Wiley signals and systems e book TLFe BO 438

In many applicat,ions, however, complex sigrials will appear as in tlie sigiisl transmission example 15.2 In this section we will therefore be extending the use of correlation functions

17.4 Correlaliori Functions 423

f "14 J I I 4

The c*ross-correlation fuiiction can also be forrned foi aero-mean signals ( ~ ( t ) - p7 )

tion 17.4.1.2):

C r 0 ~ ~ - C ~ v a ~ i a ~ ~ ~ ~ ~ i i ~ t i ~ n

li/zy( 4 = W j 4 f f - P T M t - 4 - P y H * (17.51)

tion 17.2.3 yield tlie relationship between cross-cc)vmiance function 7$zc,( z) and cross-wrrriation function plw( t)

When 'cve ~ r i t r ~ ) c ~ ~ c e d the various correlation functions in Scction 11.4.1

to rttal signals for the sake of simplicity In many applicat,ions, however, complex sigrials will appear as in tlie sigiisl transmission example (15.2) In this section

we will therefore be extending the use of correlation functions to complex random processes These are rmdoxn processes that produce r*omplcx sample fiinctions 'Fo introduce the correlation fui-ictions foi torrrpkex signals we proceed differ- ently to Secliori 17.4.1 There 'cve started with the aixto-correlal;ion function and introduccd the cross-cc~rrelatioii fuiiction as a generalization that conhined other

e h 1 ca5es

SIere we start w i t h the c ~ ~ s s c o r r c ~ a t i o ~ ~ function for ron-iptex signials arid derive the othei correlation furictioiis from i t To do this we niust ~tbsu~rie t h a t a ( t ) and y(l) represent complex random processes Ihai are joint weak stationary

17.4.2.1 ~ ~ 0 ~ ~ - C o r ~ ~ ~ a t i ~ ~ Function

Thcrc are several possibilitics for extending the cross-correl;~tinr1 function to ccnw complex random processes 'CI'P will clioose a definition that allows a particularly

straight, forward intcqwetation of lhe crofispotver spectrum There are differ mt

definitionh in other books (for example 1191) According to (17,4Ci), "e define the c~oss-cor~elat,i~)n fuuctioti for corngkx random pr webses as

(17.53)

The only difftw.rel7ce to tlie definition for real random g)rocesses is that the conjugate complex ~~11ctioi-i of tiinc ~ ' ( 6 ) is used For real random processes (17.53) becomes (17.46)