! . x • 1 .... 32,
j •
o.... 15 .
x 0 1
2
3
4
5
I
7
8
-
9 10 11 12 13 1" 1 5
1 .1 +1 1·1 +1 +1 ; 1 11 +1 +1 11 ;.1 .1 ; ' 11 +1 •1
2 -1 -1
3 -1 -1
4 +1 +1
5 +1 ;1
6 -1 -'I
? - 1 -1
B t1 •1
9 .1 -1
10 -1 +1
11 -1 +1
12 +1 -1
1? +1
1'l
15
15
17
18
19
20
21
22
23
211
25
2G
27
28
-1
-1
+1
-1
+1
+1
-1
-1
•1
•1
-1
-1
+1
+1
-1
29 -1
;;o +1
;;1 .. 1
;;2 - 1
-1
+1
+1
-1
+1
-1
-1
+1
+1
-1
-1
+1
-1
+1
11
-1
-1
-1
+1
-1
-1
...-.
-~
-
A
,1
t1
-1
t1
-1
-1
+1
-1
+<
+1
-1
+1
-1
-1
+1
+1
-1
-1
+1
-1 -1
+1 +1
+1 +1
-1 - 1
_,, - 1
- 1 -1 -1 +1 11 +1 +1 +1
+1 +1 - 1
•1 +1 - 1
- 1 - 1 -1
-1 •1 11 +1 •·1 +1 +1
+1 - 1 -1 -1 -1 +< +1
+1 •1 ' ·1
-1 - 1 - 1 +1
+1 -1 .1 _, - 1 +1 • ·1
+1 - 1
- 1 +1
'-1 +1
-1 -1
- 1 +1
...1 -1
-1 -1
+1 +1
+1 - 1
-1 ·1
+1
- 1 ~1
-1 - 1
+1 t1
- 1 +1
+' -1
t1 +1
-1 - 1
+1 -1
- 1 +1
- 1 -1
•1 +1
-1 +1
•1 - 1
~1 +1
- 4! - 1
-·
-1 +1 t1
-1 - 1 -1
- 1 - 1 +1
+1 11 - 1
-1 -1 11
+1 +1 -1
11 -1 +1
-1 11 -1
+1 -1 11
- 1 +1 -1
+1
+1
-1 +1 -1
+1 -1 ·1
- 1 +1 - 1
- 1 -1 11
-1 •1 - 1
-1 t 1
t1 ~1 -1
- 1 -1 +1
+1 -1 -·1
-1 +1 .1
+1 -1 -1
+1 i 1 •1
-1 - 1 -1
·1 • 1
- 1 -1
-"
-"
_,
•'
+1
11
.1
+1
+1
11
+1
+1
'1
+1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1 - 1
•1 - 1 - 'I
+1 - 1 ,
- 1 - 1 +1
-1 -1 +1
-1 11
- 1 - 1 +1
-1 ~1 - 1
-1 •1 - 1
- 1 ..1 -1
- 1 +1 -1
+1 - 1 +1
+1 - 1 +'
-1
, -1 -1
+' +1 - 1
+1 .1 - 1
+1 +1 - 1
11 -1 - 1
- 1 + 1 +1
-1 +1 11
- 1 +1 •1
- 1 +1 · 1
-1 - 1 -1
-1 -1 - 1
-·
--
.
..--
11
-1
-1
-1
- 1 -1
+1 ~'
.1 11
+1 - 1
-1 - 1
- 1 •1
- 1 •1
- 1 +1
- 1 +1
+1 -1
11 -1
-1 +1
- 1 ~1
; 1 -1
•1 - 1
-1 -1
-1 -1
- 1 +1
11
-'•
-1 -1
- 1 -1
11 +1
11 +1
+1 +1
+1 •1
-
11 -1
-1 - 1
-1 - ":1
.1
.....
1 +1
-1 - 1
-1 -1
-1, +1
t1
+' -1
•1 -1
-1 •1
-1 '1
'1 - 1
+1 -1
-1 +1
-1 +1
·1 - 1
.1 -1
-1 .1
-1 +1
-1
.1 -1
11 -1
11 +1
I
-
,.
-1
-1
-1 -1
+1 +1
..1 +1
' - 1 -1 -1 - 1
1
-1 -1
, -1 - "i - 1 - 1 -1 - 1 I
-1
-.
!irst four arc t::ie .fir~e fow• Wr;lsh f:mccior:s of l•ig . 2 :
1.
2.
4' ·
•
+1 +1 +1 ' 1
('YI)
- 1 -1 +1 •1
- 1 +1 •1 -1
+1 -1 +1 -1
~o elements of (94) may be considersd t o l'or:n a tt.ntrix 1
X. lntorcbD.ngi."lg rows and colUJnns yie.1.ds
:nati-1.Jc
x·:
':l'l.
11s mntrix is " Hadamard mntrix [7-91 -
the
trun:ipo~cd
282
6 . SI GNAJ DESIGN
•1
i1
( .1
+1
x· -_
-1
-1
+1
+1
~1)
-1
• 1 -1
i1 +1
-1 -1
(9,5)
Th(· product XX• yiclcl.::1 Lhc uni' matrix E nult lp!ied
(=~ :~ ~~ ~~) (:~ :~ :~ ~~) ~ ('g·~ .~ ~)
4
+1
-~
•1 -1
I
1 •1 -1 - 1
Q
Q
Q
by,,,
(96 )
+1
I. ciatrix is calla:! ortr.ogor.al if' ita product •.titb its
r"!illtposed matrix yiolds the :mh malrix :null;iplied by a
coustru:t: . .:.i:: alptabot is called orthogonal if ics clements
cnn be written liQ t.l:ic alemea.ts of an oi'l..bogona.l matrix .
'l~ •e ulphabei; (911) is Oll or~hogonal alphabc>t ; the characCC!rr 1 co 16 nf •ruulc 1 3 form
orthoi;onol 1u.phabet, as
<lo the c2an1ctcr:; 1'/ to 32Let usomt ~he thiI't! clement o f all chAracters in(94) .
An .'.llpna·oet witL i;hree eleoem;s Wld four characters is
=
ot:taicec:
,,
<,
..
4.
.1 +1
~~
- 1 - 1 ~1
-1 ~ 1 - 1
(97)
+1 - 1 - 1
!'ho pr·oduct oi' tin muLt·lx Y and Lhe tn1nupo:icd matrix
y
+1
(1
-1
- •' •')
-~
-1
.1
+1
-1
y·
r+11, -1 +1 -1'1)
-1 -1
+1 - 1
-1 - 1
v·
-1
y-iolC.s
y y• _ ,
- ,,
+1
-1/5)
- '1/3 -11;.
- 1/;, >1 -1/5 -1 /3
- 1/3 -1/3 >1 - 1/3 .
(-1/3 - 1/3 -1/3
(98)
+1
'i'ha d ifference between t;te e lel!lents on chc principal dial)Ot.ttl ar:d - he OLhe:·" .i." L~rge1 fOi' the motrix (98) Chall
ror L .e uci1; m:.tr.i.x l9<>) . For \;his reason the alphabet
(97) is callee tron~or:;ho,:;o=a_ . T!ie practical meanil'-e; o!
tran~ori:;i.o;;c=liey ir. evidenL froJJ the alr·hnbees (94) and
(97) . Bat h cor.tain roui· charact ers and elle H6lll!lling distance
6 . ;2 ORTHOGOt:A.L ALPHt.BE'!'S
283
betweenar.y Lwo ch11ractera equals 2 . llowever , the ulphabct
(9~) requires t'our eleme11 tn tUld i;l1c n l phnbet ( 97) only
three .
Let; tbe charec1iers of the alphabet (97) be
by vectors :
1.
Fo =
2.
F1
~·
'"
~eo
• e,
= -ea - e,
F, = -eo . e,
F1 = +e o
- e,
~·eµrencntetl
+ e,
+e,
- e,
- e,
Tbe end poiGCS of these l'mu· vector~ nre Lile corncr·s of
o tetral1edL·on , na ohO\\f:i j n Fie; . 20a , if tlJe 01\i,gin ol the
coordinate "yr1."i:t is placed n't Lhe center· of the i;eLr·wiedron and t:ie coordinate r;y!"t<'Jl is rot'1teJ into " J>!'Ooer
position .
'ftle ;;erms off tl:e principal diagonal of the matr.Lx. VY'
ere close to zot·o .for tranr.o:rt.hogonal nlpha.Oet;s wit.11 tnOJ"'O
t hllll four chnJO•ucters .
Let tile orthofl;Onal nlrl."l>e t ( 9") ne :<u;>p-e:aem;ed by the
chnracters obLained i::y c!:wie;ir.i; =he sigr.~ of the elen ..nts:
5.
G.
7.
8.
-1
+1
+1
-1
+1
-1
+1
-1
- 1 •1
- 1 +1
- 1 -1
- 1 -1
(99)
The ( 4 , 3 ) - alphabet cousi~~ ing ": tr.e cho.r11cterz l '"') and
(99) is called biorthogon<1l . The (16 , 5)-nlphabet of 'r$'ble
1 3 is also biox•tlJogono.1 . AJly chaJoo.ctor ot' o. bi oi ·Lhogonal
lllphabet has tho Hp10ming die~ancc d f1•011, any otlie r mccept
!or one wbict uas the distw:1ce 2d .. AJ.J. ~xqmple of n bio:'tbogonal alphabet that i~ not a Reed-tluller alp!: ..bct iz
the one shown by the o ct o.hed.i:on in Hg . 28t .
~et tho reprcoontntion o.f ch:i.ract.ern by e~emcnte or
COeJ'ficienta bo replace(! by the 1'0Pl'OOcntation by tine
functions . Consider a system 01· 16 or•:liogonsl functions
l'(j, B) . Each function is 111uJ.tiplied by one of the 16 coe::!icients of a character in rable 13 nod the products are
ll<l<led . !f the functions f(J ,e) are blook pulses , the firat
264
-o. . SIG HAL DESIGN
16 cnaract;e.t·s O'['C- repz·~!jcn ted by the 'ial~h rw1ctions ot
1''ig . < , the second 10 clluracters by tht' swr.e ~..'a.1.~L !'unctions
aultiplied b;1 -1 .
lnste"d of rnultip: yi:ig tbe 16 block pul~es by ,, or -1
nud adding the p1•otluctu, 0 11e cm.lld ,just as well llll)l tiply
one '"'tlf!h :'·.lI\C ti on :i:;..- +1 o:· - 1 and t;~.c otllt!.t' fil'teen by
0 and ad.d -the fl'Oduc!s . The c!-.aracters :;re then represented 't'~· tle co(!fficie!ltc '1, -1 and 0 ll" ,,;-,own in ~able 111
wl11n•e Lue ~i:rn~ l'OW liGts Lr.o index j of wal (j , 9) and ·t he
col= 1 l !JC~ t ho nuciber x ot l:b.c cha1·aci;er . The
~unc-io~s ·•al (j, 9 l 2re 1:1•..1ltiplied 1>y the coefficients ~1,
- 1 or C . The SU!lll!atiou of the J:roduci;s i~ ~riviol sinco
one pror.uct ouly is unequal ,,ero l'o:r each cheractor .
On" hao the curio Lts 1•cnul i; Lhnt the cernary tLlpbabe~
of ' , ble 1" and the binary al;i:-.abet of 'Pab: e 15 yield the
smr.e sig=l:; . Both alphabet"' !llUSt hove the SOJJe error rate
under the ini',ucmce of i111y kind o f dts~ur:nrnce .
:U1rteatl o l' rcpresent inp: Lhe char::.c LeI'a or the ternacy
al;•i.nbet of Table i" by 1f. Wsl sh pulses, one may use the
constnn~ f(0,9J, 8 si~e and i cosine vu:ses according to
F:. B' . 9 . ':'::ie =~:equer:cy power !;pect.ra of i;l:e !'irst 5 _pulae.tJ
firGt
are
aho\\~r.
i n Fi6 . 21.1. by the
cu1·ve!:i 11 l
·o and c . Tbe six-
chru-acLe!' would be F,6(9) •of~ sin (16118 >tn ) . ts
power spectnu: 1:ou!c be centered ni; v • 5 .:.n Fi1 . 24 . Choosing ~ = 150 :ic, wtich ~s , "'uch used ni:andard for 1'e1etJPC nignal• , one obtein:i the unnOl'tn'1lizcd froq uencios
shown t hei ·e . 'Vhe s i gn(ll l",a(e) would have it s energy centel'ed nbout ' . ~3 Hz and there 1-:ou...d be pt•actical2.y no
ene1-gy above " Hz . One should not conclude from tllis narro1·. bandwidth , tl111t the nlr,habei; of Table 14- is better
tiHtu I.hat o f Toh le 1;;; . One may multiply pulseo uccording
to J"ig . 9 by the coci'fic;.ents i1 and -1 of Table 13 a.nd
add the proouc•< . =no roau1ting 32 signals nave al.lil0 8 t
no cnore:y above 1 ,o Hz. .
Onn may con~ L1•11ct 2 11 cl1ai·acters fi·om 16 b inary c.oefti~
ci•rnLe . '!'he ( 1( , .•)-D.l pl:wbet of Table 13 uses 2' of the:ll·
It is u:;ual to sny, ;hat tbis alphabet contains 5 in.foi·t ccr.L!.
?8~
6 . 32 OIITIJOGO:IA.:.. fiL?HJ.I<ETS
rre.ble 14 . _ .he coef~icic::.t:n n.-;(J) o: -i:!""1c c!-.!l.l'UCt:Cl'!; of ii
terr.ui·:r bLorthogonal "1photct . \ : ' .. .. 3?, ,1 • c ...• 1' .
)(
0
'I
,..
,;
'+
1 11 0 0 0 0
2 0 ..1 0 0 0
; 0 0 +1 0 0
ii
0 0 0 +1 0
5 0 u 0 0 +1
6
0
7 0
8
0
9 0
0
0
0
0
10
0
0
11
12
0
0
0
0
0
41~
0
c
15 0
1b 0
17 0
18 0
0
0
0
0
13
19
0
27
0
0
0
0
0
0
0
0
28
0
29
30
;1
0
0
0
32
-1
20
21
22
23
24
25
26
0
0
0
0
c
0
0
0
0
0
0
0
0
0
0
0
c
0
c
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
c
0
0
0
0
0
0
0
c
c
0
0
0
0
0
0
0 0
0 0
0 0 -1
0 -1 0
-1 0 0
0 c 0
0
0
c
i)
0
0
0
0
0
0
c
0
()
0
0
0
c
0
0
0
0
0
-1
0
0
c
0
;.
b
?
l:l
)
10 11 1 2 1 , 1'• 1 :.
0 0 0 0 0 0 0 0 c
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 c 0 0
c 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
.1 0 0 0 0 c 0 0 0 0
0 .1 0 0 0 0 0 0 0 0
0 0 .1 0 c 0 0 0 c 0
0 c (' t' c 0 0 0 0 0
n
0 0 0 0 +1 0 0 v 0 0
n
v
0 0 0 0 •1 0 c 0 0
0 0 0 a 0 0 I~ 0 0 0
0 0 0 0 0 0 0 .1 0 0
0 fl 0 0 0 0 0 0 11 0
0 0 0 0 0 0 0 0 0 1·1
0 0 0 0 0 0 0 0 0 0
0 l) 0 0 0 0 0 0 0 c
c 0 l) 0 0 0 0 0 0 -1
c 0 c 0 0 n0 0 c -1 0
0 -1 0 0
0 0 0 c 0
0 v
v c C• 0 -1 0 c 0
0 0 0 0 0 - 1 vn 0 0 0
0 0 0 0, -·1 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 -·1 0 c 0 0 0 0 0
0 -1 .;: 0 0 0 0 0 () 0
-1 0 0 0 0 0 0 c 0 0
0 0 0 0 0 c 0 0 0 0
()
0 0 0 0 0 0 () 0 0
0 0 0 c 0 0 0n 0 0 0
0 0 0 0 v 0 v 0 0 0
0 0 0 0 0 0 0 0 0 0
0
0
0
~
0
0
0
0
0
0
0
0
0
0
0
c
0
0
0
11
-1
0
0
0
c
0
0
0
0
0
0
0
~
thnt
of' infor.·ne,.;ior: ruU 1·1 bite
:iation digi 4;s a.-ric 11 c3eck <1tc1:0:
OL'
- t ..:t~ ... r
c
0
0
0
eacl.i.
.:.''.)chnrncte1· contains 5 bit~
<IU?tdancy . A to~ul of 3" Cb.ll.L'VCLcr~ <Lay be cor: t rue tcrl
trom 16 ternary cor fficir.fl'~ ~ . 'l'he 9lp:rnbec or 'hble 1 1.1
Uses 2' of them ; 00(' rnny L1saii,;n uhc info~mntio:i 5 1 i t.n
to each character . One \dl l , however 1 be ,..clucta111. l;o njSign tho redundancy l.; 2 (;1" - 2') t;o ~hr·n . 'Z'tJ· COJICl>J t of
redundancy io u;,erul, if alphabets of a certain ordet• ~re
cor. •idored . Wili!'.!.out thic restriction ~bere is no reaaon
~hy the characters of tho ( 16 , 5)-alphabet ehould not be
conol \erod to be derived from the r'' chnrncte rs or an
28l
6 . SIGt;;J. DESIGI/
R~pl::abet of ordor r rather t:hnn
froa. tne 2" c1w1:acters of
an a.lp!iabct of Ol"der 2 .
The oonceµt of distance uus also prove11 11neful in the
general t;heo:·y of coding , no longer resLri ·ted to biua.ry
elements. For a ger.eralizat1on o.r tl:e fiaru:iing distance
consider t;.:o cherac:;ers =eprcnented b:,' tile !'uc.ctions Fx(8)
and Ff'(B) int .e incerval -1 !! a,; t · The energy required
to crans!"orm F,, (a) into i',, ( a) ia Wx, :
llt
fUx(9)-.i!'~ (a)i d9
Wn=
1
(100)
-1/Z
The energy o:· the character
~·..
c8)
.
i" u
~~-
.,
J Fi(G)d8
Wx =
(101)
-112
The average energy of all ll cltflr:llcters of un olphabe't is W,
'Ii -=
•
....L: PxWx,
(102)
where Px ia the rrobabiliLy of tr1u1s:nission of character X·
·rce cn<>"!:"Y dist=co' d,,., of the cnaracters Fx(8) and.
F.,( e ) is dl'!J'ined i.i:y normalizliltion of "Oho eneJ,gy Wu:
Let Fx(a) and f'.,.(9)becom•:rucced froaaorthogonal functions f(j,9 ) :
,,.. ,
L:
m.J
qx(Jlf(j , e),
r\,(D)
2: a,,(j)f(j , e)
i :0
jrO
One ob~ain"' l"or Wx'I' and Wx:
m-1
Wx.- ;
Z [ax(.:J)
j: 0
m-l
- ;;..,(j)]'
'n x = 2:; ol' j)
{105)
jJD
Let nll ch11rActers !lave th~ so.me energy W•
w, . It follows :
1'.l'he te.ri:1 'normalized non-aimilarity ' has been used for
, _,,_;te
energy at.ito.nce if the inLogration ill'tervul ii> ........ ~
[10] .
287
,, . ;2 ORTUOGOilA.L ALFllAJJETS
~·
L; "x ( j
1 112
1 - Ti) fl"x(B )F,,(9 ) d9 • 1 -
Ja., ( j)
,.
( 106)
m- I
2:
· 111
ai(j)
pO
..
,
2::a/ j)a,.(j ) =
~
J•O
J•C
Ax( j )a./j)
I
=
'!'Ile !olloi<i.ng e::iergy ui~tar:ces n:·e
charactez.•s of ~al'!-e 1'-1:
dx"
. ,, for x
• 0 !or
• 2 for
32
1·
x
)(
¥
i"
- •
32 -
The characters of Tablr
m·I
L a~(j)
x
-1 L'or
=t
•1 for x
0
f'Ol.,
it.u~
32-·1·+1
X ~ ~, 52-•t+1
obtai!Jed for 1:"'
( 107)
1
·I
1
"1,+ ;yiel1:
15
=
l•O
2:;a~(j) • 1b
~ax(J)a'l'( j )
I"'
dx., • 4 .for
(108)
i•O
)(
:
..
"
,L nx<.i )a'I'( j)
.
;2 -
~
·I
-1 \.>
ror
11h
fo1·
0 fo1·
x
:i2-H1
x
'(
I
'I
~!
1
32-i• . 1
.. 1
• 0 for )(
t
(109)
• 2 for x I ' 32 - ~ • '1
'he distances dx" or the charo.ctern of •ral;le 13 >1ould havo
the values 16, O or a , if Wx., i!. ( 10~,) wnre dlvic:le<i by
W/lg,:u - '1/4 rather ~ ban ·o;y W. :tiH ia ju~t eni. uurul"•r of
~lements in which tte cha:-acters U.i.!f"'r , i . e . , their Hn.:i.t1ing distance .
The energy dist;.o.nce dx.., of two characters is equ:i.l co
tho square of the vector connecti ng their signal poinco
:!.n a:Lgnul space . •r1tose vectors are 1'9P!'esented by the rode
betwoen the signal points in Fig . 28 . The term distance l1as
an OViclent meaning in the vector representation . Due to
288
6 . SIGNAL DESIGN
tbe normalj.zatioc. of dx\1-' one munt r-0-qt1 i1'e* fo .r th.e vect-0-l:I'
representation, tha;; the signa: points have tl'.o ave:rage
dis~ru1ce 1 from t.heit· com.aon center oJ gravity .
Let the E c!:l:u·acters or a ':li.ortl1ogon0l alphabet be listed in sucl1 a sequer:ce that the T'ela""Cion
( 110)
is $atisfied . it follow!:i :
t f'"
{
Fx (e)Y.,,(e)aa
~
·ll1
1 for
- 1 for
0 for
x
~
~
- ·t • ·1
x
~
xL
i' , R -
~
( 111)
-1
It fol LO'.<S r:-or.i (106) that Lhe cha1'acter x o:· a biortllogonn: alp:-:iabct has an energy Jistance 11- f rom the cha.ra:_C.t.i:.r R-x+1 and r:...'1 energy- distar:.ce 2 from all o tlter cllarf1tote~5 i x = 1 .. . . R.
6.33 Coding for Error-Free Transmission
SiiAill'!Ol'I ' s fo:'lnula for the transmissior: capacity ot a
comrnunication channel proves l;ht;,.c an e1~.ror·-1'ree transmission is possiblP as a limtc;i11l> case . Fr•oru ~he derivat.i,on.
of that i'ormul.:1 in secti.on 6 «"12 it is evident bo''' alpha':iets rr.ny be obt.:iiner.1 whicll aµ_µr·oacL t.he t.r·ansc1issi on ca-
µacit..y o~· the· channel a:l<l which have vanishing error rates
J.t r::,e p!"esence a:· a.ddi tivr~ thermril. noi.se .
Con::;itle:i· a o;,·::;te:a of Fourier expundable orthogonal runecioi;s f( j, 9) ir:c the inte!·val -! " 0 ;; ~ . Random numbers
a 0 ( j; '"'i tl:. a Gaussiau tlist1:ibtttiou ~.i·~ taken from a table
And ti:e character F 0 (9) is constructed [1 - ?J :
""' (jJf~j,a)
F 0 (91 =L:a
0
(112)
j :O
Or:e :rny agsuoe tLat t,he nwnbex·s a 0 (j) represent voltages •
F 0 (9J is chen a time variable voltage . F 0 (0) cannot be
distinci.:ished. rrom a sample of thermal :noine i.f 1a gro·~S
beyond all bou.nds .
Usins a.nor,ber set: of m random nwnbers a 1 (j) , one 111aY
constt'LLct a secoud chat'ac~et' F 1(e) . The general ona:racta1 •
6
• ~ T£RJL'.RY COfolR lilAT1011 il?l!AHJ:.'l'S
289
Fx( 9) can be constructed by moans oi r.i G11uzsirui ciscr'lb11tcd r!ll1dom numben; ax{ j) . The u."Jnormalized dur& t i on of
these character:> equals •r . fhe tl'ansmirrion capacity of
tile channel o! ( :;u) ~ol:o~·a fr":n c, ~ ru1a t~.e m;era.::c rignal-to-r:oise power ratio ?IP.,,.:
( 11 ~)
Let n be the largrsl int"gr;,r >imaller than 2ci
n characters Fr(e ) be ~ocstructcd :
x•
0,
1 , ... ' .n-"°
a.nu
lnt
(11:. )
'J'lleae n cllarnc~ers for" the !i1·ot alphuoec . Now l et J, nlphnbets wi tb u CL<1racters ~acll be eon"' LI ucted ic thll'I way
and pick or:c al;1habet at rando:n . If :; ~nd L app,·oach in! inity, tt:c P'·ol;abili-o:; iA arbitra:•i:y close ~o 1 ti.at
thi s alp!:l,,bet yio ld" Wl error ruLe arr roac11i.ng zc:-o .
Those ' r amlom ulplla.oer.s • ru.·a very 6nc.J.sfying from the
theoretical poiuL of view . l'uore a r e , bowcvci" µrect.icul
drawbacks . It in not only int r,rcs~iq,- ~o o;ee how good the
alphllbet is in ~he li.l!:iL , but whar. tt.c !'!'Obflbili::y of error is for a finite ELnount of ic.fomation µer charact<'r.
EiLIAS found tho .first non- rnndoru alpl".ahct nppx·oacl1ini:; the
•1r1•or prcbabili i;y zero for: fi.nii;e energy :per bit of in f oi·ll!&tion [4,5) . ".'he- trans!ll.ission rate of infc=ation ·,,;as,
however, much ~:r:a l l er t_an Sl\.Ala;o:; ' e limit . :::ie socnl led
co:nbination alphabets also y.~ d vrulishing error probc.bil i ties and come- very close to SHfJ-ITION ' n I imi ~ .
8.34 Ternaiy Combination Alphabets
ori;hogonal !"unctions l'(: ,9 can tr!lll!mi:; m coeffic1•nt s a..,(j) . A tocal of R • ,., charactei·e can be cori; t 1,1cted. i f ax (j) moy usswne the thr ee valuon +1 , 0 anQ -1 .
'11 ~iting ( 1+2)"' instead of 3"' yi e lds the f ollowing expnnsion :
i:.
'itit s deco1nposition divides the
IQ I l.t111'11Wi, Tr.in6mleslnn ot lnl0ttni111on
set of R ~haracter9 .lnto
2':)0
b . 31GHAL DES!G!l
aula:set.~
of cLe:.racters contninint; oqu!!tl ly rta.r:.,y .!unctioila
!(J,:J) . There is o = 2'(~) ctaracLcr con~ainiug no !°Unc~i .:>n, bccaune 821 coe!'!'icier.Ls ax{ j) Aro zero . .Furtner-
aore, 1;here are 2 1 ( : , J
=
2£1
cba:.-acte!"!J, con,isuing of one
f1Jnction each , because cn.l;; or.e cocfi"ic; om; ax( j) equals
t.he i>iorthO~';Onal alphabctn . In general , ::-he1~e !.il'~ 2"\
chnr··H-:ters , each conL o~ning l1 funct>.ons ax(jJJ"(J,eJ, wllere "x(j; equals +1 or
-1 . Si.nee {~;) is the mu:i1e1• oJ: combinoL .. o ns of h out or
+'1 or -1 . These cb9.racters .for-ll
ct ru.nction~,
f:'
these elphalJets IH'e cnlleu r.ernar;y combina-
Lion nlpt:nbnt::; ,for h i 0, 1 01· " · :rnbl 0 15 shows the number 2 ~~) oi· ct.aracters in such alptu1bets .
~nblc
15 . tlwnber or ctaracters iu lcrr:ory conbination al Accordi.nt- ~o Al.SAC<: (2), th" n•L'tbe,·s above the
1 ine drcwr: tllroug!:.. ~he tabl" b~long to ' good ' alphabets .
?!:nbet~ .
•
- • ..,.
•
~
I
!
'
I
••
'
.
• ,,,• " ' ,• • ID
l1
'
"
..." '°
"'3!
I .:~1
111
:.:•u t.r,1 "" !111tj
'""'
'"..." u·".. li'll2 ·"1JilJ 1:1-uo
~(iO
..
t-i'
_,.
11:?
:12
...
Iii'<!
Oil()
11!!•1 (
17\12 <14.(1:,!
~:1110
l(jlJ
4 1~
•'
l:!I!
u
111';1'
41)(1~
U'11.'i• ~l
:2/tl\
Zritl.1
115:?<>
.'il:.\t
,'II~
-
102·l
II)
Eq·.iation (1"5J yields, i·or h • :n, clle 2m(~)
= 2m
cha-
:-acLei-s Lhat contain all a: functions nx(j )r( j ,e) with ax(j)
equal +1 "'
lllfll8b1't'l .
- 1 . These
! l l•.,
the eha.t·octt1·,; of the binary
~or."ider gn alphabet witt. charecterc conta:u:ing h !'unc-
tions 1( j, a) . Each charact:cr contaius h coel!"icients ax(j)
eqwrtl '.. o .... ri 0 or -a. 0 and JJ-b co~i'L'icie11ts equal to zero .
Let thc:Je characters be transmitted . Cros!lcor1'elation of
ti"' rcccivod aigru;_ 1·1ith tho functiona .C(;l ,8 ) yields the
cocfl'icients ax(,j ) . Let addiLive tllet•inal ooinc be superimposed on tl:e signal. The coefficient A u( j) a.re ab·tained •
,;hi ch have a GaussiaJ\ distribution W'l the moan ei tiler +so•
0
or -o 0
,
denoted "oy n'" 0 (j) , ., 1011 j ) ~wl , 0·•1(.j):
(!!.'"'~o,j))=(a'·' ') •
<a1• •<.1 I) = (a•••)
•
•1,
(t''· ':,\il) •(n'· ") = - 1
0
llo
The v ..r1ance ~· of ll.ese di:>"Cribut1on::. follow£ fMn (60),
(70) ond (71 ):
al •
<ala1'j)) = ~>!~
•11-1 • h~1
(11 '7 )
b • nucber· of coe!Iicien;;s s.~(j; '<lith value "'• or - 11 0 ;
n • lg,2 n' = inforn.ation per c!·,.;ractcr in bit!l 1 i 1" a!l
cho.raci;crs ai~e tre.r.stiLt.ed with ftqual trobabili;y;
P 1 • average po.,.,.er o r' n o=tti.ogonBl co~ponents of the1·:r.al
'· rioisc in an orthoe;onalil.iy intnrvnl of duration T ;
P • ha~ = nve:~a.ge r.igunl po weJ" ; Af • t!./2T ;
Pd,
average ])O\-.TC!r o! t. J1 ~rmal r.oiJ:Je in a freq ue11cy 119r1d
"<"'
of
~idtll
6f .
The average noiae J>Ot.'Pr F •.- rnt.he:~ '::h:i:i Fh._ 1 or r,.., 1 is
used as a reference in order to fact l i :nte co:n_psrivon between binnry and ternary alph.abetn .
•rhe
s~
...,
2m(~) SUCIS
• 2; a(j Jay(j)
(1 18)
l•I
cuat be
produced frOIL ';he m coeff~.:ie:>c:: n(j) !'eCeWcd
nnd the larges t; one mu<'t be dc::ermiued for d"cociir.,-: according to (5 . 25) . m- t. of che coeffici cncs n,, ( .1 J a.re U l°oi'
any ~ . ConsidOJ' thoec sums for whicl 1 en rt a in cor; ff'i cier.Ls
a.( j) are O,forinut=ce tbose for J O ... o.- 11- 1 . Tr.c 1·"lll•ining h coefficients a.,(j cguol +1•o o:r - a 0 ''"d ,;riold
2• different sums s,. . Tt.e largest of theEe 2• st:l:I<' will
contain h JlOSitive 1.erms a(j)a,,(j), whi.1e the :-emniu.ing
m-h torms are O. 'rho lnr;;;estol.' all 2"(f,l swns s,. will 1.><>
the sum whose non-vMishills term a contain tho !: coe rr l oient6 a(j) with th<'.' largest mse;nHude . The swn will be
la.rgoat for the tra.nsmit~ed cnaracter Fx (a) when he ab-
..
2')2
1- •
SIGl\PJ, DESIGN
r.olut" V!illle of <:he !: coefficients a 1•11 (j) and a•· •l(j) J.e
larger than ~hal of the IL- h coer!"ieiec~" nlDl(j) , and i.f in
1
· •-araer a~d a1• 11 J i e. sci:o 11 er than zero .
a{ ldi -1
v o:- u · "<J) is
c ·) ·
Jle:;,cc, the fo 1101dr:g cwo cocditions 1>uet be sa;:isfieO. fol:'
er1·01·-frc!'o C.ccoc.ing (see Fig . 107):
1 . All coefficients
n.l'"C uo:J-nf!{;P-'tive :
a'·"= a1· • l(j)/~ 0
nnrt
-a•·•• = - a<· ll(j)/a,
( 119)
2 . None of the h eoefJicicm;a •n1" l on<I
-a•·1
>
is smalle:-
thnn the absolui;e v alue of one of ~he m-h coefficients
a" 1 • 111"( j l/ti 0 • ~'!lis eonditior. nc~d:l to be satisfied only
i f cor.·liti.o" 1 is SHliHfied :
,,,, '""'I "'
-a - la' 'I "
+a'·"-
a
0
' · 11
'
-o.1·11
<
(120)
CD
'ii:e ce'"sii;y functions w,(x) of a'· " ll..'ld ... ,(y) of la'°' I
W1 (0) oi'
arr. p;1·:~n by (.<. . 59) . '.::hi> rrohability l (ul· "<O)
cuudHion ( 119) no;; ':leing i.atisffoc! 1'QUO.l s :
~
\1
TiO
~'f
J0 e;q{ - (x-1 )'/2o 1 ) ]d.x
-
(121)
H 1 - orf ( 1/1[2o)]
Th~ µ1•ounbili ty p( - a1- 11 <0 ) htts the .:ru:•~ vnlue :
p(-n' ·"<O J
H1 -
er~·( 1 /V•?o
(122)
)j
p1." "e"o~<'5 cc.c :;>robability that the condit:ion (119) is
not sat.i"J'ic;j J'or a• least one o! the !: coefficients a1' 11
Etnd ~'" 1:
r~"
•
1 - [1 - ·.•,(01]" = 1 - 2·• [1 • "r1'(1/1[2o)]•
(125
Cor.sidC!'r 'ttc dist!'ibi.1tion of a' · 11 -laCO)J1 0 < aC·lJ< co ·
:ts Cle:ie1ty fWlcLion is given by (4 . 1>1) to (4 . 6:')) . 'i'he
N'CJl.,•bllily that the condition (120) ia not satisfied for
0 and - a'·"
E• cortnin one of 'Che h(m- b} differences i:a'- "-
la' '1
- ln' 'l 1ti
0
ri(
a'·"- l1tl••1 <0 '
;1(0<·•» o)
D
W(O)
--
J w(z)dz
.
(1 2'~)
29:;
A
D
I
•
I
Fig . 107 Dcu llic,y funct io ns of
aloi l , nl- 11, a.IC I ,
latoq
wid
a'· " - In oI
fo1· a ternary
c ol:lbin ation 11lrua\Jct . The
h.atched a1•eas 1 ndicate er:-or s .
.
.~
'
· '
0
1
.6'• ).
'l'ltis int e£!;rf.11 wan ev;.i. uettod in (ti
p~~~-• rleHotea t in! pi·oba·uili t y, that thC" condiL ... ou ( 1 20)
is not natisfierl for all h( o;- h) di!"fnrrcce• a 1"11 - 11.1.10 '1 and
-a•·U_ lal•t! :
P~ •• ~ 1 - (1 - W(O))•<m·hl
(125)
Equations (123) and (125; yield :;he erro~· probabili;y rl.'!
of ternary coabination alphace~s • . biorthoi:;<>ne l !llrhabel,;
Slid binary (m , m)-al~habe•s :
p•ll
1
(12<')
""~
a
1 _ :!"( 1 • et•f{ 1 A(2o ) Jh ( J.
f ~ + erf~1~20 )
2
"
e1•r' 1 ) 2o
~
I )ti!m-nJ
a' • hPAl I nl'
liquation ( 1 26) yields f or h • m t he error p1•obo.billt y of
a binary lllphnbet with m c oe.Cficient s a ,.(j) ond 2 m cha-
294
I .
SIGN.AL DES!G!l
rnctcrs, \\'hicL iz c:l:c ~a=nc a~ ('l·~) :
P~.:.· 1 - (1 - v~" )(1 - .?'.!.~) • 1 - (;)"'(~ •er!'(11't2c)]"'
mr., /nJ-
o' •
=
?"'
/F,
n = 1g,,.' M("')
rt.;
•
m
(127)
1
rhe errox· prc'::>abilit.Y or biortl.Of'Or."l alphabet s f ollows
h • 1:
ro1~
pO I •
M 1)
1 -
• 1 -
o1
•
(1 -
H1
p(JI
1
)( 1 -
:-ill l
•l,m •I
• el'f{ 11{20) I
l\, /J = F,,, /ill' ,
)
( 128)
q : :~:f.1 ~>if~n·} m-•
jf
n = lF. 1 2 (~)
1
•
lr; 1 2m
Fig . 108 shows the e!"ror probability for sooc biorthogonal alpbabe~e . The e:'.!'<>r prolobilityof the binary(5 ,5)alpba'bet ( CUJ."1e n = ~" m = ;; ) wid Che ( 16, 16 ) -alphabc;;
(curve n • 1o , ir. = 16) a.re shown fo!" cortp1<rison. The curve
n • .5 , m • .5 i" the sa'lle as curve ' e ' in .!'ig . 102.
1
rhc cho i ce of t~-C average nigne 1- to-noiso power rati o
p1o~tcd along; the abscii;sa require-ti oxrlanation . The mea ni.np; of t;he average signal power r iti rvid~nt . The average
noise po"'e r P 1, r o f o ne ort hor;onnl component of the;rmal
noise in an o~·tllogorw.l lt y lLLervul o! duration T is used
ae reference . Plot ting P/Pi.r would give a f alse iropreaaion , aince the character~ or the varioun alphabets transr.ii t iHrrerent amounts of informatio1: . l t is better to use
F /r. , t!:.e avergge signa: power per bit of info=atioo , ra=hr.r than P . 1'hi" gh•cs P/?,, 1
(P/n)/?,,,
c
F/n.P,.,
~ ?/1~••
•
~·t:ich is used in Fig . 108"
Plf., ;
61'
=
n/2':
(1 29)
Coneitl.,r the t;rans:niosi on or chnrnceeL·s with n • 5 bi;s
ot 1 t ··orrnation wi;;h an e rror pL'Obability or 10· ' . According to Fig . 1 08 1 tlle binary alphabat (n = 5 1 m = 5) requit·"s n i·atio P/P0 , of 1 1 cl!l aM t;ho bioi•thogon.al alphabet (11 • '.> , m = 16) one of 8 d.ll . llence , ·the biort!logonsl
al1 h~bot r equires 1 1 - B = 3 dJJ lees signal power . 'flle
price paid for this gain i~ an increnee in the number of
295
-'
r \,:JJ-1:1
. r.
'
...
~
'
' ' ,' 1,
~''
•
~t~,,
\
''
--- ' \
\
\ \
\\
\
I '.j.
\h\",.
..
;.i \
,\ \
",,
..
'
.
l
~
I
•
~
!•t
..
·,,
J
\
..:,. \
,'~\
l"o1
~
.
e..,.U_·\t
,;, . \
\
\
,.
\ \
I
~
~·'O ""\\"\.
tlol
·-
..... t'..,,
~'
• ,\
""'
~··
""
\
\
·\\ \ \
\\I. .,f. \ ,,~
- :\ .
\ ~
\.
'"
..
.,. \ \ \'
'; \\'
"' '(,~
\i I
• .../ t1 '1 i
1"•10
!
'
"" J-1I
•
\ 'I •I
"\
1,
\ \
II
" ,.. ~~'Cl~ - "
l'ig. 108 (~eft )Error J ro·oability ~ o~ borthogonal ~lrl•a
bet s . P average sicna.l powr.r ; F.. average F<>Wer o:· thermal noiGe ir. a :·r,.quency band of wid:h 61' = n/2'1' ; n ic!ormntion of the chru:accers i.n ~it ; T dllra1'ion o! the
characters ; C1 number of o rthogonal functions L'l ~ he alpliabet. Solid lines: biorthogonol a.lphnb.,ts ; dashed li1rns :
binary alphabet n (~ 1 :;i) w1d ( 16 ,11') .
Pig.109 (right ) EJ•rot· pi-obabili.ty Jl of Le rnary co:nbi nC1tion
alphabet s ; P , P 0 , , 'r. nnd m d e.fi ned in t:.e caption oi: JiifJ: .
108. h nw1f<>er o f orLi1ogona:. funcLiOtll' in a char acr.01-. :lashed lines show the error rro·oel::iliLie~ of the binsry alPbabets (5 , 5) and (16 , 16) .
Orthogonal functions required .rrorr. m • ~ to rr. = 1 6 ; a 11,/;,,bimcs la:r-ger section of the ~i.me-functi.on-domain or , nomeWhe.~ less precise, a 1( /5-tim.e" wic!ot· l'x·eque ncy bn.11d iu
required . Consi der l'w:~IJeL' GlJe trano:i.isoion o f ci'ttl"J!Ct"Ore
'1ith n • 16 b i ts of i.:i.for:riation wi;h dn er;·or pro1'nbility
Of 10" . '!'he binary alphabet (!! = 16 , 11 ~ 16) rnquir"s >t
rntio P/P6 , of 11 . 'l db ; the biorthoi;;onal alphabe; (c • 11j,
m • 32 768) one oj' ~ .8 dll . Th1Ls the bio rthogonal alpi.abet
requiroa only about one quart or of tho zignal power or ·Lhe
binnry nlpbabet (11 . 7 - 5 . 8 = 5 . 9 dB) . The number oJ' functionn required increases , holfever , from 16 to 32 71_.e .
290
G. SIGNAL DESIGN
'l'hn mm 11 er tr.e reqc.lired ~r:-or rroloahi I 1 ~:1 the more
Justified is >he use of :i biortt:ot:onal al;>tabet . For ex-
aaplc, i!; rcq:iircs a :-at;io P/Fta1
or
"i"'
.2 rlh for an error
!J!'Obabl.lh:; or 10-7 ( CU!'\'e n - ~, m • 1t. iu ~ig . 108), while
Lte biliary alphabe~ (r. • '.:> , Ill • ,) i•equires a ratio of
14 . 8 dB , a possible l'educcion of thE> nignal power by 1"- . 8
- 11 . 2 k ;S . 0 dfl . •rtiis same dil'f~rt'ncc nmounts to somewhat
moro t h an ? dn for the alpr.t;bot:; n • 1b, m • ?2?68 and
n • 16 , 10 • 16 A t an e rror p~·obobility of 1 0· 1 .
Fir;. 109 shows t he error probability of t;ernary combinotioll alphabets accordi!lg to ( 1 2.;) . A coa:parison with
Fig . 108 shows t;Lat chese pw:Uculu.r ones need a larger
r1ttiO ?/Fo1 thau the bioo•tLogor.al 1tluh1tbet uut a smaller
one than the binary (a:,m - alphabets . For instance , the
uiOl'thogonal alphabet n • ~Cl , a.• 12 yie~ds an error probab1.lity o: 10·• for a ratio ?/F6 , or 8 dB; the cooparaole
eo~bir.at1on a~phabet r. - ~ . 9, ~
H, h • ; requires a ratio of about 10 . 5 dB .
?ig . 109 sho•s that t!:crc aro nlrhnbcta which transclt
more .ullo1·mntiou n ;·1i th the HOlllc n1.11tbor u of f unctions
thw.i tbe binary ( m, m) - nl phnbet cmd neve1•thel ess y i e ld a
lowei· en'Ol' p;·obnbil ity . '.J:·he>ie ulpll/lucLs u.o more than ex-
chane:a 'lllOJ.'e functions ' for 'lonn nl11nnJ power •. Consider
lJ • 8 . 8 , m • 8 , b = ;; • A character of " binary
(8,8)-•lp~.abet <:ransr.iits wich m • 8 !'Wlctions the in£o r 1Htio:i n • 8 bits, which is le~n thnn tho :; • 8 . 8 bits oi
tit!.' cui•ve
t:hc ternary co:nbination nlpbnbet tho.t requires also :n i=: 8
functions . 'Ihc e!'ror probability of the binary (8 ,8)nl~habe~ is rep:·esented by a curve th1<t lies between ;he
curves u • 5, :n • 5 and n = 16. o • 16 in Fig . 1 0CJ . This
curve ia about 3 6.£ to the right of the curve n • a .8,
-- ~ 10-' •
m • tl , ~. • ; ror error p!'o b a b l.· 1 l.· t i es between 10" cu=
Cor..oi<icr t he error probabiliC;y pl l ) of (126) !or iare;e
vnltH'" oJ' ru ruid a . 1Jsi n5 tho
1
.1rr ( x ) - 1 - \l m-:"'
one obt1un:o :
-x ?
,
....
~ppr•oximo.~iona
x >> 1 , and 1 - y 1' e ·Y , y << 1 ,
6 . }4 TERll/JlY
....
Pill
"'
CO~ffi!NATIO!f
-Ee"'
1 - ~ -
r, •
Let n snu
lim
p~~
m- oo
•
lilll plll
1!1'1-00
Thu~,
f'l',h
.f.i
1
.for
0
fO!'
>> ,.
(130)
1, 1n [ h(:o-ll) '1ii
fii'ng]
1
a??~'09Ch
"
~
,,
297
ALPUAB:::TS
P
~
iiu·in t.y :
(131)
> 0
< 0
crt•1,1--free -cra:i::Hr..i.e;~1oc. is a.c!::.ie\•ed ir.. t.he
lia..it:
n • co for '11 < O.
Using tbe relation
D
I!"
a
>>
;i '
(132)
one may ti·rurn for·m Cho con<li cio11 '11 < I) into tno following
condition, hold.in(!' ~oi· a conl51..S::"t value of J.J:
P/P61
> 4 ln 2
(15))
A ratio P/F 6 1 lnrger ~l'nn 4 ln 2 ;yieLdu error-f1•ee
transrnission f'or infini "tely la-~r :n a!lC r:., ru1d finite h;
tlleerro rp;-obatiiit;; is 1 if P/l~ 1 _ss:i:allerthan 4ln2 .
The limit func~ion li:n p~3J for :n - o:Q , n .... ..:;o ic ~!..o~·:r. irl
Figs . 108 'Ind 1 09 . '!'his limJ t ls tho soJoe for the bio,·thogonal alvhubet" (h s 1) a!1d Ll':e =ombination dplrn.betc (I:.
> 1 ) . Hence , the combin!'!vioi:.. alphot.ie---:r: are tte su_reri.01
ones from the st.and.point o! f'1mctionr, or baridwitltb rnoui re<i, nince the nurr.bor m of !.·~ctj_o1i.ti .i:equir·~d incr1'!~tfll;!!i
cropo:::-tionnl to 2" fa= bio:-thogor.sl !1l,;>b9br.l,, but oru;;
1
proportional to ?" hh .for cottbi.ua:cion airhnbc·!.r .
Let ll not remni n constant no m and ZJ approf.)Ch infinity ,
bui; let 11; inc1·ea.se propol'Lional LO :n" , 0 « o. < 1 . ~he
1:ondition :n >> I:. or (13?) is atill nat:..,fiel fo:-largea .
The condHion '1 < O ther: yields "he Iollowi11g condition
in place of ( 1 ),) :
<-IP.,
> 4 ~
., - a ln 2,
Errol'-freo tram:miasion is possible il' u ia emu.llor than 1 .
G. sn;r;AL llEsIGll
~ot us i:ivcst!ya.t O' t.ow Shar:~on 's lic.i t of the tr~ ...
:nissior: Capacity in tllf! for:n Of (5"J is !iifr<JSCiied by ternary combination alphl.lbets . Ihe ave1·or,n nois.e µower p
41
iiuot be replaced l>y f'.,.r i:_ ( 130} . It J'o llow.a:
"'ht" concUtion ri < O boco~r.es ~
~lt.. [h\:r:-:i;~J
<
~
..
T!Je ,pproximat i o::>
lg2 2(~) " b lg,~, m >> L , is sub-
n •
:· ti i;uted on tl1e lel't nide ru:d the tei ·mn
lJl
n <
•
"'
4T02
[
~
h
lL ult - '
The term ill the
r.(lg,~)
j
11f'1•
reordered :
P
( 137)
:',,.,i
brncJ..e~i>
'::icco:tes 1 ••l:cn :ii cecomcs i!lfini<;e
and I: r~~ains ~initr; it bcco:nes (1-a)/(1•n) for h = m• ,
0 "" :i < 1 . The infor:11.ntion trane:oi~~ed error-free per unitime is eqcml 1'0 t../~· ,-ince L is the in.for:r.n1'ion of each
clwxnc•er crn.r.:;mi t • ed d urinp; a.n orthogona.li ty ini;erval of
dur·ntion T :
c
- n'f < 2 2Tm
1
,.,
..: • 't"
Tl:"
<
1-o.
'I
lLl 2
'Il
j
!':":'
m,1
1
2 ZHi" 2.o. Ti"i7
.oga:-~ ti-~
ti·M.~r.ii,;sion
Ii
I
c onota.rit
I
r;;
I
h =
IL"
I
(438)
0
~ (t
< 1.
(139)
to the bcee 2 must be u::;ed in ( ~) if the
cqnrit;; ie t:o be obtained in l>i ts per unit
t < rne :
c
~ << 1 .
(140)
"'·'
'L'he !'lght r.and slc.e o I' ( 140) i s larger by a factor 2 or
?.(1•et /(1 - o.) ;;tL;,,.:, th<' ~ight b.antl sides of (1.?81e.nci(~39) .
Ecnce, a. ccrr.ai--:1 com.b.Lnatior:; alphabet with n ~ constant
trenal!li ~s ~al:·~~ mucb in!'orlL!ltion error- free ns peraitted
t;v St.a...T"ff:on 1 s ~llit 1 J·l'"Ovided t~c signnl-to- noise powe.r
!'"'t lo
P /Pm,, ! s 31rmll . Tho phy!<ictl r.ieanini;; of the condition
6 . ;)5 ALP!iAtl.i>'l'S 0 I• ORDE!! 2r+1
~crnary
nlphabe~ hav11 on:.y thn tl:u:ee v1>iues +1, 0 and - 1 . An incre!>OC
o.Ctbe everage signol-eo - noice ;iower <"<>tio P/Fm, i is 1<01·thP/Pm.r << 1 is evident ; the coof ficlents of 3
1css onne t he cr"l:'or probe.bl lit;y ha,; roached zoro . Use oo ulcl
~ i ncrcaoed ratio P/!~. 1 onl;/ if ~he coe:-r1 oien1's could assum<' mo=e thon tl:.e -:l:u·ee va lucs .1 , 0 and
- 1 . Tbc tern ary comoinatio n W.phab,,La muo. c be :>eplac<:d by
alphabets of bie;her order .
A more de1'aile4 icvesi;igat.iot: of term•!J' co:nbinaUon
alphabet:> war. =ecentl:;• ;mblished by K.AS?.GK [~) .
be made or
6.35 Combination Alphabets of Order 2r+1
Let chuac·~err- fx(ij) be <:om;ioncd or m orthoi;onal functions f(J,9) , -! a a i; !, mul~ipiiod b;v coefficie nts llx(j) .
These coeff::.cier.ts :nay aSSl.UllC 2r+1 values :rather th8l'I )
as for ternary a .... phbbe"t:!: . /, tota~ o" (~+21~)m characters
Fx( e ) can be produced . Let ( 1+? r )m bo oxpendc><l in a blno-
mi!l.l. scrie s :
(2.r )" (~) is tne nuo'ter or cLaractei·s in the alphlibet
contai.n.i.nl'" h of tt.e a :'unctions f j, 6) . ~his 1>ear.s that
h of the coeffi cients ax<J, are oon-~ero ; 11. runs from 1
to (2.r)" (~). These character~ form o coiibinn~1.on alpll~l et
or order 2r•1 . Lf'.'t al:i thc"e c:-.arncters ·ue t,.ans:ni i:ted
with equal probnbility. The in.formation per c_arac- er i~.
bits equals :
a >>
r
~ach o! the ~" coef:r:i cientR ax (.J)
O may n::.,ll.!1e 2r values . Tte;y 0.i'e denoted by o, , p
±1 .... u• . The prol (;l'dlity of o coefficient a x(J) assuming the value a, i~ denoted by p( p ) . Let p( p) be independent of j . '!'he averngr.
power o! the f unction s f(j , 9) is P 1 :
:)00
The average JJOl<e:- of
~he
charocters
~o:r.posed
of h functions
f(j ,9 ) is P:
(144)
The following assum?&ions are made :
a) The probability Of a coefficienL P-k(J)
lue a, is inOcpeLdont o! o: p(o) • 1/2:- .
~eving
the va-
la,-ap.d is indo;>endent of p. lo, - a,.11
• a 0 • This condition is satisfied lf a, is a mul~iple of
a0 : u0
pa 0 , o = ±1 •..•. ~r .
Tho average power P1 of a function f(j 1 9 ) rollows from
b) Thr difference
(1"-3) 0-nd (1h1') :
P;
..
:l, p 1 a~/?r
•
• (a~/r)
p ••,
H
a!0
.
L;P'
,.1
(r+1)(?r+1 ) 82 •
6
0
P/h
6P/!l(r+1)(2r+1)
Lat FJ cha.rncter Fx ( O) be transoi tteci . Crosscorrelation
with the fWlctions f( J ,e) yi11lds the coefficients ax(.J) at
the receiver . Superi.l:lposed n<lditivc i;bermol noiae changos
these coefi'iclents into a( J) . '.!'hey hnve a Gaussian distribution with -ieans ,01a 0 , - 'i>l"o or O; IPI = 1 ... r. Theuft
coefSiciento IU'e denoted o;y a<-P>(j), a.l·Pl(,j) and a101(j):
<" ,;/ >:
I· p)(
<"'
l·PI( • )
•)
(al•P')" 0 I
,.;1 ) • (.a'·'')• -P
(146)
(a IOI(•)>
e:;' = (a'•I) = 0
The variance of ~ties" distributions .follows in analogy eo
( 117) :
"2
(a!(j)/<1 0)
(147)
h'r+1)(2r+1)P1• 1 /GP
= h( r+1)( 2r >1 )P•. , /6nP
= h(r+1)(2r+1 )P61
/6nP
301
LJ. numbe:- o;· non-:-.ero coefficicnt.t:i Bx{,i) ;
n . lg 1 (2r)°(~) - "lnJ'ormation rer· clt~.i:·acter in bits , i f
nll cbaractpr·s Ill' <> trw.Jsmittc(J with equal probahiJ.ity ;
2r • r.W11ber of non-?.er·o values whicl the coefficient" nx ( ~)
'IOay assu=:Je;
average i:.o"-er o: n oZ"thogonal comror:e::. zs of -:hcraal
noise in ru. orthogona_i ty in-erval of du.ratior. . ;
p
M:(t'+1 )(?.r;'I J/
nverage signa power ; :H
n/:>r ;
PAI • nverage pow a L' 0 r t.r.cr:nn..1 noine in Ti f _r equenc:y bantl
or width 6J .
The character~ or co:nbinat ~on al rnobots of hlghor tl1an
third b.der '1!·: :,oi; t-'1Jlfill.O~Led wi:b equal energy . One
ount detor>.ine the r:r;nlleH energ A!.'. according to l .. . .:4)
fo r the detection o!' Lh.e R:i.gr.al . l"~s :neans th«L the
(2r )"(r,) S MS
P.
•• 1
•
s.,
.....
;E ta(j,
•••
- n~(j)J'
(148)
be coo.pu;;ed a.'ld the one ••iti. the s"'al l·ost val •e decerr.i....neu. . An error occlo.1'S if s . . :.~ t1o"t annlli:st !.'"or '!" • -x,,
OU9t
where x 6enotcs t;he t.1•i1n.scitted
clla.tactr•!' E'x (a ' .
'.l'ho r;m11llcst value or S~ i~ ol.>~al nod if the b nmnl.1.ost
terms [a(j) - A ~ (j}J' lil'e adder.I . Tb~ 11 ~ci·:ris , f or wldch
e( j) is equal to Bx(j) f 0 in Ll:e noine-L·ree case, >:111
ba the h s:i:al:.est ;ermr. ia the p1·esencr· o!: additive tllerciol noise if the fol lo·,;ing cor:ditior.n are sstisfi"d (~~e
Fig .11C):
1 . None Of the h ooo£ficienG '3 a'·PI • 11 1·•1(J J/a 0 nu<l -1;.11 • .,1 •
• -n1·•1(j )/a 0 i s ~a.relier fJ•om its cor1·ect ·1.ean ln,,luol a
• lei than 1roa. tl:e oLhei' meanc lo ' I• 1 . - .. :r , 1 0 .
2. None of the !:. co..,rricieni;s ;i' · •I a..-,d - a'-F'
~rom one of' the cen.ne 1 ... .. r li!:nn tt~e ab!io_ute va.lu~ o:·
on~ or the !L- h coefficients a 111 • •rhi= condition mur.t be
satisfied only i f condition 1 i s soc.Li;fied .
These two Condit.ions nre essentially equal to tho condib ions ( 119) and (120) foroernsry combination o.lpllabeta .
'l'!::e caiculation of the err or probability is much more
6 . SIGNAL DESIGN
30?
0~11,y
complicated .
the r esults wi!J be Gtated t.e r·e .
Le Pm. ti
denote the err·or· pi•obobility d.u e Lo themai
noise of n combination alp!labet or order 2r+1 , using b
out of m functions . For large values m, 1... 1 L' arid smal1.
t
( .?roll
val ues of o 2
,
m >> h > > 1 ' r >> 1 , cr l
11
= lg 2 (2r)"(m)
h
Ol1'2
Ju·' P., /3nP << 1 ,
~
11 le; 2 (rm/h),
=
:·o lowi ng
obl;ains tl:.e
fora:Ltla:
~- 1 - e - e"l1 1 e - •"~'
r; , =
1
n
lr:
h -
(150)
'P
8Er'll'0 ,
Lc-c the inf.orm.:ltioll n per
ch~racr.er
lim p~·~i>
1 for
1'1 , >ct 1'11
lil!I
0 f o L'
fl
.
~~
n-oo
pl 2rdl
m. h
'!'he car.ie
T] 1 <
0
1
1
n, = [r: ln(m- h) -
2:
<
c,
grow beyond all bounds :
arbi t.r·a..r·;,·
(151)
"1 , < 0
11 1 > 0 i~ not possible because it holds :
-p
1
chr~ Poi J + [ 0 ln li -
The term in tl1e seooud bracket
j
s equal
>p
( 152)
8li.r' .l'.,
J
to 11 1
It followa
•
Ii·om '1 1 > 0 th-..t 11 1 must be l arger chan O . Hence, 'li < 0
,yield8 the error pr·o·oability 0 and 1'1 1 > 0 yi.eJ.dn the erra11
proba"ui:i~y
1 for
condi Lion for
4!j
11 -
co. !lewriting 11, yielc!s the Iollo\·ling
e~ror-f~ee
2
transmission :
·-
> ?n r. l n ( :n- u )li
Suosti.t11tion e r u
''I ;M
(153)
S:i:om (1119) yields :
ln(m-h~h
> i5 -c.2l- 0 2 1L1(rm/h
(154)
Lee us investigute how Shannoio ' s limit in the J:or)ll. .c;;11·)
can be approached by combinaGiou alphabets of order 2r+1 .
1'he average noise power
must be l'eplace<i by P.,, 1 in
fOL'utUla ( 1.50) fo!' 1'12 :
6 . 3> Jd,i'HABBTS OF OJiDER 21'+1
Pig.11 0 CeneiLy
f"LLnc~ions
303
of
a10T , nl·'' , 1• · •1 , ...11p1 , al· o• , ·:i(·tl
and s1·•> • •r11e h(ltChed areas
indi.cato
<'T<'OI'" .
a
n
-Cl
F~. I
One obtain" ft·om (1'.;1) :
1im
n-oo
11
p""
f'fl,I\
• 0
J
fOL'
r.
"/P,",r > ~
1' 2 I r.(m-h
Hi
( 15~)
Using tho 1·clutluo
.!!
.r ii 21\/h
n. •
><bich followa free
" < h l-
~
~,
(1·~Q'
fo::- r ""1, one obtnins fL·oc
I[''til in,a:.-h)hJ
~...
- F.....F }
1
('l;i~J :
l157)
One 111ust choose t = tl{ct) so ":hat -:;hr. ri.;.hl tumd r.idc
or this inequolity become~ as large us posni.bl'1 rnr a certain value o.1' m a.nd a fixed rati~ ?/Fm,f . 11'1'\4"! r>..tire~oion
is too complico~o(I to find a m~ximwn by du'1'cron~iaUon .
One may aoo , however , that tl>o raci;or h tr. l'l'<lr.t or t he
loga.ri~hm shou ld bi> aa large as possibln . U iJ becomes
too large , tho term in the brackets bocomen .Jm!llltir than 1 .
6 . SIGNAL DESIGl\l
l hi::; tc-rm 11o ulc! then becowe arbitr1arj ly
small
11
wivh in-
crr-,qsicg m. Bence , h 1.s chosen sc th-tit tl1e equation
ii = con <tam;
4h' ln(rr.- h)h
(158)
is savis:"ied . 'I·l1in suggests bbe choice
ll
=
m/qln
r.i
(159)
•
It follows ft· om ( 157) :
m
1
n < 2 (ln mJ"'
(160)
lg , (K.P/P,, r)
'l':hc inl'ormat:ion n/'r t..1·ans1n.itt,ed r.;P.r unit timo bacomes :
C
n
m
"i
= T < 2T (ln 1n)'"
[ lg 1(P/Fm,) < lg1K]
(1G1)
This :formula differs froo Shall.llo!i ' s limit (51f) fot· lru:>ge
- 1/}
values of P/Fm,; only b;; ~he fo.cccr (ln m)
. Tbis small
difference is probably accounte<I for by having chosen tul
equal C.istribution for p(p) rather than a Gaussian distribution . 1'he pl'lysical meaning o:: the condition P/Pm.i >> K
i"' L'eatlily Lwtle!'8tantlable . r > 1 bad been (lssumcd in ( 149) ;
many different •ralues for the coef ficier:~s a ,. ( j) will per-
mi"t an error- free transmission onl.y i.r the ave rage aigru:tlto - noise po 1·1el" ratio is largo .
References ordered by Sections
rntrod11ction
1.
MA:Jf\ , r . 1·1. . ,
!Jc~· ~.nll:1bla1Jf
var.
H·11ta~h:'.pa!1H11n1"'e11,
Nac!•r . "~chri l: ~··1(1r''"'•) . 18~-18') .
2 · S:'IJJ·UD~;-;i .? . l ., Tl~o?or:r of r·req1..i llcy
rno:iul~t.io11
1·;1 .
roisc,
Proc . rn:. 5c'i1';--·tl),1• 1'1-1 u<)2 .
; . VQ:.;L.;K.J:;.':l . I . ~ . > rowttrJ a u:i~Jied -~co:-yof l!O::IJlAt!on,
f!"o e . 1?.ll ::1t{"i=)1:>1- J , :1l "-; -; ., /5::>- 7r::i:; .
# . RJ.lJl..~ACHEfi, H. , hi111gr_ s.:i.t~~ von nll5~ur::ir:r.-n t::-t ...oP'..... nol!0Utik Li"n~"· ' r·:ntr. . Annalen 3"('1'122J, 12;· -F.8 .
.5 · l•OV!I E , F . F . , l 1he t.t·au!Jpc~:.~ior.~ or' e-o.,r!uctDr::·,
r·rn.liactions AL£E ~;.t1·J! • ) ~ t-ri;;o--:..8? ..
\;) , Of'llOi'H:l , H. S . ' I'll• rJedi;;n or trnn""ouitionc [()!• f •Hl'l<ili;l po·..:~i· e..;:id t0Lr:r110.H~ 1 inc ci:.·c·Ji I ... , 1l'ra..'1sacL ~one J\GE
37(1 )1d ,I ,697-ft3c •
7.
PlN'Ki.:H~ t B.. S . , Ira1 Jr:-tio!1~..:.cl t~-,.., :;'ii: Fe::n.~nr~crile-it.H~
S:~n,
.. cl-""f:i·ar~-~11- u::d I ern7.;=ecbtocunik , ;.·. 3on·-1~r!1e!t
(1 ')19 , 1':E- '1-' .
a. KLEI::, ..... . , Jie 'lrnnoric dt-S ·,abf'!"'.Cf !'~C'1(!f!.:'.; -~J::.ll' Leltnn gcr., kfYrliu/:te ..., ·:art'.: t1r1 r:g":'.'r ·1 • .
9 . WALSH, J . l1 . , A close·.:1 !;t:~ o! ·Jrt:hogor..a_ t'unct.lour.!,
Amc1-.,J . or l1atllt:1:1CJ~i~:!J :1 ~( 1123) , ~•-2'+ .
10 . HOWE, P. ',/., 'I'i.e 11~0 of Lae;ue-.:·ro nntl ...'ulsh fur;ct.ioz.. i3 in
mtJtnl'i~il; pro::ilertd o r ·a=:..'lble ~Df.lt1ir1r
at. 11ii;t :;c1tq..1e1•a turo TecLuic':! RC'r1ort, A!J-4Y=i1?Z 1 ,, ~ ' .
11 . I·RA:ICE . t·: . J-" . , ~.Jnl:-h f\.Uictions, co1·i-9l n111"z::er~ ST'd [Er~udo- · mdo~ fll!: ... tionn . .. ("'c!wi•;:.l fi..-•rort. Af1--r C·"=)(C(1 1 • ) .
1 . 11
1:':3._GQI·tI, .F . , '.'V.!"• ~ 1ng,...n ·}t~l' 01·t1 EJ:;cnal=eiher. , f'\f ·li ./New York : Sp ...!.!ie;t". ·1·1 L,s .
2 . SANSONE , G . , Or·tho1·~on;;i_l f unclin11 s 1 Hew York :
lnt;oi~scienoe 1')r•<J .
·
3. l1ENSE, J . , ReiJH·n·· ., ... ,-;icr:_ui11~t--~J J 11 .Jc1· ms..,;i1em~~1 .... c ~~n
Ph;vsik, ·.,r.... in : de U1...i·.·1·'=. r 19~:, .
4 . ;1rtrrt·-T-iOi:SO~I , .. . :1 . , r!::..e e:.;;lc 1.. 1u:: of !"initr dif!"c.:·ences , l.01~dc>n: ;-:cl'i11lnu 1~ 1 .
5 · h~RLUlf.J. r:. :::. , Vorl enr~r.ge-::. :.:ter :it r:·r·r~uz.~nrec.._'l\U1f't ,
R 0 rl in/:r<:!w Yor.:: : 2p.r.i.r • ..Jl' 1.._•2L .
1 . 12
1 . CO IJRAl:T , R . a"ld 11. :il ..,bET ~ (.1r·t•i('h,l"'r .. Je1~m;;:=11emn.t,1:-.chen
I'hysH., Berl l n/:lew York: Spt·iJtfer 19:'•1 .
2 . MORSE , F .. :·: .. ru1<l tt . ~·'J.ft:p_~;.;1-: , 1·eL•.r11,.:. ~f t t.en 1·1 1.,i.:al
Phynics, ~Jp· Yori~: !1CIJJ.'ff_-J- :iil l 1' r 3 .
LENSE, .1 .. 1 TI:ei1.cr.cI~twi..~l~lu.r1P:t-!'l 1.!1 ~e:- a:or"tth~mati. ct~::
!"'!:Ysik 1 Berl!..:.: : d·~~ lJJ"Ujtor 'JO_:-: .
4 . ::-_ ~, F•. , Si~1.'ll8Jl.a.1yne =.:.~ l.agu(jr ...... echen foll-no:r.erJl
-A.re• ••• elek .. Ube=tr~~~u.
?:";~19'6b),16~-1)'~ ·
. • ,,,.., '!TA.KER , E .. T . ar.d 1, . :J . '.,.iJ\.TS01r, ;.. t":O ll'~~~ of ·1, 1· l'n l."ltllyai(j r-harter IX, Wondon: Camb1·i<lgn II . J'res:s 1J~J .
. • 'l'!TCfll•lA.RSH , F . C. , 'fbeo1·y of ~ho r'o1irier - i11t•Yl"ll ,
London : ox:rord u . f'ro• ,. 1 957 .
? . 11Ll:;Xtl'S , G. , Konve1·genzproblc"Jo dor Orthoeonol 11 ~111.'n ,
Bc1•l in: ~eutscho.r Vt lag der 1..1iaeenflCbtJften. 1:)0::""0 .
!:
1 . 1'
!.1.l. • ..i)L:t!f9~.',
.:s,.,,rl1n:
~· • - . , ~ti..__"1:art5 <lei~ 1.0.nor"~ r.~t.~!~mnt:~ , Fart
<!er 'n1 .,;Br.e.:tlo.! ~·:-n 1961 .
2 . i'ITC:!l1/Ji!;E . E. C .. :1.eor:r of th" Fo.1rl <'r- i r.i:ef"ral , Londo:i: O>ti°o!'<! Uni·;c::-s:.::;y ?:e::~ 1-;157.
3 - P'RAJJ-.~.,.t.;J] , R . , 'IL.::a .Pot;rir-r-t"ra.n. ;-OJ'n. e:.d its annlicaiion , :;ew 1or¥. : i·:cGr!i·,·1- i-ii:l 1~1_ r. .
-11 . HEitfi.!::.TT , ~.. . H. , :u..d .:.R. JJ.h.V}...! , Dttt;l;;.. lt·~i:J::::nission
New
"fork : l'lc<;1· ,i-lal.:_ 1·1.c-.
'
)_. . \\rIEHEH, :I . , "'he foU.rif.!r-iJ1tr.cr ·.l nn:1 r:ert.ZJ.iIJ o: its app.l.J.-.:nl.::.on:"' , Lo rHJon : Jamhri dc;<' Ur.iveJ': lt,Y j- i ·c-::;n ..,.J33.
:>".:·11ts.:.:1er ,·,.,.rlnc
0
1 .111
.'.\P ~ o I' ,1 rtho.,;cn;,.~ f.'u.11ctions , Amer .
J . of' Muc:.e:nr..tics 115(192'.:°) , )-24 .
~ - RAI.D·IA~L:ER, E . i .Elr....:~e S.Htz.n von nl lr;emei'1en Orthogo -
'1':"lJALSH, J . L. , A c 10.!l•?rl
r.:• ... 1'wlkt.ionen1 r:ntt . l\.r.na4f"'f~ 8?(1 lci:.:) ,1 22'-1 ;~ .
3 . HZ:JCE330~! 1 t: . 1.-.: . 1 So:ae n, tr>c< er. t.ho ·w~-t.!.al'".-l"ur..ctionn 1
·rran:fnc.:tion1" I~ ~...:-1;'1n• ,L) , r. ·- 2 .
4 . I,lFDI , H. 1 U'"cer ~ine 5fe:.:ie_lc ~:L.o!'"r.~von stark mu2.tipliY.utl·, o:·tnogon'1ler. ?ur.f.;1on"~'"Y'"tcc:en, No1.atshefte fiir
r--~ c~.~m.otiK f-,(1t"•i..:.--_ ,15: - 1.:•1 .
W&lsh- I'i.mktioner. unc endlicncicer.ciongle Hilbert;rau:r.e I ;Jon,,::si:eU;e f'.ir lfathemutik 70( 1'}0(.) , ;;42- 3'18.
Li · -, tt\:er reni.:lse f'u:ik ... ior:alo i:D Rr111~·u Cf•) L0 , 1) und 'W alshFouri "!"kon;· f i z i ,·.,-.t.,n , J1on" t ~1,~ ;·L,. ! ~ r !·1:. :..li~"'"til: 72( 1968),
s. - ,
;.a-1;1, .
'/ . Wl·,J~.t;, F . , ZusLJ.!I.!!.e._w,- vou ,.,'· l ...l.-Fcurie r-Reihen mit
Polyaowc11 Non'itsbefte fii !' Jfott.r-irweik '11(1CJ6/) ,1 65-179 ·
s . fI •~JlLEH1 , F ., Syn~hese l in1?¢1rflr poriodinch zoitvariubl.er
l•'i Ji:~r mi- vnrgeschrie·oenem Sequenzve.L·hulten t Arch . elektr .
tluo Pt l'a~.in~ 22( 1968), 150- 1&1 .
.J . - , f::.ul Syst;e:a de1: sal- will cul-f1u 1I-: Llonen als En1eital'Wlt" des Syatr~ma 1r·~ i·.rn I ~11 -.F11nk-:i c:1nn und die Theo1'ie der
.n - 'HHl c.~11 - l•'o nrier:;r annfor:na• i.on, 1l't1er i r 1 Dopt . of I·tath.e:nu · 1 c .. , {Ju1sb1·uck UnlVe!."tiLLy , J'~us~rin 1Ju') .
1r . "r [,f;:nO:It~ , :·I . \..' . , On n ·~'!.:iaa or \:Olu~!.eLe ort..bogona1- sy~
~'!CI> 'in !bs3i'1fl; , Izv . Ak~d . llauk . Sr·r . l'hch . 11(19"7),~6;;t~{
•
11 . rl!IE, r••.:- ., On ti:e '.;a:.sll- IWlction11, ~rans . Ame;- . Na-.h .
~c . c,• l 1 "19) , ~72-•1 11.
1- • - l rhr• 6Cr.cr;a~ 1 zed 'A'alnh-l°unc!.ions I Trans . Amer . Math,
Soc .
b'i(1 •:;.0) , ~;: _77 .
1; . FM.EY , F. . E . , ;, i·ei:..arks.b.e r9riC!'l of ort;hogonal iUJlCt.1.or.s , r r:>c . lonnon :·!::>th . Soc . ( ? ) :-~ ( 1 )32) , 2q 1-279.
. .
111 . SEJ.. ~R!.tG=: , .R . G.
G"'ne!."a2.i:.:1.Jd Wa.ls1 r:r'il"!-~forms , Pac1f1c
J . o.f f'lat.h>;'e1tt~ics 7\1':9;i.~ J , 4~.1 -'-8
1 ;. . ...OHT , S . • Su un r:.o tevole r;i:Jt(·rr.ti ortl~ogona.J.e di .r-unz1:
vni , Atti Accad . Sci. TsL . llolo['.nn , Cl.Sc i.fis . , /!n1l . 2~b
1 .
Ti~:id . Xl Sl"l~;i No . 1 (1958.l ,22~ -2::-;o .
•
11 , . MOltc~J,o;l\'l.'HAI.,ER 1 G. ltf . , On Wul~h-Pour·iel' f:IO!"ics , Transa-ct; 1.on' Amcr . l'l»th . Soc . a11(19'.J7) ,1172-'.;>0'/ .
~1
1'/ . Wlr;fn:h, N., Nonl inear problems in t•1uido;i theory , !' ·" •
lliiw fo"rk : MIT Press and Wile;,• 1')58 .
18 . li'O~JLE ~ I·' . I•. } '11!'°1€ trn.:J 'ipOsi r.ior. or conduct.ors, T-i·nnaBct.ioi.tl AI::.:E 7 r·1JO~ ,'..J >4 1-•·,57 .
19.PL..J:.!t~ON, • . *;;. , · !"ror- cor1·1:cti:1g codPs . Hen '!oJ--V:: J11L1
Press :-:.nC .tile.. ' 19,_.,1 .
?f- . L.OOKIS , L . f; . , -~ i;"t"'.:r-od;.:.cvion co n.Jst.r·~ct hn.rmonic •ua·1iyois, En1, Lewood Cl il' f: ;,J: 'fnn li.~s~~·and 1 9~;;; .
21 . H.Af'IMOND , J . l •. ar;,J B . G. .JOHNSON, A revi"w o·· ortho1•onaJ
squ!lre ~1avc ru.rH.:~ ior:s t'ind thei:e h~pJicatioJl to .Linear H<tl. \IO.l'lts, J . of I, ~ f-:-anl 11.!J f:r:.s~ltutc 27_3'1 J•':l:'>) , ?11 - 225 .
22. \'IJ£:U:IIJ, ll . ../. J Or the Ll:eory -.;if Fcu1·ie: l.nte1-~1·!J .. : .)Jl
top~l~;.;~c:.croup,; \iu :i·:s::;i=J, >ii.r . s·oornil.\l •. S. ) ;oc72J
(19J2J 1c';r; - .....1-1~ .
23 . Fl:·lE ~ G. J .,
.-!-.e 11/nlsh f1J..n<:tious , Encyc lopaedic ul.;tionar;;· of l··:-,-aic~ , Q) l'ord : F~.:.·1~umon PrcsM , in prinL.
1
24.HlLYAY., ll. andYu . A. $1UlEIDLR, The arplicutior. oi' 1,fnl:;h
ru.nction~ i.J1 ~rirrox~11te calcul11tious , Vor':"'O~Y Ieo1'i t·:!jtcm!itiche?kix f'hsshir. 2, Bazilevs:kii, !1osc<>"-· : Fiz.wat&i:?. 1':-o2.
2_ • .;;)Ol.JI/!'O::, l . 1 . , Sr.i~ti:.·ing tecluLiq1 .o;.o~ :·or pti.tterr. !"'~ 1..:os.
niLior- E!:iC~IJ'l.~l"d -'lf'r.t-~h tran!:i rot·:r1n Lion; 1 'f'ht~uis) Uni Vi"I'9 .
or TO!'Ol1CO ' Curw.da ( 1 )bB) .
2& . SYLV!!:S1'Jll, J . J . , 'J'to-.1ghts on inverse O!'L iogonal 3•ttLl'ices, !_";iclul ti::111e~lL~ :: it.:.- succe;.o:sioJH! , a.-r:iC t.o t5alsted ;•~ve
men ... .::; in t~·o 01· ;ao1"e colours, ~ith n::plicat-: on:s to r.e~-.·ton ':1.
rule , orn"cir.-nt.e: tile-w:>::-k .. a!'.d -:he t.hcr)l':'l o..: nu.n(;ern,
Phil.i'lat; . ~·~(1"<0'7) , 11G1-'·'i'7 . T!.i» puper Ee Ls al.read;; Lhe
pooitive .'llH.! r.of~t1tivf! :~l.1 1 nn wr:icl1 n.t.·e cha.rn<.:Lt l'irti.ca:. .fo1·
the Wal.sh _f~.. ct. io-nr..
1. 22
:r:r:iORS=., F . 11. a..:::! I!. l':::S!IBA~H, •. e::.o1s o:f tt.eo1·.:i;~::"1 pllysic:::, Vol . 1 , ;·i~C-""...-? i J,r·~·; lor~: ril'G!·.~~-:- Eill 1 ·;3 .
2 . BRACE'1f}~!.J1, R •. The l'curie.:.·- r.:;r·r.'!rt:JforiL fln1i ilB ciµ1 Licutions, Ke" Yo:·k : "1cG1·uw-'1i 11 1'}~~ .
3. t:A...'V~ORO\..' 't'~lGH 1 L . t\1 • ·1.tJtl G . E' . /J{IJ,Q'~' 1 P..irtkL1.on'llai.t~Jy:Jiu
in DOI':I:l_:_erie!.i R8UT.lcn, C'.J:.t.;~tc•r 1.: __ 11 . Sectio;, 1; Eerli·-1:
1
.\ltodel!lie 1') 1 .
': ••.YI
i/er~l gP-1nei11i:?r1;ng Ji:s You1·itor- lntl'"·e1·:1le:i
u.nd des Hegt·il'!'es F1't: iur~nz , Al'~hiv el.e?. . Ube:.'trai;u11p; 18
1:""1IA.ru·1u•1•E 1 H. 1
{ 1C)($l.) ,4'.'J-...· 1 .
.
2. P-CHLER, F .. , Da:;; S:, t.~1 d·~r e:sl - 11nC ca_-FUJ1kZionet1 alt.
E":"eiter...iug df:'.- S:,·~"tr· ue!' 1.·:91.·!.-ro.r.r.-:ior.en ur_d die 1'hcorie der snl- ;.mj cal - l•'ouriei_~trs· .. · rorm.aticn 1 'l'ht.:>315 , Ca~ t. .
o.r f'Jn\;be:naticD l Inn~lll'U1:k- Unlve.L·,,i ty- , .Au.:::;t1·in '1·)67 .
1 . ?~
1 . GREE:l, R. R . ,
;. Mrial crtl1or;o:."1 decod.-!', Space F!'O-
F'..l'&11s S-:.;.~en.:-y, Jet F •opulEion J.a'borato!'y- . f~tia<l:eDa , Ca!. .
llo . ;:;7- 39, Vol . :V(19- J, ,.7- 251 .
2. POSllER , !:: . C. , Combinotoriiol structuL'"s in plo.neiary i·c!(Or111aissanco, Symposiwu on e:t'ror-correcting codes , ~I~ tit .
neaeac·ch Center oj' the US Army, lJuiversity of Wiscon:Jin
1'1b8.
3 . .ID.CH, L .B ., Co:iputation o.r firite Fou.i•ier series, Space
,..
7,
1l'Of;.!'a:l~ s~n=:.-,
Lal . ,
!lo . 37-~9 .
4 . !h.-i-Tj,
1
,
J\;t F:-or.:i.lz.io!'l J.;;.;borntO!"Y,
1
't'o!. . .:~ (1·~, ·
t.. . K. , J .Kh!l.l
anC
i 2'3~ -2o// .
:1 . : . A~ll1f..E.'\,~·; ,
::nr.adena ,
Jiar!nreord trans-
fC>!Tl i:t:i[;C Cl)Oir;g , lroc . ::J::JB ::;7(1C,VJ\',8-· 8 .
~ . •,n-r~LCEi:.:J. , .J . ~ . ar;.d T: • •"'' · GUII\I·~, i.+.r1rt. fo'o·..!r·i-:r- f.adamard,
t:rur:!if01'..:r.. a.ud _ ~ r u J(l i.L ~igr::t I _re{_>:'HSl-'Htnt:i en and classlf icncion 1 1'.hSCOh ' qf~ Iir.cnrd. {19Ci1J , ~;: 1- . '/) .
1 • HAJJ< , A., Zur 'lheori€ 1e1· or•tr.ogonalen Fu.11ktionensys~i1t1tt, Ma":J-t . J.-"'lna Lc:1 1}!~ 1~!1~
, .551 - 7i7'1 .
'( . g;IlJlXS , J . 1. ., <:oi.1rut::tio!l of t.he fr1:t '...":ils!i-.Fourier
~2'Wldortt , IE!'·
'fr'"'" · on Co1:r;iute:-s C-18 (1')6')) , 457-"59.
1.'1
1""':S~u11::;::ns 1 f . L . , lt.~01·j· of _:'".:-c<r1cn,::t a.oduln~ion noise ,
I roe . iRL .::<>11 Jc.8 , ·1081-1(""? .
t:! . ll1'U~J, J . A ., Le.:.· Zcitr:1tla1Jf von "F.auncll~p~u:.u:nu:gen , :El .
?1nc11r . •rec'1llli< 2f;('l 11;;:),"l!:F- 189 .
~ . t'Af~'PER , F· . F . , "loJ'JluLion , .noir:r.autl ;.~1ectral analysis ,
llew Yo1·k : McGrnw-llill 1'n: .
11 . l'.JJtMll1'ii , lJ . , l. e;ciH• L'~•lize<l concept o C f1•oqurmcy and
tJODli°' r.p11lica.t.lou:-1, rLhf•, 'l'ru.nc.oc tionr on [t.fO L'W Otion Theor7'~ l'J'-1 1 1(19f8)~37 1 -;·ti2 .
~
·1 . 'a'"'Jt~SCE 1 G.,
• r- r ·t~.. -2 .
t·: ... J.-1·nc
S:;!!;te:it;heoric, Lcii:-:-ir :
Geese &
. 11
~EE::ER, H. Vi . , R,,_1,rnr.(\ntati-011 o~ sii;:::-.1 .. ~; Jeaign of sigJH l !! ; in j.ei..:t"..orr n on r:oll!.!l.u.:ii.cn.tion s.ystr::u Theory~ New
1-~r'~· : ~CGTa ·•.;-.~i l l 1 1 I•. 1 .
l: . 1.0-u, E . D. , f}iniii·e o·rthogor.~..le Si~nnl(llpnnboto mi t spe... iel l r·n Ko~r·c lul ion~o.z e;erJ$Chai..ten, Archiv eJ ek . U-oertraisung 20( 196t.),?11C-;."lu .
3. AKIYJU'lii, 1'1 . ! Or!;iiOf~On~l PCf'~ t.ra.~smiaaio::. witl1 weigl;ltod
l.•lt. ~cr~gc!., J . o!.' t;t,f:" lnGtitu--.:;e oI Elnc:;ricnl Comrru.c.J.oaLio:-:!' Cr:F;inec::s o!' Japan 4~l(191~.6J,11~~-115'J .
1. ;:;r:u·::.D, I . L , H. ::. DU:>L::.l' ar.d S . E. si;r:a;::;p,, Partial reai.·c:.a" ::i§;~.sl f<H1L!ltl! for rsr3.ilel d:it:> trQll:m:.ssion , 1968
IEEE ~~~ . Co!if . o~ Co~1:111nicst~ocs, necor<! rr . 811- 816 .
r . ~h;.ttG, ; . -...' . a.~d H. A. GIEBY,A~hr:orntical :~tudyo~p~r
!'n1~mur.ce o~· rin ortho "Ollel n.;u_ tiplcxir.1; dtit
tra.nmdssion
;rherne, 1:rs.S 1}.1'.L lJ1t. . Conf . o:-i Communico.t.ion. • J\ecord _pp .
>7 ·3-B37 .
1 • LAN(i·t-; , F . E . 1 Si. r;:~ri 1 ll u11d Systeme 1 , Br:lun!lchweig_: -Vie we~ 1 'J(..(. •
? . 1'
~::'J_tJ-:£H 1
J . T·j .
,
.rnteL'µolato:r-y function theor:r,
C~
:>r"d;-c ':rscL iu N:.tliemat.i.cs and Nnthc11nticnl Fbysics ;3 ,
::.On·l,;n: t:ombr-icgc; Uni vc1·~i ty Press 1')35 .
2 . l.fXr:;sm;, H . , <.;111 and den:::i vy ti:.eore:nA, Amer . Nat;h . Soc ·
Coll . ?ubl. a:.c19qo) .
3. S&'AJJ;o:: , t: . E . 1 Comr.iunica•ion in ~he vreeer-ce of noise ,
P1•oc . iRE 37( 19''':1) , 10-21 .
Jl.EY.ERENGl!:fJ
.1 . GOt.Ll!M:I., . , -, ~oit111tion ;;J,e.:iry, £ni;le,..oou Cliff 'IU:
Pre"t~ce fl•
1 • ' .
; . L~~:~s1;, J . n . , A .. i.;;c:J~z::ioc 01 ~n=!r! i:-i~ :;ht"'Orci::;, :-roe .
GE "7l 1'1'.> • , 1 " •-122':.
&. t~O!CT.Ei:Bl:li , ;.. .
.E>:~ct; b t i:i_·~olet..t Jt, of b 11·.!-lic:.it~d
fu.ncti~-:·r... , J . ,\p_;>l!.cC 1-t:y~ics 24(1J:-;.),1-"2-1h;+ .
1
?- n'-·vArt·..;, I . ,
Gai;.1lin,.r r.::~eo1·e-n:i !.!1 uL~t;t•.a.:t. hu..:·uor:ic
anaJy.si~, f'i .it~1e:r.~1t.icko !7.-zk&l:!~.· Cnsori!~, .Slov~:i . A.~:orl. Vied
15( 1
s ' ' ' "' ·-4~ .
2 . 1:+
1. L.A f. ~;e; , L•' . H. 1 Korro)a1;ior.sc•!o}:trl)nik, IJ,nt'lin : Verlati;
'l'eclmJ.k 19'.>') .
2 . BlffiR· llROWJ\ R~i::earch Gori: . , Haneboolt of or.~i·at1on:il aa:plifir.r _i'lf'Jilicotior.s , ~·ucson , Ar1:':onn: 1-,11.-;.; .
;> . P iil ,;,;11<.:K H.::Si::AH<.:l-!I::S , inc . , Applicnt ionr. r.i .. 11ud J:or cou.-
pu.... iHg ::ci.pl.1J.'i<:!1·;.,. Dedit!J.lU, l<as:s .: 1<~ .. L .
4 . JOlUJSO:~ , ~ . 1': . , A.n:"j,lOf; corr.._:;,uler lec;u1iq1.oot<'.;, J3'·d York :
!'lcGra·.1-Hi 11 1')' ; .
5 . KCRJI, G . A. ~!ld r . 11 . KOR:I , Z:leCtl'OlliC tmdog lllld ktbric
oo:nput.er; , ~le~ 'J."ork: J-h:Graw- 11111 1C.•.:,I• .
b . SC!iHl!J, r . ' L . t1C\.;:J: 'lnd ll.h..:-3}:c.'!'H' !'oo~C'Ction O!" or<;;t.ogonal !';inP. an~ conine p·.:i].:;e;:; CJ li"1cttr ucl;1Vo Hi..: r.ot-;·.orks~
?roe . 1CX./ In, . Tel":t"t"ri.r;g
Co~· .. "·a~=..:ul:':Lo:..
;:i.; , 21C- 22C .
7 . SW!Ct: , D. A., \oial.sL-fu.cccion ger.erst1on , U:U:
':'r:.·.~
aci;ions on In!'or·cntioc Theor:r IT- "5( 1~·6':1), 167.
2 .15
1 . .bA1LARO , A. H. , Crthcgr;nul t:lr.Ltiplex.lt:I-.. , SprJC:f! ~ira.l 1\e.r-0nautlcd ,
·r~cb.nicnl Er:'.'cronce Seri~~ 1 1 A~ro .. :p(liCf'I EL<ictro-
nies, /u1vant•f\l1 Cnmmunicar;ior:;;" ~.(-19C2J , ;·lnv ., 1.1-1Jn .
SANDE!IS, 11 . ·..: .' '"'he digiloc~o1•tllojJ'OJ1al 110<.htla',lon GYStem I Adv1:111"'1.1 ._ 1 n !ucllll.w:ilcatio..:1 S;~;;;. l.ern:... ., ( 1°1·/), r/·1- 'l ..
z.
3.
KUllN, 1' . G., l\.li . HOHEY and
'il . B . SMa'll ,
data tra.Lu:.imlasion eysterr. i ll.EE '!'r~nr::nc;,icn:-:
t!'Onic~' nnd
,,.'i·""'t)rtr•tr~1 SE.T- t}(" "~ 7 1 ~ c,:: _,l.) ,
!le
01•Lllow~tcli
OJI ~:1
ncP El ... ::-
4 . VITll<f:BJ 1 A . ,J . ~ Oncoded_thr:ise- co11erent co1:i:111r1ica-tion:;,
1RE TransactiotJa ou Sr.nee 3lectronic.li Wld :;:e..,1•31(.<ol..J.·,v SE:- 7
( 1961).
5 . MOSIER, R. Ii . nnd H . ii . CLki1hU~:J 1 r:1r:-:plnx 1 Ft bnnd-.:ijt!'le!"fic lent b1nr.ry zranemiosion Sj·st.e1., 1 t'3!)8ttCt.1on~ A"i".c..;:.,
Coamunicotio11 and r.1..,;~r::C..:c.o ld19';7),72'-72'/ .
6 . JA??E, R . I~ .,
Digiloc!': telc~ctry :::ystc:r. foi· t..!1e A..:.1·
.For<_:e opecinl "'·eoror.n center'_-::: hlr:._c Scout Jr . , lltE l'rar.~
ect ions on Sr,ac'l .::.lec~1-ocic~ .wd Tt:leneti·;· O::tl'-'.l(1'JIJ2} ,
3-"'' .
44-50.
? . SANDE.RS, R. 'N ., Communication t'"'i"!"ici(lncy compnrinon of
several CO:runw.Llcution sycte:ns 1 froc . IR£ 48( 1':•1:.0) , .1'/~ ->S8 .
8. llOEt,z, M. L .• J:: . i.iiEA.LD ancj D. L . MJill'Hr.,
lllnu"·;y dal"
•(ranamiaa.Lon ~<'chniques !'or li.noar "Y"tem:; , Froc . TRE 45
1957) , G5u-l.>b1 .
DigH...:L d~.ta comJnun.lcuc1.on ~eclln.iques ,
hoc . lRE 119(1901), 1 9&-209 .
10.F!LIPOWSKY, R . F . and E . I.NUE!ILDORJ: , Spoce coounu.nicatioos Dyotcmn; Space communicati oos tecl:niqueo ; f;ngl ~woou
9. WlEll , J . M. 1
:;1 0
Cli f fs ;1J : Prentice liall 1 965 .
11 . KA~'SUNARU , K. ' 'I . H. YASHI , Y. TAKAlJA and K. OGAWA , Super
muli;icham1el carrier tolcgrap!l syst;em by pi.tase mcaulntion
(Rectiplex system) , Fu.ji ts u Scientific anc Tecl.ulical J .1
( 1965) ' 261-279 .
12 . ll.AR11UTI1, B. , On t!1e transmi!;sion of inJoI·:navion by orthogonal tii:r.e functions , Transacl. i ons AlEE ; Coll!munication ana E.l eccronics 79(1960) , 21;a-25~ .
13 . - , Radio communication wj,th orthogona! ;;ime ftw.ctiona
Transactions AIEE, Goaununication and Electroxti.cs 79( 1960)'
221 - 228 .
t
14 . SCl:ll~ID ,
P . , 11.S. DUDLEY and S . )': . SKINN:ER , Partial response signal fo1·mats for parallei C.ata transmission , IEEE
Transactions on Communic:atioll Technology 1969, in :print .
15 . osc - A bandwiat:~ - efficiimt fr~qucncy o u ltiplex sysLem for non- equalized voice cha.!mels , Allen- llradley Co . ,
Internal Repo,·t ( 1968) .
16 . 0SA't'Al<E , T . and K. KIRISAWA , Ali orthol:lonal pulse code
modulation system, Electronic,; and Communications in J;;i.pan 50( 1967), 35-'~) . Translated uy Script a Electronica Inc .
:? • 21
'i":""WAGtiER , K. W. , Elek~romesnetisclle Wellen ,
ha.user 1953 .
!iale : Birk- ·
2 . S/Hllr':OW , \~ . I., Lehrgang der hoheL'en r·rathema Lik , Teil Il ,
Berlin : Deutscher IJ'erlag d<:r 'tlissenschaftcn 1961 .
;l . 11EACUN1 , L . A. , llegati•re impedance boosting for 2-wire
bilater al signal ~ransmissiou , 1 968 r:EEE Jni; . Con! . on CoJnmu.'lications, Record pp . 165-168 ; also Bell System Tech . J .
4'/( 1 968) ' 1 0'19- 1 041 .
4 . RATH3lHl , D. K . and li . J . J'ENSEN, Nuclear test instrumentation \;,:ith aii niature supercond.uoti,re cablco , IEEE Spectrµm 5(1968) , 9 1 - 99 .
5 . ALLEN , R . •J. and N. S . NAJ·!MfaJHI , Analysis and performance
of supe!·conduccive coa.xial tra."Tlsmiosion l ines, Proc . 1EEE
52( 1 9611) ' 11 47- 11 54 .
6 . SANDY , G. Ji' . , Square wave (Rade:oac11er- Walsh .functions)
analysis, J"iit!'e Corporation , Worki ng Pape:r WP- 1585(1968) .
2 . 22
1 . SIEBER'l\ W. I'! . , Sl.gnals in linear tiile invariant GySten::J, i n Lectures on Com.i:ouJJi cation System Theory , New York :
McGraw- Hill 1961 .
2 . 21
.
1 . WAGNER, K. W. , Spulen uncl Kondensatorleit=e;en , Arc~V
fill' Ele!<trotcchnik 8(1919) , 61 - 92, ~·eceive<i oy the publlsber oo 1/7/1915 .
.
2 . CAMPBEI·L , (; . A. , Physical theory of' the electric wave.
Filter , Bell S;ystem ~·ech . J . 1 (1922),1 - 32 ; US- pate.ot ap
plied for on 7/15/1915 .
.
;:; . ZOi3EI, , O. J . , Thoory and design of. uniform and compos:i.te
electric wavo filters . Bell System Tech . J . 2(1925),1-~6 ·
4 . BAR'l'LJ:;TT, A. C. , Ttie theory of electrical a_...t;i:fic:i.a.:.
lines a"d filters , New Yor•k : Wiley 1 950 .
REFERENCES
5.
CAUER , \\1 •
311
Sit?bschairungen, ::5e!'lin : VDI-\'r-'rlag 19:;1 .
Coceunicatior. lictwork:-; Vol . 1 : TI:.e
classical tt.nory of lu:nr~d constant net"orks; \'ol . '" : ~!:.,
c:ae.sicttl Ll.t!Ol'Y of lo::g line,,, filoer·s, and !'Cll.l~ed necvorkc , ~le~ York: Wiley 193-. .
? . CAl..TER , W., 'rheorie der linearer. \'l'ecLeeleit.romoc. . . a.!tungen , Le-i;:~lg: l..k:u.lectische Verlrsg~ge:-ollt:~hu!t. 1f)l·1; English editior. : Synthcz; i s of li.nea.r comcr.uroicaci..on nf..'tworks ,
1
6 .. IJt'llLEl·iI:J, .E . A
_.,
1;ew Yot•k: McGraw- i!UJ 1958 .
8 . LICfiLER, F . , S;vnthese linear·cr
f'"rir.iC.l~ch ieit.va..riabler .F'ilt:cr mit vorgcschriebenem Sequer~zvorha ~on , Arch.iv
elek hL' . OberL t'l.lf'iUJlf5 22( 1968), 150 - 161 .
9. iiARJ'IU'l'JI, !J., Sequenc;y .f ilters banrd on Wn'I ah !'LtnCtions ,
IEEE
'rrt1.n ~actiOn!l
( 1 968), 29:1-29~ .
10 . - ,
Scqu~:ic~,·
on
Eloctrom.agne~ic Gomp11~i bi
!ilt ers , Proc.eeainga of
t.he
Hy S!"JC-10
Sun-JDe1~school
on Circu:l t l'hr!ory 1968 , Czec11oslovak AcAd~my of Science .
Prague .
~
1 . llOliAK, ll . J . a;id P . ::. SCJiVilD, Jn-rml:iction
filte:-H, -EE:: Trane.actions on Elect rue:.. rnetic
lity
E!iC-10(19(,8) , 2~0-220 .
2.:.:!1
1 . COSTAS , ,; . P. • Sync!"lrono•.is
conounlc~ ti ion •
(1956) , 1713-1718 .
2 . BLACK , H. $ . , Modulaticm theory , Ne"• York:
19S;l.
~o di;;i:atl
CoL'lpat~ bi-
l roe . IRE l•4
vur.. Nostrand
} . SCf!WAflTZ, M., In.formation i;ransa:isHio11 , modul11tion , nnd
noi s o, Now Yo r-k : i'lcCra>:- Jiil l 1959 .
i~. PAN'l't-;R, l . F . , Mo dula~ion, noir.ie , anO epecti1•ol tu1alysis ;
New Yo i:-k: Mc«r.. w- liill 1965 .
5 . vr•rERJJI, A. J . , lhinciples :>f cctennr. COIW!IUJJJ.cucion ,
New York : l".c(;Mw-!iill 19c6 .
~
1 . CROWLEY , T . U. , G. J . ilkRP.:E, S . E . MILLER, J . ll.l'lERCE a.r:d
J . P . Rtr.TYO!I, l1oder'- co:ru:iunica~ions , l':~w York : Colum1'ia J .
Press 1962.
2 . ~~ •' . 1 f'i~glichke~ten de~ E:-ss tzea VO!: ln<!ukUvitiiten wid ;:;chw1ngkreio~n in zukiin.:"Lie;en Systc11en d<>r J·;ult.:.plexi;echnik, F1·equ.enz 19( 1965) , 297- 307 .
3. FLOOD , J . E., Time division :nuli;iplex cy,.teme , ElP.ctrollic Engr . 25( 1953) , 2-5 , 50 -6~ , 101- 1().i~, 1 111.>-1 50 .
4 . ROBEllTS , Ji' . I!'. and J . C. snam:ms r-:ul tlct: .. nnr. l coa:.:nculication sya~ea..s, ;,oireless Engr . 22h91,~) . ~38-5'•') , ~{lt•-589 .
5. BEtfNETT, W. R. , Tioe division o:ul tiplex ~yr;LemG , Bell
System Tech . J . 20( 1941) , 199-221 .
6 . RAABE , II ., UnterauchuogeIL e.n der wech:iolr.c.li;lgen f'lehrfachi.ibet"~rogun15 (Multipleriibertragtmg) , E'I (oit t:ri.oche }lacllricht~n-'l'och nik 16(19}9) , 213-228 .
7. Technico.l Starr llell Telephon e J,aboroto:ries, •r ranamiseion S:yntcms for Cor.ununicatio.na , Win~ton-Sn.J.cm NC : Western
l:lectric Co . Technical Publicationa 1 965 .
8 . '!·echr.ical Scaff S:e:i:ens Cer.t~al Lnborntori.:·.c , Nac!::richt.c~t.lbert :-o.g-.u:g 1
9. scm:;.rr.r;: ,
N. _,
B.e-:::-l1n/ :~e11 York:
w . R . a~:;rz;rr.
Sp1~1u~e :· ~P:..:vu .
anu ;,> .S'T:sn:,
Co=;inication
sys leas H.!.-d !.ec::nique::.;, Jc·,, Yo:-k: ~~cGr~:-.- Hill 1~66 .
1 •. LAIOC~i l V. D. , Theo1·eticu: a:ial;veis of \•arious s~ysce:ns
or lllUl ;ipl,;x t~»ms:uissi;;:.. , RCA Rcvic-.. ';i( 194<3) , 287- 551
I~ !°'8-l.:-8j~ .
)
11 . tiii!iJ·IU1:!!, 11 ., 5eque:Jz.-r-:ultlf·lex11y~l.~J.1~ .::iir lel~phonie
w1i.? Du t.enu·be!·tragu::ig) 1 . Quildrr1tu:-r.iod ·.J 1 ~'t 1 ::>n , 2 . .!o..intieitenl>1md-~·Jodu Int ion ; Archi v elei<tr . 0u<'1'~''""w''" 22( 1968) 271
.;)5, 1u2-10d .
i
12 . - , Tl·Hger•s;ys Leo .fii.r die
pnton~ 1 191 416 ( 1963) .
1; . lll.1Ai!'J'll, Ji. I ., VerfeJiren
:·fi;1 chr:i.(~111:"r.t1-:chn:l
'
I{, 111 . Ge~an
zur tli1.o;Hu.len Gotlicrung von
F'tJ1•u~.-..8~.,1·iicheu ... , \.J . Ge:-mnz: poten~
1 : •i14 8~~ ( 1 96~) .
111 . PICEI.ER, F . , Dao Sequcn7.v:.clfscl1, e1n nettes Sorechwe-
f;t nn---z fUr vol~C!lcE.i;ronische .l•'ernc11rechVeJ:m tttlunE;sa.m ter
x...... . lnt.. . ~'l'iEs . Kolloculu.m deL' Tectulicchcn ~.J_:hr.-c!"lule I.ki:t_:.
ua", En&t Ge:-o:iny 7(1%7), l!;i - 20 .
•
, • 11..
:-:-rro~:E'i , '" . r· . :t."ld D. K . '···'EhVE:l . A::. ~ tro iuc t ~o!l to single
a.debQ.!,d co!ll!:lunic~"ic~, FToc . IRE ~4(1956),1667-1675 .
e- . hOHGA.ARD, D. E . , The P"~~.e- ei1i!.. t. aet..hoC: or n:.nglc-sidebend cii;nnl gene=a~ic!.l ; The 1 l1u..,f:<'-~hi..:"t.. u.ctt.od o.: singleHili• \Jic.d ~i rnq 1 reception, Proc . !!IE I.I, t 1 ')'.j6) , 1718- 1743 .
' . S1'..RAGA ,
'1•,' . ,
Single- ni Uoland t;CJ1f'!"fl v ior_ , Elect;1•onic
Tocl1r.::loc;:. 3C1(1CC2) , 1&6-17~ .
11 . \.1r;A'v'r:H ? D. K . 1 A tr.irf! method oJ i:;c:uer·stion and detec~io 11 oJ" sini::le - ,1 i,cioba"a r;ir;nnls , Proc . rru: -'14(1956)1703-
17V) .
5 . NYQ.ll 1S•f' , H. , Certain topic~ ill i;al A!J:l'Flt'h transmission
ti1AO ry , 1.Pr+::i~!Jnctions AIEE ·17( •Jt}28) , 1 ·1'/-f·,1111 .
( . EE\:Ki·:i., l". K ., J . H. !JA'.'E!'. 'l!l<l B. 11 . s:.ur:!!Eltll , An A."! ve-
:1L.a.J..;i.:J- siJcho.nd clC1.;;a
Cl''anzmic.J1011
scL
using nynchronous
iotcctiori. fa.?.· ::erlal t.1-an:;:n ic:.i.or. :.lf Le. 3000 bits per secon 1 1 rrrir.n~ctions ~:EE, Goun.unict.\t;iO:l nnd ~ectronics 81
Cl ~.,2 J , 9/- 101 .
.
? . 5~C!::'.:!'T, \·.' . R. and ., . R. , JA '.FEY, Dr.ttb tra!"'~tission, t~e\ot
York:
tlcl.ir:iw~Hil:
19;3 .
8 . 1::.RNUTH, H.. ? . E. S::illii!>&n<i IJ . L . t:o~.AK, Ti•b.Jlsposed side~ t.:;J a.odul~t.ion for dq-ta transmierio:1, !EBE Ti·ansactions
u" Co=~c::iLion Technoloi;;; co;1-1!>(1')t>7) ,868-870 .
~ . 1'
1:'"""f.A•ill1JAD1, E . J ,, fo_r1aloi:; r.iodulation nym;ea.s , ~n. Lecoures
ou '01w,unicotion Sysi;em Tl::eory , Hew Yot·k: l'lcGraw- ilill 1?61 ·
'. Vl'JE!ff I, A. J ., P1·inciples oi' c0Lu1'ent col!ll!luru.cation,
;·Jow Yo1·k : McG1'ow- Ilill 1966 ,
' . I i <:H.LEH I•' .
Das S:ys•cm de1· ad - u.11d c:nl-l'un lctionen els
E•·"Pi.t<!.ru.n~ lleo' Systems dP.r Walsb-Fu11ktionen und die 'l'b~:
1
udr d~r· st1l - und cal- FouriortL·t111urormscio" , pA7 ; TbeS De;·L . o.l' l':«thomatics , Jru1sbruclr. Ol1ivereity , Austria 1967 ·
~ · 1
~
- , b . , .Frequenc:r.. moU'Jl:tt:io11, Proc . lllE 18
1 . ~JtJ :;EE" PC.u
(1;)00),11 "'·-1205 .
2 . £it.H:J, .h . • ~ Fraq·.t,.ncy tLv:lt; I rit.-<Jit , le-,, York : ··icGrn..-ilill
..
·t<..1L.2.
'!J . CUCClA , C . r, . , HRrm.<J1ticG ; ::1itl1:1banc1n , a.'ld
transieut~ in
cor.11:n..uticat1o t. engineP.rir:~, he•.-1 Yor.r.: r:cGrnw-Hil l 1 r1~\::> .
'' · B:...~CK , Jl . S . ' :-iod.Jl~tion t:1cory, -~i!".:CPt.Cll K~ : V:JJ. No-
.. r!aJld 1
··r
3j .
-, . 22
1 . GRCSP.Y , J-1 . G., Go:nr:1unlcatiar. Uy phase
llli 27( 1'i 'i'l) , 12G- 1 )b .
"" l'UCY. , 11.S . , 1".oJ•lotion
LL~oP:;,
s--rand ~J~3 -
3. PAl:TJ-:1., I . F . , i1o<!.ulnt: ior.. .,
tiew
Ye~~=
!'lOlSe,
~oclulatio n,
?.t·oc .
Princutor: '.-JJ , Vwi
and
:~o
::p~ct:_·.:.L a.--ialy~-H;
l'rGrm·.-llil l 1<-e:o5.
" . 23
1 . OLI\fi::h , ~ . :1 . , J . 11.I'IERCE ar.~ :> . E . 3HA:l:'10J·: , 1'he ptJl.losop~y ·j:· PC.Tl , Pr.:ic . Jh:-; ;6(~ J-'~·$ •,4132~-'13 7 1 .
4 •
fi),!£. 1 l! . J: . , l"!'::.nciples cl EJUlse cod,. ...vdu at.:..ou, Advnn . :Ele(":rro ... ,
i;~~
lnt'r:.: Acodan: Le
P1·~ss
1':J51 .
3 . PL00:. 1 .J . r; . , : ittb li·...-isiou ir.ul~iplex ~yr.tea:.s l fa""t 1~ ,
J,:iec·~ronit lfogr . :?'.:·(1·J~;.),·1'f<,-1JO .
4 . GOODJJ..J, , \~ . 1·1. , Tr-lol•hony by pulsC'- code- JOO<'l ula-i;ion , tlrll
Sy st.err. 'l'~r· r, . J . 2'6( 1)•'17) , 39~·-4' ) .
; . de JAG!-;1., ? . , Lt:l~fl :iodulG ... ior.. : A :J.et:Jiod _:; rc~1 ~z.·a:'n1 i.-.,sio=. •.i.;J.t;.f -hr- 1-~.it code , Ft-.ilip.s ReSCliI'Ch Report.' 7
\ 1952 l , '+'1?-'•u(,.
6 . BENl'fE'P'l' , Yl . d. ., S~1ncto1•a o f f'J.UWJt.ized B.1.1->JH:t.l!> , Bell Sysileo '!'ecu . J . 2'l(1''"•tl) 1 44&- 1172 .
,, . ~1
1 . HE.:RTZ , 11 . ~Die K1iiftc e:le:..:tri:;cLer Ee. '"'J.u~mgc:: l:t:han<..t:lL nacL c!t: K::i:.:i.t~~ .. 1 .cner.. TI1...-<>rif' . k.n . Ihy8 . , .. lE'.Jt Fal -
ge 36(12A~1 ,1 -22 .
2 . ZUJ-tn•J1 , !, . , Elektro11n1:,net;i scha Sti-ahl.ungs.Cel<ler , Be1·lin/
l~ew York ' Sr·rine;er· 1 ')5:'> .
; . SLATER , J . . a!".!d 11 . l'l~AtJl-'..., Elect..romng::H:t.is::r.1 ,. Kei,J Yo1'it :
HcGra~-:i ••
1047.
4 . \o!.:..R£ , L . .h. .
Pitman
1 ·-~·~ ·
i
Elecent ... of el.er:r1•os;iagne&:. ... :.<ave8 , :rew York:
'.) . BECKER , ll. a.ndF . St.l''l'ER , Tb>Oi'1cclcrEltiktriziLiiL Vo l.
1 , 18th <:LI . , Stutti;1u·t : Teubnnr 1':i04.
LAKDAU , L . D. and :E . :·J.LIJ>O:•:! ll'l'Z , .;..ehr1 Jchdert heoreCichen P!".y!'"ik Vol . 11 , KlassiscllC Fc:Ctheo1"i,.. , Berl in : M~a
de;;rie 1')&>.
~ . K!U.US , J ., A:i~erulaP , :Tc~·1 Y rk : r.cGrtn·.- H1ll , 1qr: •
8 . Scm:D<.lllO!F , S . A. , Electrome!.".netic woveo , ;!ew Yo::-k: 'ftm
tloatrand 1 'J4} .
t=. .
~
1 . P".AID1UTH , I! . , Grundlegen und aogl icl:e An·•erulungen der
REVi:JlF.JICES
St!-quenzt~c:mi .--: . Bullotir~
11i ... ct:~n Ve.r·ei::~:;
de::. Sc:..r:ci zeriactcn
5''1( 1t..~·J8} , 1196-12C·5 .
f-l~ktrotect.--
11 . 11
:r:-i;RAJ-1ER, H . , Mu~l:&JOnti cal method:; or sLat i r.t ca , Princeton : Pri:iceton UnivcrnHy Press 19'•6 .
2 . KOLMO(;OROPF , 1'. ., Fo1Jndations o f the thoory of pr·oo1.tbility , New '!ork: Cl!dMA 1 <)50 .
I~ • 'S2
1 . ~Ctt_.IT-1' , H. , Sy~tE!ut.heori~ f'U:- rte;e ... ~O!:'-? '!o:l'tY~::ige , Be:-lin/:le•.< Yo!'.'I<: 8µrillf<'?" 1';•)(. .
2 . ~::>:.:a=::ro:t , D., An i:.1::1-oduc-tior: to ~t.:o.ti~ttcal co1:1Dwticntion c-!.leory, i:<'w York : Y.cU:-9·.-1- ra11 196C .
~ - LEE, Y. ·...1. ,
St:1ti!itical tl:eory o: coDDunica.tti on , Ne•,,,
Yorr. : Wiley 1960.
11. llE , Y. W. , T. LClfEATIJJJ·l and J . ll . 1tiIES:mH , lop;;Jication
oJ' cor!•elntion anl.i.l.y:;i.Ln ~o C.he detec• i or. of poriod ic signals it> noirle , Pi·oc . ~!m 58( 1950) , 11 .. 7-117 1.
:; . GRJ::};:-J , P . E . . The ouLput signal - Lo-I10.lne rac!.o of cor1·~lntion tletectorr, fRI T~"cinsaction::s on tnfor·:nntion Theo ry :T-'1(1958) , 1C-18 .
b . ',{[ElfERJ :: . , Gcnerali•~d !:armonic w1"1y:li.". , Actn ~latlle
~ntic~
55\1930J,11?-258 .
7 . ?:SCJii:R, F . A., Die ;imt~-<>1tatische E·,hnm!lwig ::u.falliger
Vorgfill6e i!l der Schwingungstec!Jlik \ YerLlleldecechni.sche
Zeit~chri:'t ~(1J'2),151 -1>S .
s .11
:;-:--Jioon , J . L ., Stocwrntic processes, I. cw Yoi•k : Wiley 1953 .
2 . VAff DER ZIEL, A., Noise, Englewood Cliffs NJ : Prentice
llull 1954 .
3. RICE, S . O. , Mat11w•11ut.icnl <malysis or
t·nn.lou nolae, Bell
Syntem Tech . J . 23(1c,1,1. ) , 282- 3;>2 , 2'1(1"'":>) , ••6-156 .
'~ . SMIJLLIN , D. !l.l:d JI . A. HAUS , !lo i se in elndron devices,
r: .. w York : Wiley 1';'5') .
_
r • .Es.t:r;::T-T , W. R. , f':lectt·ical noise, lie.,; Yot'k : f'JcGr&h'- Hill
19E-O.
6 . DA\'E:l?OR::, 11.E . Jr . anC. W. L . aoG'_ , An introd:iction t'o ~he
U.eory o: randOili sii:;nA IA and ::ioise 1 :lew Yor·k: licGraw- Eill
1 ·158 .
7 . scni-.rAR7Z ,. (\: . , Jr..fo1·a.11tion tre.nsw.issiou , moclula-.;ioo , and
l101 se , Kew York : l'icGr,iw- P.i: l 1959 .
8 . ROO~', ~I . L . a.orl I' . $ . P r1'CllER , On tho 1''ou1•iei·-se1:ies exp1.n:oion of I 'IL'1<lom J\UlcLion:; , Annal" of l'lu·t h . Gtn tis tics 26
(19?5) , 31 3- j18 .
.
'" · fL~tJS , !!. L, :i . o . llif. scandru:·ds o!" mc·t ltoda of measuruig
noise in Encar ~•opot·t,; , Proc . IRE 48(1'.)60),60 - 68 .
.
1' . WR.I\ , K. , Di" otuLiatische Verteiiung der llulldurcn
i;iinge von !lausdJepa.nn·.lJlgen, Archiv elek . ilbert!·agung 1,0
(1965) , 685-698 - .
11 . S'lALAY , G., Die Vorteilungsdichte der !nt:erve~-e ne1
ciner.i Rauschsignal mit Schwellwert , Archiv clck. ltbertr,,_
i;un1£ 18( ~904), 31&- ;22 .
REHl!l:NCES
12 . t.OH:~SON, J . li . , Thernal afSitalionore:eclr1citj,.i11cond:.Jctors. f1:y!:ical f(evie·... )2{1928 ) ,ul- 10Ci .
5 . 21
1 . KOTEl1 1NIJCOV 1 V . A. , Th<J 'LlleorJro£ optimum r_~oise irin11.u:i ty ( vrn.csl~tion of the- Hun,r ian or•ieinal put 1 ieheJ in 1 1 •47,
by R . A . SI:.V"ERJ1J.~l ) , llew Yor>: : l"lcC;·:.• w- llil- 1_.~ J.
2 . S::EBERT, ',.' . ~ . o.;:<i :1. 1 . Roo: . St?.~1st:.c ..1 ~·'cision tl:eo-
ry and 'Ot[oJ[llln1cntions 1 i n Lectu!-E·r. on comr.iunica:;J..or c:rsteOJ tht>ory , New Yor~ : McGraw-Hill 1')61 .
3 . l11DDJMC l'f, '.l ., An introduction Lo ste t iat.Lcal comr.1uulcntion LLeory ; Jlpw York : l"lcG;•a·•-Hi 11 1960.
'- · .-!Al ~:;:u:::, L . .:. . and V. ll . ZIBAKO'I, ='xt,-action of si ·i.al s
from nvrne , Englewood C:i:·re !\.:' : ?rent:.cr llal:.. 1')62.
5 . fL'cRl'IA.t;, W. k ., ?r·inciple~ of ti:" sLaCi<ticnl Cheery of
communication; New York: McGraN-lltl l 196:3 .
() .. 'riIEl\ER, N. , Extrapola l i.nn, i.11t.r.:rpolatiou and oooo~hir.g
o;~ !Jtat.ior.:ary t.i:1 e seri~'t, :~ew Y"1'k: r-:~r PrP.~s a.nct '111'iley
19'-9 .
7 . l!A!':<.:OCK, J . 1.. ., Si511a
det;ection tl:eox·y, 1:ew York :
f•icGraw-ili 1l 19GI, .
8 . LEVIJISON , K. 1 TLe Wiener Rl'":S n1·1·or· c 1•i torion i.n J"il~er'
design and preil1ction 1 .J . of :·Jat1>. and Fhyrics 2~(19~7) ,
26-'i-278 .
9 . KOL!".OGOROFF , 1• • , Interpola-;;ior wio ext1·apolation of
ata'tionary ranOo:n sequcnclt!s , Bul l~ci.r. cle l ' acadllr.iir des
scien c<"a de USSR , Se r . Math . 5(1941) , 5-1 11.
10 . SHEilMJ~l'; , S . , Non-mewl 0qunr e ._.r1"or c.r:. teria , IRE 'l'ransactions on In:oraation ".'.l.eory 1'!'-11(1');,9 J , 12'>-'< 2£ .
.,.., . BO:rE, L . ~,· - t A .;it:ip:!.i!'i"c! der-i~nL~on o~ lwee:r l~est
squaT'e e::i.oothi!'llr and predicti on t~.eory , F:-oc . ..RE 38
( 1950) ,1•17..ft?6 .
1 2 . A.R~mrns, E . ttnd H. DYM, On the opclrnum da l ect iou o l' digitaJ. SJ..gnaln ii.. :.be pr~til:!nce of ..zhi-: n Gu·..;.r..si an n ~isc,
IRE :'~nnsaction
on ~ir.:nunication SySt;C!!!:S ~S-10('1<--.. 2) ,
336- 3,...c' .
:; . 23
1 . JlORTH, D. O. , Jin analycis of th~ factor" wtlici1 ilctc1•mine
signal/noise Ciscri~ation in nulsed-c:u•i•icr system~, reprim;ed ir. ?roe . IEEE "1( 1"1>3) , 1::.1e.-1027.
2 . TURrt; 1 G. L . , An introductio!l ta r.1atched rilteo·:i, IRE
TransncLi.ons on Ir•.ror•n:atio.n Tbeor:1 IT- 6(19.:.0) , 311-329 .
3 . SCHUSSLER , >I ., U'ber· <lo n Entwurf opLimnlot· Su chfilLer ,
HTZ 17(19i;,C.),60"-61 ;, .
4-. SC!iLITT , H. , S;;stel!l- tieo.rie f'iir regeLose '.'orga;:te , Berlin~lew York : Springer 1)60.
5 . ZADEH , L . A. eno I.R . RAGAZZIIH ' Opt i mum t"ilLeL'~ for Lh«
detqction of signnls in noiceo , Pro c . !RE 1.0( 1')'.J?) , 11 23-11
& . PErERSON, E .l., Stati~t.ical analysis und optimir.otiou
o.f systems, lfow York : 'Wiley 1961 .
:"1 .
~
1 . CORRIIIGTOM, M. S . and R . N. ADJU1S, Advanced analytical
end sie;nel processing t;eclU1:i.quco : Application of Wulsn-
REPl:JU'.!iCES
funcl;iOHC to !"IOnl i.r.car O.llOlysis , reel lliC:J.1 liPpO!'t AJJ27'7')42( 1 0,:,2) .
2 . h'E .. SI-::R, F . E. , 'ih'e.2.~h r·•. uction ane-:r-.:.nof J..r.!·..-·u _qneous
c~nl l ne !..:' !;~Ocr.&st.ic f roblc:t:- ~ '!'be~itl, f Olj~echni.• lnsti,;u te of ;;rook.ly11 ( ~ JC4 / .
2.:.2.1
1 . EA(rJ-fI>ADY, E . J ., Iii·.,· e:-si~y to-:::lu~!'i'.J"'!t 1 in L·cCl.ill..!·es on
co:ru:1unication ~ysteo • he..::-:; ; ~·le'.of ·10.:-k: ;':cGr:r...,.- Hil L 1961 .
. 1. , Li::ear I vers:.t:1 comtiL.:_z:g ti;:.c.nniques
2 . lU-IEr:?;_~: ,
F.::"Oc . :R:.. ~7'1~ ~~ . , 1~J7~-1'i ,.:.. .
,
?· Hl:.11CE , J . N. md S . S" i;;, N, Hult.rle .l1':.,roit;y ·,deb non1nd e]Jt1Hd;>m; faui ug , ? 1'oc . 111;; '18( 1960 J , B';l-104 .
1,. PRICE , R. , OptitnWh do1'oction of runcom ni[l:unl~ in noise
wit.h nppl icatio:-: t;o scau:;er alitipat..L cor.i.mttnicnLions , IRE
'::-nn.~'tlCt:ions on Int"orzatio:: ':'heo-:-y IT- 2( "195~} t "'27-~35 .
5 . ?R:CE, R. a.'1.d J . E. GREE1. , A cozu:.uni"'!•1tion ~e\.!1uic:_ue .:or
cr.uHirnth cbaruiclc, ?roc . JhE l•6(1 ':l'.>'1) , "~5- 5'/0 .
6 . GJ.J•;JJ , A. H., l:ornpr.irisou
of
PSK
vc
.'lSl< .:ind
PS1C-AJ·1 vo
FSK-Al·J binary codr:d l;rrut ~mi~r.ior. uyotcms , IEM. TrMsac tion~ on Cc-!lllilun ic!a.C ion Sj·nLems CS- 8(1+1 J) , 87- 1 00 .
/ . RlDOUT , f . ;.r . 11.ll<i L , K. ·,.;;(UJ'.E!i , Choic"' of "ul ti- channel
telegraph ~y:--;teue ~or use on HF rudiv lin.its , Proc . EE 110
( 1<j..,:; ) , 1•!-02- 141v .
9 . TURlll , G. L . , Ou ·optim~l civer~i l.y rcceptiou I, I1IB
Trllll!l£\Ctious on rd"orma.tlon Theory i•r-7 ( 1901 ) ,1 '.i'• -1 66 .
'.) . - , On opt h ial d ivcrsUy r r'ccpt i oi; 11 , lRE T-~nr.actione
on r.:o==ication Sysl;ett.s ,;S-10( 1962 J , "2- :;1 .
10. Llt.'1l, H. R . , 1'::~ C.et.:ctnb .. it.j· o.r t"ed111F. raciotelegrap.
sicn•I!" in no1co;, :'roe . IEE 1~•b(195?) , 13u-1ll() .
·
11 . VOJ-:LCAEE , H. Jj , , !'base uhif" keyllli;: ir: fadir.if) c hannels,
I-roe . IRE 1 07B{ 1 960) , 51-38 .
1 2 . P!ERC.E , J . N. , 'J'heoroti cul d l verdty Lmprov~mont :Ln frcQUency- slli.'."t kcyi 11f: , Proc. TRE ho( 1 '>'\8) , 903- 9 10 .
1;1.AVIA':'T, J . ... . , E. D. JOI;E!l And ti.B . ~A'n', .!,.requency divers~ty in t!:e reception of s,.l,.ctive .'."nding binary frequency
- 'llodalAted sii:;noJ;c , Pree . HJ·. 1li'~B~1':!5'/J , 98 - 111' .
111. BE1.: .o , ? . A. ''"d ll . D. Nr-;r,JT( , 'I'be ~ffecL o f frl'.>quency se"!.ecti ve !'adi ng on the binary orror probnbili ti en of incohe1·er:t and dln·e rentinll y cohei'enL matched filter receiver!" , I:!.EE Trnr·sactionc on Coa:..i::.unicar~ on Syat.e:ns CS-1"1
(~"l63) ,' 7G-186 .
15 . l!~:tsr.:c;i ,
Ph . ,
Fch1er~·!liJrscheinlicOkeic bi.tllirer ltber-
tragwig~verfalu·en bci Sto1·.w~cn durcli Rauecben ur.d Schwund,
Ai-c.:J;iv olek . U-bC'rtragtmg 17~1963) ,1 8~·-1 9'7 .
.
1 6 . -, l'ehlerwah rsche i.nlichkeit; biniircr Obertragungerr bei
:·iebriacteapf'ang und frequen2-sel cktivoc Schwund , ArchiV
ele~ . tlber"ragung 1?(196;),271 - 27'7 .
17. ZUl!liT, H. , Die Sw:imeohoufigkeitokurven der exzen"1'~
sc!Jrn Ra:rl eic;h- 'Jorteilwii;; Wld ihre Anwendung Dur Auabre1tung.;me1rnunge n , Archiv el ol<. Obertraguog 11 ( 1 957) , 1~ 78- IJ81L
18, lrENZ E, E . , 'r heo::-etiocht> Untersuci1uogen i.lbP.r e in:i.i:;o
Divereity- Verfal1r'en , Archiv elek . Ubertragung 11( 1 957),
1a;i-191._
19 . SCH\..'AR'!'Z )
x.'
\..1 . H . BEN!~h'TT and S . S11't:..N,
Com.municetion
RE!•'£P.:ENCES
31 7
~ys;e;r~ a."ld technique.;;, 1~1".':~ York : tlc~r,~-Hill 1 11-36.
20 . GF.O:"t:KOH•, .: • , 11 . SCHOLZ enc K. VOGT, Korrol Rt:.on:ollcss\Uigcn ia Kurz.,cllcnbcreiclo., !:TZ 11(1 •!)d/, J1-')~ .
f' . 1'
~YQUJS1, H. , eerr.arn ~op~c~ ic tclcgr3DL trw:smiBsion
t:.co1-y , ':ro.::.sucLiuns Al.:..r. t.1!( 1 }28 j , t·17-•·.4i. .
2 . Ktl~Ff·i'.Jt. t.ut, K. , tlocr i:i!lsch>:i!•e-vo1·t:'i11.i:& in 'tlellen ~il 1: crn, Elt! Hr i "Che r·;«cr.:ri c!n:en- I'"c'"' i.k 1 ( 1 ');:>11 ) , 1111 - 152.
3. - , Atlt'§Leichtivu.r·t;iiage u.nd ltrequr_.nzci~nrnf."t"f't1·istike.o in
L.incuren w,ynr-ro:1i1"~ I E,lr-ktI'i ~che Unch1·ic.:lit..cu - 'l'echnik 5
(1028) , 18-~2 .
11 .
llAR';{,J:;Y , R . 'l . L ., Trammissi 'Jr.oI lnfol'lml.'lon ,
!c.itea~ 1recl1 .d .
~ell
Sy-
'/(1·)28) 1 535- 563.
5 . KtlPFMUJ,J,f.;R, K. ,
Sys ~emtheorie <iel' t>lekt1:iscl1en
St'.1t tgarc : Hii·ie 1 1<;;.2 .
A mn~heJao-t i cal ·_ licol·,y of co:1.n.11ni cation 1
Die
Nuchrichtcn(;b~rLr•1c;uni:: ,
6 . ~llIAlHlOtJ , l' . r: ..
Bell Sy: ~l·m 'J'..;.cl- • .., • 27(1948),~7~'-·123, L~!--t:.~t.1 .
7 . - 1 Con:m~1.lc!l.lit>n :,i, :he pre~o:nco of noi"''• !roe . IRE
~.'(1''"<~9).~
-?1 .
i- . 14
1:'""""T;.HO, R . li . , :raL.slJi3s!o11 of infon:tntion . :1e,., Io=k : r:I·~
Fred< "nd ".: cy 1 '61 .
2 - SC!lllID.."' 1 I .o. , 'ior::.chlP.gc- zur frerech.nv.h~ tJor "lll.rklichl'n !:nunltr
...... i..;it teim V-o-rt;inf',..nsein vou 'le1·lur..ten au.f
dem Otct·trc1f:U.."lf"t.!'h·,..get /ccl.ti1l e:e.r.: . Uto:-i:1·n5ung 61. 1954 ~ l
1e} - 21.'." •
5. ZEr·:A~.iEK, ii . 1 Eler.ientare 2:nl'ar.:ru;1 l ionf"t;.HOOt'-.(l t \.lien: Ol denburg 1'1'.:-9 .
'-. l•'l!:Y ) f' . l ..:ni'ormution.st11eorie , Ber- tr: A:.t!i<l.e:ui.o 1903 .
5. SOMMi~HVll.L1';i 1J . M. Y. 1 Art introtluctior t(') tn~ E:;eoa~etr:y
oJ N dlmf'ltnH i.on~ , ;.Jew -~orlc : Dutt or. 1 t,;2~; .
G. l~ADEf,IJNG , :::; . , Diu 1111;.tllemati~cl.eu Hill',. ,,ittl'l dP;; Physikur:: , l4·C:rl n/Nr"tl Yo:·t:. : Sr.:1·inge"' 19r7 .
h . 1~
1 . hA..'tl·lU'l'H, H. , :Ji, t'bei·~~·ap;unf:!llrnpn;,~ tii ~ voJJ Nacr.:-:.cntcnkanillen nach dnr 1.'~rnllge:rei:..erur:t• Cc-:.:; Be ·i.·iffe:; !t'J"eQUeLz, Archiv el~k . Ul.ert~~g 1~(1~t;),12~-'i33 .
2 . SOHHZRF:::,D, A. , l:'-cer Jie f'or~r.il:ui2"..Uf.8 t!•. Lichtes in
di::;pnrgi er.,?!<!<·!; l\<!J~t'll . Ann . Thy!' . 1.4( 14111) .1 7 / - 202 .
f. . 21
::;-:--_:;A•,:.f.r'OH ! , v.' . :1 . Jr . n::d '.i . L . F.OOT, .kn i1~t.:oduct1on to
tt P tbeo::yof !"WldCm ~i~n.:-'l..!s a.'"lC t.~l~e 1 II,,._.. York : &·ic(f!'~iW
!:ill 1·~ <,9 .
2 . HJ..Ji::.-;;,:1l "tr.1 . ,,: . , 1r1nci1=:es of thfl !-:tolis~lC~l t:hco:ry o!
cocu:nuni c ."ti.or. . t f>;('H l..:Jr.f': • ~:cG.i.•r-iw -r!!ll 1c11 -: •
3 . t~•.1ir.;1s•ii.t:r:~, Ti . J\ . ~Jnd V. D . ZUJ>.AKOV, Ext.ruction of sie:nnla rrom noine , 5ne:l~wood Clirfs Nt.1: l'rl'lnLlc iial~ -;902 .
6 . 22
1:"fftu"lNUTH, n., P . J;; . $CJil1ID and H. S . JJUDLEY , 1-tult;iple acces$ communication wit;h binary 01.. thogona.l nine und cosine
pulses usi::g !:.env-y '1lllpliLaC.e cn;>r•ong, 1968 IEEE Il:t.
l:on:- . o::i Co~ur.icutionn Hecord pp . 7::,4-7'} J.
? . '!All VLECK, J . H., w:u.!. O. !'IIDDLE'I'Of; , ':'he ap,,cLt"amo.fclippe.d :wise , Froc . ri:;BO: !/1(1966) , 2 - 19 .
; . SUNlH., E . D. , ldf'al liinarypulsc t!·nncml.eaion by A/"! und
Ff". , Bell S:yste"1 To ct. . J . 38( 1959) , 1 ;b7- 14?C> .
h . :>;
'f°:AlKENS , A. J . and D. A. LEW:NSii:I , Evt\luai:;lon of :acssage
ci"c~it noise, Bell :iyaL<11l 'lech . J . 39(19,,0) ,!l'l"- 90') .
2 . Sl11Tli, D. i:. . 'Uld '•' . E . BRJJJLEY , rr.e th<'ory of imJ>ulse
noi:.;c ii;: iC.eal freq_'.l~ncy-.ciodulat:ion receiver:; . I-Toe . IRE
}4(1946) , 743- 751 .
3. BErn:E'.l'., W. F. . , E!!'cLi·ical noise, r.e..: Yor•.: NcGra'1- Ell
1960 .
S'l'!,~l!'EP:S , F . 1 ., On Lhe calculatioe. of J..C1pulse- noise
L:-:i.t1sie1"t;s ir. frcquoncy-codulai;ion recP.1varD, fhi l ips Re~Cllrch Repts . 2( 10'1'/) 1 4t,8-474 .
' . 111.EoJUTH , H. , Kodi~1·on uti t o r thogonolen FLUlktionen , Al.'chi v a I ,,Jc Ooort :-ne:ur1~~ 1'/(1 'J(,_;) , 429- 4j7, :;oo-.)113 .
11.
I
• ,,,,
'171Il~'nul;;;,
R. W. , Error detecLing and nrror correcting
codes, Bell Sy;;teu 1'ecri . J . 29('<950),1 1l'7-1GO .
2 . SLE.Pl/U; 1 J . , A cla.s:; o!· b=rary sl.gualing alphabets,
Bell Sy~~e:i l'ec!: . J . .55( 1}5( •,203 - 25'- .
3 . ~!OZEf, ~R,<.E'f, J . ~: . 11.lld D . J\E~Fl'El; , Sequ<>!ltinl decoding ,
N~;; !ode Ml'l 1':"<'"'~ ncu Wi!ev 19l1 .
4 . ·;ALLJ.:}B, ll.G ., tow-dendty parHy-check codes , Cn:nbr1.dGC , !':a~" · : IHT-Fre~s 19o5 .
c . ··1u-r.1.rn , D. E. , ApplicuLionof Boolean nlgt!b.t•a to switching circuiL o.lesi8·n anJ •o cri·or detection , J l!F. rra nsactions
on Electronic Compule rr C:C- 5( 195/i), 6-12 .
•J . .'E'J'ERSOh , W. \.I . , Ei·r·o1 correcting codes, New York : !-!IT
l'L'<Hir lill<l Wiley 1'" 1 .
? . -, Fi.•og-1"ess of i!"lJ'or:un--ion cueo!."Y 19i:.t;-l.J~, IlJIB T.r ansuctionr on !::.for:r.ullo11 'rh.,o:-y r ·r- 10(19c3),221-2r-,1t.
o. 1.:2'., C. Y. , So::c p·operties of ::on- binai·y en'Or cor:ecL:.!l§ ... codes , U~E !'r.fln:""ac~:.ous on lnforantioD '.illeory ll-4
(19:><>), 72-82 .
J. ULRICH, ''-. .
ilott-binary error co.t·recting codes , Sell
J.veLein Tect: . J . 3<(1~' 7,l ,-1541-13$8 .
1 ~ . REED , [ . $ . , A cln:'itt of m:J.ltipLc- errot·-cor1•i:cting codes
nnd the Uecotlln~ schono , IRE T.ranse..ction:-J on Information
Tlleory l'l'- 4(19:>'1J , 38-'19 .
1"1. 11iEISS, P . , Ol'!cr tli " Verwendung von 1dalnhfunktloneu in
~or Cod i e,·ungstiieorill, A.c·cbi v o.l.ek . tlber•trag\J.Ilg 21 ( 1 967),
2l.>~-c£58 .
• . ~::>
~OWiili, S . '.' . , L . 'J . ~ AUHJ::RT, f"'..F . :EAST:sRLIHG, J . J . ST!.FF-
1.E!l and A. J . VITERBI, Digital collllllunication11, EnglewooC.
Cliffn liJ : Prentice 11'•11 1964.
2 . H.AflNUTil , !! . , Orthogonai codes , FL·oc . IEE 107C(1960),
242-248 .
H.EFER:ENCl~.S
319
} . Al<ONSTEI:l, li. P. . , Comparison of 01·tl:o(!'onal an1
code~,
Proc . lEL
110(~165),~96~-1967 .
block
:.- . HSifJl , L and '1 . Y. ESIAO, Se•1eL'al cloc~e~ of codeE 6"ne!\5ted !rom ortt:ogonal fw:.ctions, IEEE Trn::n:iction£: o!l
Ir.fol'!Ultion Theory l'r- 10( 1';6'-) ,80- •1 .
S. F;.1;0, ll . Collltu.nieation i.r. t!it pr.,ae::.c" of ~Jdi~ive
Gaussio.r. c.o.iee, in Corn.mur:ica-cio!l fhco:-y, :1e1i York : Academic f''!"er,!'\
1955 .
6 . 1. .0.CllS , G., Optittization o.r signal wavefo!"l!ll!, l:EEL T:-;insnctiona or. Informntion T1'1cory IT- 9(19r->3) , )~,-97 .
7 . L'ALE'l, iL : .. , On ortl1ogonel ma~ricos, J . f'lnth . and Phy-
sics 12(1933),;11-520 .
a . S'!'Ai'!1'0N, n . G. nnd D. A. SPROT'l', A rami l y or dirrerence
set<> , Canadian J . of nacli. 10(1958),'1.:; - 77 .
9 . !JOSE, 11. C. tuld S . S . SJJRTIL.UIDE , A noLe on a t'eauH in
the theo1·y oJ: code construction, lnt'ornAtion and Control
2( 1959), 183-194 .
10 . f\l:.. D11AJW'r, L J nf or1lati0Dstl!~orit Wld au ~omatiaclie In-
~orma Uonavor:.u.·bei~uni::;,
Berlin:
V~rlng T~cb.nik
19G4.
2..:.22.
1 . WOOD, H., Randoc norJJal aeviates, Trnctn for Coaputers
25, London: Cambr~dge Univer~i~y Pre$S 1948 .
2 . l!S De1:a.rtaent of CoM~rce , f!andbook or aathea!ltical
fu::ici;ions, 1;ation&.l llu:eau o.r St=ia.rds Apµlied Nuthe"1atical Series >5 , )oi"asl:..i.nc.,-ton DC: OS Goverr112ent Fr in ting
OHic" 19r.1._
3 . The RA..t:D Corporation , k mill ion rnnclom dl.gitn 1<ith
100 000 norrr.c<l c.l.eviat;es , Glencoe Ill . : Tha !"ree Press 1955.
'1 . PErERSON , W. \I . J. Zrro,_. cor•rectini; codf· ~, Ne>1 Yo1-,c ~UT
Pren" oncl \.Ii I cy 1';164.
5 . EI.IAS , i' . , Ei·i·o:•- free codin g , IRE 'l'r<.1J1Auctl ons on In-
f ormatlon Tl1 eor;y
IT- ~ ( 19511)
, 29-37 .
6 . ?4
1. !IAJW.Urfl, If . , Kodieren mit ori;nogon .. 1 en Punctioncm , L .
Ko!Dbinntiona-fllphabete und r·:itimw:n-Energio-Jt.lptabel..e , l..rchiv olck . Ubcrtrngung 17(19..:;3) , 508- 516 .
2 . KASAGKs U., Kor=-e1ationse:opfang von liuchr.tabl"n in bi nhre1" bzw . ie.t·:.1Hrer Darstellu.ng l:ei Pandb~grerau.ngen und
gauJlschem Rou~cuen, Arc!:oiv elek . t:icrtraL;WJ& 22(1960) ,
487-ll'f~ .
Additional References for Second Printing
1 . ..l~F3W5, 11 . C . • .1 r• • L . C.l.S.h\R:, A ge:...~r -~z~<l tech..'li·1u"'
for .:..-:re~tral :L'1!'Jl:,: .i , IJ.,;o;f. 'l'r.?.n~ . on Coor.·ute:.·~ C-1 1~·(/')?0),
11.J-,".'l~ .
2 . /dfDREWS , H. C . tmd u - ~~A.Jn~ ? Xror.ccker ll:ti.LJ.'iCeQ , compute!.·
lrrJ;Jleuu;!nta.tion , und jSCrtcru.lized spectrn , J . A~i~:oeiation for
Vomi,utiq;; :·bc!:iJ.er;1, 111 r rint ( ec;heciulell ror April 1970) .
3-. AL:r:XA.QRIDIS, :; . A. , li1<daoar c trr.n,,,·.,1·m ir. template
u.ut~f y;attern !'r.cog.Hitioto. 1 I'l·oc . :.i:1•d ll:1w1Jii ll1t . Coc.1' .
on ~~:.-rtC:D Scie1lc~ (1<;r/ ),127 .
tl . CAH.1.1 1 J . '"' · 1 t,;t·::Jor·1li::ed .ar:nonic -u1 1lyEis !or :pa..,~er::..
rccoi;uition ' -~ bi nlogicr.tlly derived =ooel, Ma~tci•'" Tl:e:o:is,
Depar~:tent o!· Electric~! ::ngi.-.ccr::..t:g, Air Force Ir stitute
o~ Tech=;.o o;;:-, t ·rto1., 01.i.o 1%3.
~ - !:l.!:CEB , J • .'. . ru-1 ~••Jl.\RWIT , Ex:ue:t•i. entH l Of.cro.tiori of t!
H•1dem~ra :;rec~t'or.ioteI' , Ai-.. :'JlieC. Ortic!l U( 1r.,1•.-< •) 1 25~2-2554 .
Li • .P:GJI!-'OWSK1, R. I•' . , Viultlorthogonnl C.11·.. 1 · ~.::'..:l..nsriission
.-:'.-'r.tcm:. , Digec L 1 1)1 .•( 1EEE Int . Conf . on Corr:!rlunicat i o:is ,
Mi1U1eupo_ i ,; , p . 3;, .
7 . 1-"RANC3'0H~· 1 E. , f:..vt1 l.ur-tiolJ a.:.id or•ti ·ni zr1tlo!l of per l'or1.nnC'c c:ri°te!"i!:I ir.. ~ i 1!0r1l" !l:;:::tcaz co::ituinin1~ , uncort :.:.in
parao:ctcr . J·!:i,,tor ' :. 1'1.o.-is, J:oy.;;rtmeJJt of ·:J i.ctrical Ene;ineering, :-olytcclu.ic IL.stitute of Brookly::. 1~1...8 .
8 . GIBBS, J . E . 3.nd~ . A . GE3iEE, .11=;..Lcation of \·hlzi1 function~ to trar~ for:o :;pcctrozco~y , ·i,,cure 22'•( 1o/>9), 10121'>1; .
9 . GEES , J . E . 0 ne ~i.J . f'l:II.iLhRD, ;,.'a ... sh functio11s ~E solutions of a logica,I. l'!irrot:cntia!. equ?:tcion, DES P.cport ~·lo . 1
(·191)9;; 3or.:e neC1tOd!! Of so.:.ution ol' :.l::ett.L.' Ordinary logi~, l dif7crcnti·al equHt;ions, DES Rorort ~lo . 2(11)69) ; Sorce
l'ro;>erties of l'unctiOJJU on ~he non- negacive .L11~e~;ers less
b!nJJ ;::n , DES Report No . ::;(1969) ; Katioi:aJ T1cy:;ical Labor atoL·y , Divisio!l of 1':1 octl"ic,11 Scier:ce, 'l'eddington , J•Jiddlencx , England .
·1i. GCLO:'fC : 3 .. t~ . l!.:l l1 . D . jjhUJ·iLR'i , 1.rhc ~ear..;!::. .:'or li•ida:na.rd
i.atr·ices, )_'!er . M~t:, . J-lontlll:; 70( 2)( 19o5), 12-1( .
11 . lf"JGG:!l:s, W. 3 . , ReJ,ronont~tio~ !illd analyois o~· sig:ialn,
• nrt 1 : T'::.-, u"" o~ oi·thogor.ali::ce e:q;onenUale, roc::nical
P.e;.>o:-t. ;::, 20e1'~c19:;a) .
·1'> . ITO, T . , :{o'tE:- on n cln.::::; of :::tal;isticnl :-ecog!liti on
f\lnctio::,- , I!ZC T:-w1s . on Computers C-18(1~t·".l),7( -79 .
I ~ . KACZVi1'.RLJ , ~~ . nnrt li . f:·l'Er:n:AUS , ~heorie dt:r Cr~t-.ogonal
rnic.cn, New Yo.d; : r:Ji~ l "O'J ?-.ibl. Co . 19~1 ( or.Lgi.na Uy pub1 isherl ~.':Jl'.'!';Zt...o.;ri-Lw6w : ~lonogr;.,.!je t·l::.:Lerr,n tyczue , ll1 , 1935) •
·11• . liOWJ..LGZYK , :E . , Somn 1.roblems o:r nppl;vine;_ ox·tnoo;oual
runct.LO.:l" i n teLeCOlllJJUu.ications (in Pol.L~h) .
Rozprawy
l•:tel:ti•otcch.rilc"ne 1"(1'Y .- ), 11&9- 1+8') .
1"> . ~ ..i,. BAF.R::.: 1 ;: . B ~1. • , A transfor:n tectni,1uo tor linear,
t.i..u:.e - V:....l"~·i!'J.b, 1i-crete-tin.c cyste:?Js, fl.: .'1 eo ·i ,Depill"tme?:.t
or E~cc~rical r.n~ii:oor~ , U o:- l·:ichig•ui: nr.n ~i>or 196?·
1<. . tl!::LTZER, 3 . , •. • !: . SEl.RLE e..-id R . BROW~l 1 Humerical specitic·.tion or' biolob.cul for:n, :iature 21c(1%?),'i2-3f; .
17. 'IA!·:FL'-'l, £..K . , J.. aotc on tl:e Walsh !'unction", :;:ELE Tran,; .
ou J·:lectronic C0'1JlUl:Cr~ EC- 1;i("i964-) , 651-o'~ ·
ADJ IT lDllAL REFERENCES
324
18. ?ICHIZR , P. , Walsh- Fourier S~ntLesc ortim•ler Filter ,
hi·c!.iv de1· elck . ttbertragu..'lg , i!l orint .
19. SC,-:REIBER, H. H., Bandwictl: requlree1ent" for Walsh ;Ur.c~i.onP, J}:J:E Trans . on Info:::'m8tion '!'t,eory, in ?rint .
20 . };EARi.Jo:, !. .ti ., Shape analysis ~y use of\./• <1t !Unctions ,
hoc . 5th Int . Nach.ine Intelligence Worf.cho;. , Edinburg!:
1969; Edinburgh Uni versi;;y Press .
21 . SirAW , L . ur.d F . CJU:.ot; , An expans ion for ev•il u&t i ng s el'.is itivi~]i too rnn(lom parai:ietcr , Automntica 5(19(•9) , 265- 273 .
22 . SIE~US , K. and R. Y.Ii:'AI , Digital Wnlnh-l•'ourior nnnl;yni;:;
of oeriodic ~:aveforins , IE'.El' Tr:ms . on lnst.rument.&tion and
Viensurcment D·l-18( ·1959 ), December .
23 . STPJ>l AT , D . 1
Onsupervised lcnrninG of mi>.-tu.J.·ou o.: J_Jro1
\ .
,
babilit;y .rw1ctions, i n .Pattern Recognition , L. 11 . l<:<nal eel.,
lfaohir.gton D. C . : Thompsorc Book Co . 1968 .
24 . SZOK , W. C . , Wavaform charncteri"otiou in term,; of 1-lulsh
functions, i·bster' s 'J:hesis , De1wrt ment of Electrica l Engineerinr, , Syrucu~e U'1ive::·s i ty 19.,a.
?.5 . TAJU , Y. "nd l': . HATOR:, R:11 Con::nunic~tior, nyc.teo using
HadrutA!'d trunrfor1llation , !:lect:ronica ar.d Comnunications
in Japan 119(11)(1%6 , 247- 2'?7 ?6 . W.IC!:JL~I , C. , On decouposi-tion 0£ W; lsi:.-Fou.rier series;
Mu!tii::lior:: !or Wo:sh- Fourier ~eric!: ; Mcr..n convergence of
~Nals.!1 -Foui"ie: series; Best ap~xi.!'letion by Welsh polyno!tia2.", Tchoi<u ilutl. . J . 17~1965), 76- 8< ; 1.,(1CjGI•) , 2;;0- 2c;1 :
16(19b'>),183-188; 15(1963,,1 - 5 .
27 . 1,10NG , to; . und l:: . f.ISE!iliERG , Iterative ~ynthc:iiooi"tlll·o"
hol d tunct.ions , J . f'>util. AnaJ.y,.is and l.pplics.tions 11 ( 1965) ,
226-2?,'.'; .
28 . '!AHO , s. , On lhl.Gh-l•ourier serioc, To ho\ru MMtb . J . , 3(2)
( 1951) ' 22!-2'•2.
S:vmnoaiwr.., on Applications of. Walnh Func1;ion".
Jo'or a rerort on tlie first sym;.•os iuJtJ ( ;>O lfoy 19L>8 , J'crnnoldetecll!liaches Zontrnlamt der Deut ncl1on llu.ndeero.ot , Darmst,,dt , Wo:it Gor:rrnny) see li . h"lJBt!ER , Sequenztec!:..uik : Bericht
iibor eine Diskussionst~ , !:achriclltcntcci.:.::U:;cLe Zcitscb'i~'t !ITZ (1968) , J<o . 7 , IWO .
~he following papers 1;e~e Pl'e sente<! at tbe i:ocond ~yn~o-iw:;
(1l.pril1969 , ?:aval Researc!: :.:;:iorotor;;, Wa~hington D. C. ) :
1 . A::JREWS , 11.C . (:left . of 3"-ectrical Engir.coring , U of
Soutt:orn C~J ifornia , Los Angeles) , OrthOt".Onnl .!'unction decott;iosition fol" data i:rocessing .
2 . 11/JU".UTH , II . , Ap;ilications o::: Wr.l;:,t "unctioa" in co=ui:ic&tion!" Crubliohcd in !EEL Spectrum 6( 19· ')) ' r:ov. ,82- 91 ) .
;;. s,;:my , (; . F . ('l'he r·:r-n<E Corp . , act.ean , V.L.t·g1.u1") , Specul ot iona on 1;0,,sible applications .
h . VAND.LVJ::ll.b, 1':. F . (Tolcom kc . , :-Ic Lean , Virgin.Lu) , Signal
symmet ry o.nd J.ogica.l structure of' Walsh fu.Dct.Lon" .
The following ll<>Por s a.re s cheduled Iol' t he thii'Cl sy11po>iium
and work ebop (31 Mo.rcl1 to ;; Apl'il 1970 , Naval Reeea:rc!J
Lnborato1'y . Washington D. C . ) . Proceedings will be published
ADJITIONAL REFEREt:c:::s
322
fo llo 1Ji ng tl1e meet il1g and will be avail1.:tble f:r·oo Code 51+35 ,
1
Naval He search Labora·tol':Y, Wasting ton D. C. 20;;90 .
1 . l[A}fDI'IERE , E . F. , J; flexible \.falsn filter design for sii:;rtal s for moderately 10•.·1 ,;equency . 2 . LEE , ·:!' . , Hard•are
approach to Walsl1 functions sequency filters . ;; . RO'l'll, D . ,
Speci al fil te:r·s blised on. Wal!:ll:: !'Ul':.ctiOn::! . 1, . ?IGEILER , 1i' . ,
1,.Jals.11 i'unction!1 and optima l linear system~ . 5 - DAVIDSON ,
I. A. , '£he u,se of Walsh functions fo::! o;ult;iplexine; s i gnals .
G. LEE , J . D. , Revi eh' of i'ece:nt h·ork or:. r;i1)plicutiOn!:: of
1\ra1~11 fwiction.z in communication a . / .. EAGDJ.-8ARJ.U'~Z , P. and
l~ . l.OJlliTJi.N , Theo:'etictl >Uld exoerimental studie s of a scquenc;y multiplex uystem . 8 . lrti'lHIER , H ., Ou the tranwnisGion oi' Wabl1 multiploxcd s i gnP.ls . 9 . SCHREIBER , !l.!i. ,
B~ndwidtll
E' . J .,
requirements for WeJ.sl: .functions .
\~alell
10. Iif::BER:r ,
£unction generator i'or a million differ·ent
:'un.ct i ocs . 11 . \·llLI.·lCER , J . E . } Parrunetri c amplifier based
on hra1s11 ..t'v.J1otion.s . 12 . PERLi1.~J; , J ., Radiation patterns
(or antermas >:iti1 1.falsJ1 cui·rent inputs . 13 . HAi·IBL'.:R , lLK.,
Approxima'Cion ru1d Repre!:ientat i o~i of ,j oint p:-obo.bility
dist.t·ibution of bi ~'lary randou: -....ari ablee by h1nlsl1 f'u.nctior:s . 1it- . Al:IDREh'S , H. C. , Degree of .freedom and computa i;ion requirements in mat'i.'C mul ti)'.ll i c&i;ion for liadam11rd
and o·l;her t:i:·artsfol:·ms . 15 . i,'Jlli-LCHEL, J .E ., Properties of
mixed r:!d~: fnst Po=ier-Frada:nard tr<'msforms . 16. LECffiIBR,
R. J.• , Invariant properties of Had..mar<l. transforms wider
a ffi ne groups . 17. Sj>AJU,f>, lL il . , A ' logical ' \~alsl1-Fourier
transfo1~tt. . 18 . GEBBil , 11 . JL .. , \.,raisr. i"u.nctior:s und the expeximentul Epoct1'o scopi st ..
19 .. DECKER , J .. A. , J!ad!ll!lard
spectrometer. 20 . GIBES, J . 8 . 1 Discrete complex Walsh
functions . 21 . OH:}TSORG, F . ~ . , Application of Walsh func-
t ions to co:n.pl e x sigrutl!l . 22. ITO . T.. , Application of \·lalsh
functions to pattern 1~ecognit io11 a.11d switchi11g tbeo1....J .
25 . BHO\·nl , C . G ., Signal processing t;ecLniquei; using Walsll
functions . 2 11· . CP...'lli , J . W., An npplicatio:!l oi' Walsh funct ions to image clasoification . 25 - P.APJffif , w. A., Digital
image processing aspect s oI t he \·!al sh transform . 26 . K..•_'IE ,
J . , '."latrix inversionbyilalsh functions. 27 . \\'ELCTI, L . R .. )
!in.damard mo.trices and. llalsh functior1$. 28 . 1./P.T ARI) C. ,
.i\pproximci-tion of f'unctions by a ~·lalst.-1',ourie1" sei"\ieG . 29 .
Flr:B , r·; . , Walsh- Fou:r·ier trans~orm . 30. PlCiiLER , F ., Walsh
functior.::.s rw.d linear systea theory .
31 . A.NDRJ~11!S, H. C. ,
Di gital imago processi:Jg . 32. CASPARI, K . , Generalized
Gpect;rwu anal ysi s . ;:;;:; . HARMUTH , IL , Survey of analog sequency filter~ based on 'tlals!l functions . 34-. BOESSHETTER
C. , J._11olog sequ.-0ncy :wulysis and synthesi s of \roice s i g n,. ls . 5~ . CAl·l.!'MU:;LLA , S . J. a."ld G. S . ROBilfSON, Digital sequenc,y decomyosition of voice s ig11.als . 35 . HU'BNER , J( . ,
Ann.log and d i git::tl multi p lexin;; by means ot: W<1lsll functions .
57 . HAR.'lUTH , H. , Electroll!agnetic Walsh waves in commnnications . ;;a. GIBBS , J . E. , Sine 1<aves and 'tlalsil wave" in
ph:;· sics .
Index
171 continu«i;ior1 of .f;.mctiom: 27
134 , 268 con~inuous vnrinhlc
188
ndditio:1 ttodulo 2
2C , 25 corrclntion fW>ction 15?, 214
air~raft collision
162 cor!"elation cofl!'f!'icier.t 212
8.llp:ilude clippin&
270 co set
120
runp:itudo nnmplinK
122 cosin~ che.nriel
135
angl 0 diversit,y
238 covarianc.,
211
anr,le men~urcment
173 croornt:il k
120 , 132
nnt.-..nnn , act,ive
17 1
- ai;termu~lOIL
1 05
aAtronomlcul ~ elescorc 175
- oat!'lx
')4
aLLenunLion r.oo r ri cient 9>
ouclio nic;n11l s
222 O cl ay mat rix
91+
avo ra t:e oscill1;tlon
deneH;y fu.nct l on
188
perl.od
4 , 166 dl!'l'rnction rro ~ 10f•;
2?2
avernc" .,,nv"l rni:;t:n
11 diode quad mul ~ i J,lier·
78
nxion .."' o f "'l'Olial;i:l t.y 18.Lt diJJo:r.
~'10
- mo:nont
161
Bann ell 1 _ '":l11 Orf':u
13
- v<'ctor
161
bn.lnnc~J nyr.te:1
26' ci:::tortion ft'•.•e line
87
5e~noulli cistributioL 19() C.istribuUon ru..uction
18L
- 1telhoi.!
'38 <iist~·otution den~ity
188
- ro l yno<r. I el n
9 .:>or·nler effect..
172
Bc!1!':Cl t" :.~ct ions
203 dyadic correlal1on
53
- inequ,.Jlity
11
- p·oup
2b
Vinu:::; chru·uctc1·
C6
- :_·ntionr.il
LJ
Activ1• nntl'nne
ac~ivity factor
.,~
- s!".iJ't
hio-rt!-:Ot~On~ l
codes
Do l LztntUlll .stn ~iBtic
·a1o~k
Born! mo"1:-ur;.1bl o
Ca'Jcll;y ' .s
148
t~'lCO l'~r.I.
prindp~l
val tH!
,,7 ' 280 E ig-0n!ur1ct ionn
277 electrical l ,y nhol·t
218 enerc;y dis Lwic c
18'/ er::J:iea:bl e nv0rnec183 cq_ur::. gain nu111mc1t. ion
ergo<ii c b :v po t;he :;i.,
Gauc11y d isii·ibu Lion
c ent i·r;J. 11'.H Lheore:o
cLw:u1ol routing
character group
ct.a rae it~ri:l t.ic
func::ior.
c i'·cu. nr 1:01 ari ::11tion
clo~od syste:i.s
coaxinJ. cntlc
codt oo<!u l nt i o:-.
colli!'::on wr1J.~r..int,;
combi r10 t.ion
companuor·
coa.pletcnccn thq11·re111
compl et<" n;y:ltoinn
compl'eLif·JO t'
er1~oi.· COl.' L'OCl.10 11
35
- de Lf'ctj on
- functio n
?06
197 .Euklidiar Ap,.ee
127
?6 Fcr:ni ~::•t~~1.l c
.i·0 !"!I. ;'l!l t;: s
1'?3
- ,, ;:;equeiaC.Y
16<) .fourtl:
12
88
15')
16.?
181:
270
12
11
octt.od or SS."
channel
~rcqucr:cy
- di ve1·s~ly
- divi!'icn
- .filter·::
- li:nited
- rr:o<'lulaL1on
- Pl1 Lft...ing
- s ,ynL l1esi:.:1·i·
- t heo1·y
comp1•oflnion or in.tor- t rac kiHC: J'il1iC!'1.'
inution
45 .function detector
coJJd i tionC<l p1·obabili t;y 186 .fu.nc Lion l imHe<.l
55
87
:086
2"-'/
238
2'•7
279
279
1 ')0
183
218
91
~21
"jl.J.1
., ; r:
2~~
Oc
5f:
58, 2'•9
155
181
7(
~'.:>
1 1~7
79
250
Q. ~,·~i&h
19,; w.:.i~ber thca:-y
213 t:yquis~ rate
p;co1tctr1c optic"
1'/o
g!-oup code
2ac. 0 :.-ofr system
- delay
2o1 or<'n ;;fr,.. lin"
- ~heory
1?.0 oµerulor, J.ifferential
- , cie;c:!l'anc~ions
Haar- Fourleo· L.:·.:i.ns:"om
4G - , l i nnar
!:olf adder
?1 , 76 - , ti!n~ v:u·iable
!foll mu~ tiplior
78 optirnl t< l•·ucope
Harr.mine; d ist;once
2'/8 opt lrr.ul r•o I cct con
Hankel l'\lnction '191 , 201 , 203 oi•tho1;;0 11 t1l di vi:,;io n
llc 1·mlte pol:ynomia I s
18 outph11s.Lnl" 1t1eL!il}<.l or SSM
<lir.tributio=i
fiEll'tr;im 1 Jlp<lle
277
82
265
87
55>
75
54
54
260
238
62
1'1-1
163
Pru·ubolic cylinder
110
l"u:.ictiOJl3
18
inco~p~~tc syoLe~
12 , /2 y.,1·•'bolic reflector
171
intebr~l, 1r.lali:::h i'u.nct.ion 1G4- l'"r:m.etor incrc;;-ation
1 99
iuter:r,1tor
70 pa.rHy Cl!UC<! digit
278
int.cllig..,:it interference 22~ Pnr~C"V!l~ !I theorea.
12
intar~rction
18~ µartin! r~~pocue
8~
io t. r•tal
183 ?Cl'.
11f. , 159
periodic contin"'-at1on
29
JohntK>11 uoise
21& pne-" cht1.U11el
135
'"·rint. di~-ri':J.utio:i
1i3' - ir.odulation
157
- shift mc~hOd
1}7
Ler ml re 1o 1 ynorr.ia! s 9 , 38 - JUl!lpa
85
liHf:''ll' 1nC.upende:?cc
6 , 2'13 ?1 a.ncht.· !'I;' 1 th eore:n
14
- opei·a~oi·
162
54 Poi!l:ting •ts Vt!c:.or
164
Lorent~ trnnsl'or..matio!l
1'/3 µoltt.ri ~n<'l \..'olt:h wn .res
lowei· sideband
108 powe1• loudtni;
85
pI'ism
222
M•T'(!;iMl dit;tri buLi<l:i
18> probabi I ity , uxloma 18Lf , 185
moth~mnt1c11l expectation 189 - , dnfinod
18'<
maxin1ul l"Otio nummo.t ion 238 - !UJIC Lion
185
l'lnxwdl ' "' C'QUttlions
1c: prod u c L of i·andom
mcnn ~qunr& deviation
19~
vw:ilibles
199
261
mettr \'1:tl Uf'
191 ;.iropa.get i or. t i:oe
cixeU nio:r.en;
?11 pol~.ri~~cio' di vor=ity
238
- vector
259
mobil" radio
l mnge
~ignal
1
1
corur.unic3.t.:.on
11odifie<1 Hankel
f·1nction
11odulet.1on index
«ol~-o 2 ~tldition
16/
19'o,~O"
155 , 1S7
Q .lac! •a-Jrc modulntion
n~oticnt.
20 . 2'> -
rr.OJll<·nt..
114
123,138
ol" ra: .. -Jom
v•u·iables
204
191
268 RutlemachC"r runc:.lons 19 '121
166
2? r~1d ia 1.. ed pow~r
mult.trliorn
77 , /8 rodintior. 1•esi:;::::ancc 163 ' 165
rndio commu1ticotion 1
167
N 1ttlJ' zOJIO
161, 162
mobile
82 , 271
Noutr.nr:.n. !'unct;ior.!1
203 r.si,,ou coolno pttlsc
180
non-ayuc.11.t.•onizod g roups ""t2'1 ra<lar I 111•g1•L
231
r.o r•mnli~i::d ayctcms
G receiver fllto 1·
:rnl~i1>le ~cceos
xul~iplicu~ion theoremc
325
DIDEY.
rclabivistic inecha...ajcs
random alphabez
- ,_,.ariablc deJined
1 '/2
T ar~eu
anal ysis
28() tretcking
1 80
1 80
68 , 280
1 83 transol·thogonal
87
201 i;elcgrapher ' s e quation
8-<•
237 teletype Lrans 1..iiss Lion
85
r ectangular refleccor
1 7'/ '!'EL.EX
R"er,l-r1ulle1' alphabet
280 t!Jermal noise , clcfinicion 21 8
resoluti.on r e.nge
141
17" t hird metl::oC: of SSM
r·eHo: vuble angle
51, 1 55
1 /5 tj,me base
rioc time
1 21 - diver :>ity
238
rol l - o .i'.r .factor
6 1•1 '130
271 - division
time- frcquoncy-domaiJ1
21.~9
21-19
S aop:inio tlleoi·err.s
71 , 9'/ time- funcr;ion- domain
scalar 'Potc nt iaJ
160 t iroe- sequenc~-domai.o
249
'167
Sc.'.:ll:tiu iuul tiJllier
79 ti :ne- sh i ft s
second mei;ihod o!' SSl'!
1 37 topolor:;ic i:;1·oup
26
111-'t
RBGR
1 83 transposed SS~l
Sequer11~,y alloca t ion
1 24 two- dimensioni>l r ilte1·s 1 05
- bnndwidtl1
99
- definit i on
~ Unce:r;;ain cy relation
25'
- filters, 2- dimcnsional 105 J.nsyncb1·onized g r·oups
1 28
1 08
- l'o 1:·mant.1:1
91 1 221 UPI'er oidehand
- limited
58 , 249
1 9.)
- mult iplexing
115 Variance
1 60
- 1'e:>poase
99 vector potent:la:
62
- s'1if Llt<g
1 81 - repr ost:nta tion
- Epcc~=·n
1 01 voice 3ign<t l G
90
- trockinf) :'llte:'
151 vocoder
91
si~nal clftssi!'icai;ion
1~5 vol tae;e coc1pai•i son
229
- det~ccion
225 vestigial SSl"l
- delay
261
- space
6?. , 63 , 60 Wo.l.i:;h fwic.Lions , ir~tegra! 1oL
sbii' t t;t·1eorer.i 1 s :.nc 148 ,1oSB - multiplier
~;6 , 77
- , Walsh
27 , 148 ,150 - t:racking filter·
1 !>4
single sineba.nd
107 ~ 1CJ8 - \\·aves , polarized
1 69
t:-l multuneous
wa\•e eq11at;ion
89 1168
tr.ansmiss i or.
83 - gtl.ide
250
siue ch•..unel
135 - optics
1'/b
skit< efl'ecL
88 - zone
161, 1 62
Rnylelgh disLribuLion
space divo!'sity
- probe
special shi f~ t heo r.,m
speech unal ;ysi s
stan<llnts w:ive
nt~ I, i sci.cal i nde11e nclcncc
- vu_:_·lable
Stuacn~ di!:'itribution
SWIJ <>1' i·~tloru 1JaJ~ i al1len
S\tperconduct.ive ca°Dle
s uper1,o;rOltJJ
swiL.ched tele1'hone
net \.:ork
synchr onization
systematic co(le
238 weak co.n ver.gence
3)
171 wideba.ntl anl.ennt!
1 65
1L1 ~! i.·! icncr-C!-_inLChin Lheorem
17
)1
1'/0 zps defi:'led
186
183
206
196
88
125
87
121
2'1') , 280
50