Wiley signals and systems e book TLFe BO 427

41 2
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17. Describing Naiidorii Sigiials
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A ranclom process is callcd statoonnry if its statistical properties do miot change with
tinit.. This appwrs to be the case for 11ie two random processes in Figure 17.4,
wliilc in Figure 17.2, it is clew that tlic linear expected value changes with tirnc.
For a precis(>definition we start with a second-ordrr cxpccted \-due as in (17,!3).
If it is lorrrird froni a signal whose statistical properties do not chaxigc with tirne,
then t h e expected value does i i o l charigt. if both time points tj nut1 t 2 are shifted
by the. billne amount At:

The expccted value does not depend ox1 the individual time points
irtstcacl on thcir difference. We c a i itse fliih pxoperty to titifine

tl

amid

12,

but

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23:

Stationary

tdom pi*oeesc as stationary z.f I f s second-order ~rpecttduulws only depend
of ohswved tinre poznts z = tl - 12.

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In (17.7) we cvtisidcred dctcrininistic signals as a spccial case of random signals,
for which the linear cbxpec.tetl value is eqial to the current functrrm value. This
means that dctwninistic signals can only be stationary if they are rotistant, in
nrinistic. signal that changes with tirnc is thcrr)fore not statiorrery.
for tiirite random signals, as i?, signal tlial i s z ( m before or after H
ical properties and cannot be stationary.
certain point in time changes its s
From this cicfinition for ~ C C C
d valurs we can derive t l r c
properties of fiist omder expected v d
l ( . r ( t , ) ,r(t2))== g(r(t,))are a special case of see