Wiley signals and systems e book TLFe BO 430

41 5

17.3. Stationary Raridtrrri Processes
The ~pperentlycumbersome limit T

4

oc

IS

nec

ry because thc iih rgral

CG

f f ( . ) c t ~ , does iiot exist tor a ~

~ f oi i),~~ ~ ~ ~t i o random
n a~ r ~ bignal.
~
The
i

o

--oo

average is thrrefttre fomicd from a section of the signal with lmgth 2T a d this
bngth is then extended to irifiiiity.

It the tirrie-av(irages (17.18). (I
ngree for all sample functions of a raiidoni
process and tire also ecpal to t lie
idle average, we can then express it with
the time average of any sample fu
. Random pio~essrsof this kind are called
eryodzc.
lll_

I__

~~~~~~~o~~
24: Ergodic
A s ~ ~ t mnilom
~ ~ ~ wss
? ~lorn which
~ ~the &me-nvcragcs of each Jar
the sunre as the enscntble uvcmqe LT talled an ergoiiic raii~fomprocess.
It must Ite prown for iridividual eases wfiether or nor a s t ~ t ~ o nra~1 ~~ ~ ~
pro
d(~ir~
is ergodic. Often this proof rannot be exactly carried out. arid in these cases it tan
be nssiinicd that the process is ergodic, as long as no indications to the contrary
occur. The big advantage of ergodic processes i s that knowledge of ari individiial
saniplr function is sufficient to r;3culate expected valiies with the time-average.
Similar to stationary processes. thexe i s also R iestricciori for certain random
RCS. If the ergotlicity conditions orily liold for

but not for general fimctions
.), &heraiitbm prnrfsi: is then weak r-ryod~.

Like the idea of weak stationary, it is used €or modelling arid analysis of random
proccwcs with miniilia1 rcstrictioii.
f i e .

Example 17.7

A random proccss produces sample functions

wkiere L J ~is fixcd yuarAiL;v, but the phase pzis couipletely random. All phase
angles are equally likely.
The prates i s statiomry because the second-order expected wlues (17.12) oiily
depend 011 the differencc betwcen the observation times. The ACE', for example,