Wiley signals and systems e book TLFe BO 439

17. Describing Rnridoin Signals

424

In geiicral, the cross-correlation function for complex raxtdonr processes i s also
neither symmetrical nor comrtiutative:

For uncorrelated rmdoin process we obtain

17.4.2.2 Auto- Correlation ~

~

~

t

i

o

~

The auto-correlation fuuvtion of a complex raritfom proccss can be obbiried from
the cross-corrrlathn furlclion as in (17.53), for y ( t f = rc(t):

As in Ihe red case, tlie a ~ t o - c o r r ~ ~ l fmctiou
a ~ i ( ~ ~ icolisifits of a symmetry relation at the transiliori fiotn r to - - E . We cmi obtain it dirrctly from (1’7.56) by
substilir(irig t’ = t ir and by uhiiig tlir c~alcuiattiomrules for conjugafr cornplw
quxitit ies:
pf&L(Z) =

E ( z ( tf r)*c*(t))= E{T(tf).c*(t’- z)) = E ( [ x ( l f- .).*(”)]*)
= [I? { r ( t- z)nqt’>}l4 = p;, (- r)
( I 7.57)
I

or inorct concisely

(€7.58)

The corijugitte syrnrnelry here van be recognised from (9,4i)),and is expressed as
r w n rcid part and a;ll odd imagiiiaxy part of pLZCf
r). For real random yrtrt-esstxs
the imtcgiiiary p x t of p k i ( T) i s 7 c and
~ tho cvcn symrnctry holds accordiiig to
(17.36). h aiiy case, the odd iruaginary part tlivnppears at T = 0, so for coinplex
random processcs, pS5(0) is also purely real. With I h e same reasoning as in
Swtinri 17 4.1.1 it bolds that the ~ i i a ~of ~p7~I ( ri ) its rnaxirnd
~ ~ ~ at z = 0 , and
i t can be exprcssed by the variance
and the inem j c T :
itlt

I

~2

+

pr2(r) 5 p&%(O)

= E { . c ( f ) T @ ( i=
) ) CT: 4-pTp; = 0-4 l p r 1/2 .

(17.59)

is in general eomplcx for a complex random pmctw, tltc
J&”hilcthe ineau
variarm is atways a rcal qnant ity

as tlie squzire expected valw is formed in this cane with t h e magnitrxilc-.squsiecl.