41 17.3 Stationary Raridtrrri Processes The ~pperentlycumbersome limit T 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,