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 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 = , and i t can be exprcssed by the variance and the inem j c T : itlt I ~2 + pr2(r) 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