Xf such an efIect!. i s observed when measnring an ACF, this Tnea,ns that the pre-conditions for a.. As .he value of pz!r z only depends on.. For the Lehdviour of the auto-correlation fu
Trang 1418 17 L)cscribiiig Raiidorii Signals
If we use (17.25) on c p g g ( r ) : t,he ACF of the signal is prCT( z) 2 -cp!i.~i:(O) i- p: =
~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):
-Vm(O) + 2pz 2 = -4 + p; 5 v m ( 4 5 cpZS(0) = 0; + ,2 ‘ (17.28)
In the relatioiisliip (pZz ( r ) 5 cpz:c (O), the equals sigii represents the case where
~ ( t ) is 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(t J- c) = x 2 ( t ) and also ( P ~ : ~ : ( Z ) = yzZ(0)
There is 110 sltift,, however, between ~ ( t ) and x ( t -t- z-) for which the expected valiie
f ~ c o ~ i i ~ s E { x ( 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,
t h r c
to ZT = 0 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 t o be uncorrelafed
This property is expressed in the expect7ed valiies 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)} jz*/ -+ OG (17.31)
As the linear expected value of a si atinnwy signal does not depend 011 time,
E { s ( t ) } = E { x ( f - z)} = fL, (17.32)
and therefore f‘or the airto-corrc?latioii funt.tion
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: