hndoxn Signals Next wc extend the idea of stationary from Defiirition 23 t o cover two randonr processes.. f t Z.yft=:E{zt-yi-d- z ' 17.46 Finally WP introduce the second-ordw joznt tz
Trang 1422 17 Describing hndoxn Signals
Next wc extend the idea of stationary from Defiirition 23 t o cover two randonr
processes We call two random procescies jornt ~ ~ u ~if their z o joint ~ ~second- ~ ,
older exxprctccl V ~ L I ~ S only depend on tlie difference z = tl - tz For joint station-
ary random processes the cross-corrrlntion function then takes a form simifar to
(1 7.15) :
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 ~ s , whcre tlic corresponding conditions,
are only fulfilleci for ~ ~ ~ ( ~= ~~ ) ~ ~~~{ (~~ ~~~~~= ~)3;(t1) a d { ( ~ ~ ~ ~~ ) , ,~ ~ ~ t ~ ~ ~
Thc cross-correlation function performs it sinrilar fiirictiori for two random pro-
cesses Illat the auto-correlation ftinrtion does for one random process It is a
measure for the rF~atio~sliip of valnes frorn the t ~ 7 o r a r i ~ o i ~ praccss 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~s can be uncorrefateti not only for large time-
spaiis hilt also for all ~Tlfues of r Their cross-correlation fiinctioii is then the
protlurt of the linear cxpcc%Xl valuos 1 - 1 ~ and puy of the individual random processts:
TIicre i h also the caiw that two riuidorn processes art? riot uxit:orielated for all
values of r, brit at least for 174 - 30:
F t ~ r ~ , ~ e r ~ n ~ ~ ~ ~ ~ the cros4 corxelatiori function clors not have the even s ~ ~ ~ ~ e t r ~
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 cross-
correktion €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