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lect09 - Lecture Notes 9 Stationary Random Processes...

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Lecture Notes 9 Stationary Random Processes Strict-Sense and Wide-Sense Stationarity Autocorrelation Function of a Stationary Process Power Spectral Density Response of LTI System to WSS Process Input Linear Estimation: the Random Process Case EE 278: Stationary Random Processes 9 – 1 Stationary Random Processes Stationarity refers to time invariance of some, or all, of the statistics of a random process, such as mean, autocorrelation, n -th-order distribution We define two types of stationarity: strict sense (SSS) and wide sense (WSS) A random process X ( t ) (or X n ) is said to be SSS if all its finite order distributions are time invariant, i.e., the joint cdfs (pdfs, pmfs) of X ( t 1 ) , X ( t 2 ) , . . . , X ( t k ) and X ( t 1 + τ ) , X ( t 2 + τ ) , . . . , X ( t k + τ ) are the same for all k , all t 1 , t 2 , . . . , t k , and all time shifts τ So for a SSS process, the first-order distribution is independent of t , and the second-order distribution — the distribution of any two samples X ( t 1 ) and X ( t 2 ) — depends only on τ = t 2 - t 1 To see this, note that from the definition of stationarity, for any t , the joint distribution of X ( t 1 ) and X ( t 2 ) is the same as the joint distribution of X ( t ) = X ( t 1 + ( t - t 1 )) and X ( t 2 + ( t - t 1 )) = X ( t + ( t 2 - t 1 )) EE 278: Stationary Random Processes 9 – 2
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Example: The random phase signal X ( t ) = α cos( ωt + Θ) where Θ U[0 , 2 π ] is SSS We already know that the first order pdf is f X ( t ) ( x ) = 1 πα p 1 - ( x/α ) 2 , - α < x < + α which is independent of t , and is therefore stationary To find the second order pdf, note that if we are given the value of X ( t ) at one point, say t 1 , there are (at most) two possible sample functions: x 1 x 21 x 22 t t 1 t 2 EE 278: Stationary Random Processes 9 – 3 The second order pdf can thus be written as f X ( t 1 ) ,X ( t 2 ) ( x 1 , x 2 ) = f X ( t 1 ) ( x 1 ) f X ( t 2 ) | X ( t 1 ) ( x 2 | x 1 ) = f X ( t 1 ) ( x 1 ) ( 1 2 δ ( x 2 - x 21 ) + 1 2 δ ( x 2 - x 22 ) ) , which depends only on t 2 - t 1 , and thus the second order pdf is stationary Now if we know that X ( t 1 ) = x 1 and X ( t 2 ) = x 2 , the sample path is totally determined (except when x 1 = x 2 = 0 , where two paths are possible), and thus all n -th order pdfs are stationary IID processes are SSS Random walk process is not SSS (in fact, no independent increment process is SSS) The Gauss-Markov process (as we defined it) is not SSS. However, if we set X 1 to the steady state distribution of X n , it becomes SSS (see homework exercise) EE 278: Stationary Random Processes 9 – 4
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Wide-Sense Stationary Random Processes A random process X ( t ) is said to be wide-sense stationary (WSS) if its mean and autocorrelation functions are time invariant, i.e., E( X ( t )) = μ , independent of t R X ( t 1 , t 2 ) is a function only of the time difference t 2 - t 1 E[ X ( t ) 2 ] < (technical condition) Since R X ( t 1 , t 2 ) = R X ( t 2 , t 1 ) , for any wide sense stationary process X ( t ) , R X ( t 1 , t 2 ) is a function only of | t 2 - t 1 | Clearly SSS WSS. The converse is not necessarily true EE 278: Stationary Random Processes 9 – 5 Example: Let X ( t ) = + sin t with probability 1 4 - sin t
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