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[RP]Lecture Note XII

# [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   X t with the same temporal separation, i.e., X t   and X s   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   X t is said to be strict-sense stationary (SSS) if all its finite- dimensional distributions are time-invariant, i.e., , , , , , , , , , , , , N N N N X t X t X t X t X t X t f x x x f x x x    1 2 1 2 1 2 1 2 (XII.1) for all choices of N , time instants , , N t t 1 , and for any . The process   X t 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 t X t X f x f x f x  0 . For the second order,   , , , , , , X t X t X t X t X t X t f x x f x x f x x    1 2 1 2 1 2 1 2 1 2 where t t   1 2 . Hence, the joint density of   X t and X t   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   X t 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)     X X t X t   . (XII.2) (ii) Its autocorrelation function depends only on t t  
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[RP]Lecture Note XII - Kyung Hee University Department of...

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