17 Describing hndoxn Signals 422 Next wc extend the idea of stationary from Defiirition 23 t o cover two randonr processes We call two random procescies jornt ~ ~ u ~ if ztheir o joint ~ ~ second~ , older exxprctccl V ~ L I only ~ S depend on tlie difference z = tl - tz For joint stationary random processes the cross-corrrlntion function then takes a form simifar to (17.15): c p & ) = E ( x ( f t Z).yft))=:E{z(t)-yi-d- z)) ' (17.46) Finally WP introduce the second-ordw joznt tzrne-average and call two random processes for which thr @hit cq)cct.ed valws agree with thc joint timwiverages joznt crgodzc There ?axe also weak forms for joint, stai Liona.ry atid joint crgodic randovn p r o c ~ s s ~whcre s, tlic corresponding conditions, are only fulfilleci for ~ ~ ~ ( ~= ~ ) ~ ~~~ { (~ ~~~ ~ ~ =~ ) 3;(t1) { ( a~ d~ ~ ~ ~ ?dfz) .~~.~(~~~ =, ~ ~ ~ ) } Thc cross-correlation function performs it sinrilar fiirictiori for two random processes Illat the auto-correlation ftinrtion does for one random process It is a measure for the rF~atio~sliip of valnes frorn the t~ or a r i ~ o praccss i~ at two timcs separated by z The extension to two random processes cat1 some (jiffererices t o the anto-correlatioii function Firs! of till, two rimtforn pror.es~scan be uncorrefateti not only for large timespaiis hilt also for all ~Tlfuesof r Their cross-correlation fiinctioii is then the protlurt of the linear cxpcc%Xlvaluos 1-1~and puyof the individual random processts: Y.&j = i*L Ply v r (17.48) - TIicre i h also the caiw that two riuidorn processes art? riot uxit:orielated for all r, brit at least for 174 30: values of F t ~ r ~ , ~the e rcros4 ~ n corxelatiori ~ ~ ~ ~ ~function clors not have the even s ~ ~ ~ ~ e of the auto-cmreiation functioa, as from (17.46) and s-cvagping n: and g, wv omly obtain The ai~to-rorrelat,ioir function pTJ( z) can be obtained from tlie crosscorrektion €ianc+ion pfy(7)% a special case y ( t ) = it) Then using (17.50), we cmt find Ihe syiiimotry propprty (1736) of the auto-correlation function