Wiley signals and systems e book TLFe BO 425

17 Jkscribiug Rmdorn Signals 410 Tlre expected value of a dtltcrminist,ic siglial d ( t ) is the value of the signal itself, a5 d ( t ) ran be thought of as a realisation of a random process with identical saniple furictions: I E{d(t)} = d(t) I (17.7) Example 17.3 We use both rules (17.6) a n d (17.7) to express the varia.nce witli the linear average and squa,rexnem: 0, ( t ) = E{(s(C) - p:c(t))Z}= E(z‘((1.)- 2z(t)p,(t) pzj2(t)} + E,{x2(t)}- 2p2(t)E(~(t)} + pZ2(t) = E{x2(5)} - Q ( t ) = (17.8) It is therefore suficierrt to know only two of the three quantities - L2ie linear average, the square average and the twiance - a i d then the third can be calculated The relationship between them is often iised in practice tro calculat,e Ihe xuiance If N saniple filnct,ioiisare available, s i ( t ) and zf(t)are summed and the rcsult, is divided by N The qu;tnt,ity z i ( t ) - p z ( t ) cannot be averargedbecause iii the first, p a s , the linear average p c ( t )is rmt yet, ava,ilable A secaiid pass can bc avoided by calculating the vmiance in accordance with (17.8) First-order expcctcd values hold for a certain point in tirnc, and tliereforr tl-rcay cannot register the statistical dependencies that exist, hetween diffcwnt points in c2 signal With second-order expected values, however, this is possible ‘I‘hcy link the signal at two different points: A’ The auto-coxrelation function (ACJ.’) is an important second-ordrr c.xpecr,~d value, which is obtained from (17.9) for f ( p , v) - p i / : IPrz(t1, t ) = E{Z:(Ll)4 t ) ) (17.10) It describes the relationship between the whirs of a raurdom signal at t i m e s tl a n d t High values for the auto-correlation function indicate that ~ ( tat,)times f and ta take similar values For (1.1 = C2 the auto-correlation functioii becomes the mean square I P u s ( , t l ) = E{r2(tl))‘ (17.11)