Wiley signals and systems e book TLFe BO 438

423

17.4. Correlaliori Functions
f "14.J. 4
II

C r 0 ~ ~ - C ~ v~ ai ~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( 4 = W j 4 f f - P T M t - 4 - 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 1 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 do this we niust ~tbsu~rie
that a ( t ) and
y(l) represent complex random processes Ihai are joint weak stationary.
17.4.2.1 ~ ~ 0 ~ ~ - 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 ~ ' ( 6 ) is used. For real random processes (17.53)
becomes (17.46).