Wiley signals and systems e book TLFe BO 433

17. L)cscribiiig Raiidorii Signals

418

use (17.25) on c p g g ( r ) : t,he ACF of the signal is prCT(
z) 2 -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;

5 vm(4 5 cpZS(0) = 0;

2

+ ,2

(17.28)

‘

In the relatioiisliip (pZz( r ) 5 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 2 ( 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 = 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 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: