h aiiy case, the odd iruaginary part tlivnppears at T = 0, so for coinplex random processcs, pS50 is also purely real.
Trang 1424 17 Describing Rnridoin Signals
In geiicral, the cross-correlation function for complex raxtdonr processes i s also neither symmetrical nor comrtiutative:
For uncorrelated rmdoin process we obtain
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 a ~ i ( ~ ~ i fmctiou colisifits of a symmetry rela- tion a t the transiliori fiotn r to - - E We cmi obtain it dirrctly from (1’7.56) by substilir(irig t’ = t i- r and by uhiiig tlir c~alcuiattiom rules for conjugafr cornplw quxitit ies:
pf&L(Z) = E ( z ( t f r)*c*(t)) = E{T(tf).c*(t’ - z)) = E ( [ x ( l f - .).*(”)]*)
= [I? { r ( t - z)nqt’>}l4 = p;, (- r) I ( I 7.57)
or inorct concisely
(€7.58) The corijugitte syrnrnelry here van be recognised from (9,4i)), and is expressed as
itlt 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 cand 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 1 7 4.1.1 I it bolds that the ~ i i a ~ ~ ~ i t ~ ~ ~ of p7 I ( r ) i s rnaxirnd at z = 0 , and
i t can be exprcssed by the variance ~2 and the inem j c T :
pr2( r ) 5 p&%(O) = E { c ( f ) T @ ( i ) ) = CT: 4- p T p ; = 0-4 + l p r / 12 (1 7.59)
J&”hilc the ineau is in general eomplcx for a complex random pmctw, tltc variarm i s atways a rcal qnant ity
as tlie squzire expected valw is formed in this cane with t h e magnitrxilc- squsiecl