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~essrs of th
Trang 117.3 Stationary Raridtrrri Processes 41 5
The ~pperently cumbersome limit T 4 oc IS nec ry because thc iih rgral
f f ( ) c t ~ , does iiot exist tor a ~ ~f oi i), ~~ ~ ~ t i o n a r ~ ~ random bignal ~ ~ The i o ~ 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~essrs of this kind are called
eryodzc
CG
oo
~~~~~~~o~~ 24: Ergodic
the sunre as the enscntble uvcmqe LT talled an ergoiiic raii~fom process
w s s lor which the &me-nvcragcs of each Jar
It must Ite prown for iridividual eases wfiether or nor a s t ~ t ~ o n a 1 ~ r ~ ~ ~ ~ d ( ~ i r ~ pro
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 f i e ), &he raiitbm 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
A random proccss produces sample functions
wkiere L J ~ is fixcd yuarAiL;v, but the phase pz is 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,