[RP]Lecture Note XII - Kyung Hee University Department of...

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) XII-1 C1002900 RP Lecture Handout XII: Stationary Processes Reading: Chapter 9.1 Any pair of samples of the process  Xt with the same temporal separation, i.e.,  and Xs for any , might have the same correlation. This kind of “statistical time-invariance ” which is termed stationary is in fact a ra- ther natural property of many physical stochastic processes. 12.1. Strict-Sense Stationarity The strongest notion of stationarity is referred to as strict-sense stationarity and is de- fined in terms of the complete statistical characterization of the process. Specifically, a stochastic process is said to be strict-sense stationary (SSS) if all its finite- dimensional distributions are time-invariant, i.e.,        ,, , , , , ,,, N N NN Xt Xt f x xxf x xx 12 1 2  (XII.1) for all choices of N , time instants N tt 1 , and for any . The process does not have absolute time reference, i.e., its statistical prop- erties are invariant to a shift of the origin. From the definition, it follows that X fx f x  0 . For the second order,  , , x f x x f x x 1 2 where   . Hence, the joint density of and is independent of t . There are many examples of SSS random processes, e.g., diiscrete-time white Gaussian noise process and random telegraph process.
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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) XII-2 12.2. Wide-Sense Stationarity A weaker notion of stationarity but one that is typically simpler to verify in practice is the wide-sense stationarity. Specifically, we say a process  Xt is wide-sense stationary (WSS) if its second-order characterization is time-invariant, i.e.,: (i) Its mean is constant (does not vary with time)     XX tX t  . (XII.2) (ii) Its autocorrelation function depends only on tt   12    , XX XX Rt t X t X t R  . (XII.3) Independent of t and only a function of the time difference .
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[RP]Lecture Note XII - Kyung Hee University Department of...

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