Wiley signals and systems e book TLFe BO 438

423 17.4 Correlaliori Functions f "14.J II C r ~ ~ - C ~ v~ ~i i~a t~ i~~~n~ The c*ross-correlationfuiiction can also be forrned foi aero-mean signals ( ~ ( -t p7 ) ) and ( p i t ) - p g ) slid this leads to the cross-couanance fur~rfto.n (compare Section 17.4.1.2): (17.51) li/zy( = W j f f - P T M t - - P y H * As with tlie auto-covariancc fuiiction in (17.39), the calcnlatioii rules froin Section 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 ~ the ~ c evarious d 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*omplcxsample fiinctions 'Fo introduce the correlation fui-ictions foi torrrpkex signals we proceed differently to Secliori 17.4.1 There 'cve started with the aixto-correlal;ion function and introduccd the cross-cc~rrelatioiifuiiction as a generalization that conhined other cox relat ioii fririctions (cross-covaiiance auto-correlmtion, aut o-wvariance) as spc'e h ca5es SIere we start w i t h the c ~ ~ s s c o r r c ~ afunction t i o ~ ~ for ron-iptex signials arid derive the othei correlation furictioiis from it To this we niust ~tbsu~rie that a ( t ) and y(l) represent complex random processes Ihai are joint weak stationary 17.4.2.1 ~ ~ ~ ~ - C o Function r ~ ~ ~ a t i ~ ~ Thcrc are several possibilitics for extending the cross-correl;~tinr1function to ccnw complex random processes 'CI'Pwill 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 ~ ' ( ) is used For real random processes (17.53) becomes (17.46)