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, 2 ( 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 / :

1

IPrz(t1, t 2 ) = E{Z:(Ll)4 t 2 ) ) .

1

(17.10)

It describes the relationship between the whirs of a raurdom signal at t i m e s tl
a n d t 2 . High values for the auto-correlation function indicate that ~ ( tat,)times f 1
and ta take similar values. For (1.1 = C2. the auto-correlation functioii becomes the
mean square
I P u s ( 5 1 , t l ) = E{r2(tl))‘

(17.11)