17 L)cscribiiig Raiidorii Signals 418 use (17.25) on c p g g ( r ) : t,he ACF of the signal is prCT( z) -cp!i.~i:(O) ip: = ~p,,,~(0) + 21.1; The value pzz(0) can also he expressed by the variance C T ~nix1 tthc lirieas mean pT;,in nc:cordance with (17.4): If we -Vm(O) + 2pz = -4 + p; vm(4 cpZS(0) = 0; + ,2 (17.28) ‘ In the relatioiisliip (pZz( r ) cpz:c (O), the equals sigii represents the case where ~ ( tis) a periodic function of time If the shift by z is exwtdy equal to a shift, by one period or a nirxltiple, then s(t)z(tJ- c) = x ( t )and also ( P ~ : ~ : ( Z= ) yzZ(0) There is 110 sltift,, however, between ~ ( tand ) x ( t -t- z-) for which the expected valiie f ~ c o ~ iEi {~xs( t ) x ( -I- r )1 > E ( z 2( t ) } Xf such an efIect! i s observed when measnring an ACF, this Tnea,ns that the pre-conditions for a weak stationaxy process are not, thrc A further property of the aut;o-correl;ttion fuiiction is it, sginnietry with respect, to ZT = As (.he value of pz!r(z) only depends on the displacemciit between the two fiinctioiis in the product :c( (t r) we can subvt,it,ute = t z and this yields -+ E { X ( t ) X x : ( t %’ + + T)} = E {s(t’ - z)x.(t’)}= E { X ( t ’ ) l ( t ’ - z)} (17.29) (17.30) is then iaimediately obtained For the Lehdviour of the auto-correlation function for r -i CO, 110 general statemrnts can be made In maiiy cases, t h e is no relationship between distantly separated v~tlties.These values are then said to be uncorrelafed This property is expressed in the expect7edvaliies of the signnl, such that thr serond-order expected value is decomposed into the product of two first-ordei expected values: E {J’(t)Z(C - z)} = E{+)} * E {r(t - T)} As the linear expected value of a si atinnwy signal does not depend E { s ( t ) }= E { x ( f - z)} = fL, (17.31) jz*/ -+OG 011 time, (17.32) and therefore f‘or the airto-corrc?latioiifunt.tion ip,c(z) = r-19 IT1 + 33 (17.33) For signals tvl-rose values are uncorrelatd if Pm apai t , the only relationship between t hmn i s the (time-independent) linear average p Figure 17.6 shows a typical auto-correlation function with the properties just tliscixssetl: