Wiley signals and systems e book TLFe BO 431

416

17. Describing Randoiii Signals

1
= - cosfw0tl- w&)
2

1
2

= - coswgI z mit

z -= tl -- t 2

I

The integral averages all phase angles between 0 arid 2n.
The process i s also crgodic because the time-average (17.19) agrees with the
ensemble average. We can verify this with the rxample of the ACT, where

__

-1 cosw(-Jz.
2

If the random process tdso has amraadom peak vdue P in addition to the random
phase p7so that
~ ( t=)2 sin(uoO%-t- 'p>I

not ergoclic. For cxanlpJer while E{z2(t)} iE{,22),

the t,ime-avera~geof tlic square of a cert,ain sample function i is, however, zP(t) =
it i s indeed statiotrary, brit

~ = l

~

Now that we are comtort&le with expected values and Iiavo learnt ways of c a l m
lclting t h i n , we van return to the. task from Section 17.i a t the bcginiiing of this
chitpter. We want to describe t,he propel ties of random signals with deterministic.
vaiiables* This will IWW be doric with first- and second-ordei expected W ~ U P S . We
start with ergodic random processrs if needed, so that, the expected values c m be
with time avrragc~s.
The most irriportant c.xpectt.d valiie e the linear avm age, tlw auto-con.rlation
f~iizctioiiand a generalisation derived froin it, tlre cross-correlatiuu function.