Wiley signals and systems e book TLFe BO 426

41 I

17.3. Stationarv H mdom Processes

Example 17.4
Figure 17.4 shows .tvliicli pmperties ol a rarirlorri signal clre represenl ed by
second-order expected values. %‘lwrandom processes A arid B hm-r identical firstorder expected vatraes. in particular their distribution function Fl,jt) (0)= Pf,!ti(@)
(Example 17.2). Tlie sample frxiictions ttf random prnrcsh A chmgp mach morr
slowly with timc, however, tlrtzri thosc of process B. We can therefore expect a
much greater t.aliir of auto-correlation firriction pZ5(tl,Cz) for A’s neighbouring
e ~ a ~ iqoy~y (~f l ,t z > for 13. This i s
values tl and 12 than for the ~ ~ ~ t o - c ~ ) r rfmction
confirmed hy the measured ACFs shown in Figure 17.5.
rand^^ process A

rand~mprocess E3

.-

t

*

t

Figure 17.4: Illustration of two raiidain processes A4and XJ with iderititat Emt-order
expctt,cd values ancl differing becolid-orcler expected values

Expected valucs of first- aiid second-order are by Iar the niosl important in
practical applic‘ations. Many descriptions of stocbastic funrtioiis rely d e l y on
these two. However, advauced models of complicated random processes resort!

a140 to higher-order expccted values. This ernergiiig branch of signal analysis i s
therefore called lazqhar-order.stnttstacs.

In Section 172.1 we considered various possible methods for calculatiiig expected
One of i h s e w a b expressing the ensembte inem with the timemerage.
The conditions tinder which this is possible will be explaitird in this section.

WI~LE~.