Introduction to Random Processes (6): Stationarity ?· 2/1/2012 · Introduction to Random Processes…

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Introduction to Random Processes (6): Stationarity Luiz DaSilva Professor of Telecommunications dasilval@tcd.ie +353-1-8963660 Intuition A stationary process arises from a stable phenomenon that has evolved into a steady-state mode of behavior we make a distinction between strict stationarity and wide-sense stationarity A non-stationary process arises from an unstable phenomenon 2Strict-sense stationarity Defn: If for all n 1, the nth order distributions of a random process do not depend on the shift parameter t (for all t), then the process is said to be (strict-sense) stationary. I.e., for all n 1, all time shifts t, all choices of sample times t1, t2, , tn, 3),,(),,( 1)()(1)()( 11 ntXtXntXtX xxfxxf nn ++=Notes Intuitively, for stationary processes the randomness of the process does not change with time a desirable characteristic In general, stationarity is a very restrictive requirement. Often, we are happy with a weaker definition 4Wide-sense stationarity Defn: A random process X(t) is wide-sense stationary if E[X(t)] = x mean function is a constant E[X(t1)X(t2)] = E[X(t1+t)X(t2+t)] autocorrelation function is shift-invariant Note: strict stationarity implies wide-sense stationarity 5Properties of WSS processes Can write RX(t1,t2) = RX(), where = t1 t2. | RX() | RX(0) for all . RX() = RX(-) an even function Note: RX(0) represents the mean square value or average power of the process 6Can a random process be WSS but not stationary? An example: Let X[n] consist of two interleaved sequences of independent RVs: For n even, X[n] assumes values +1 and -1 with probability . For n odd, X[n] assumes values 1/3 and -3 with probability 9/10 and 1/10, respectively. Is this sequence stationary? Is it wide-sense stationary? 7Ergodicity Defn: A process is said to be ergodic if all orders of statistical averages and time averages are interchangeable. That is, (as said above, this also extends to higher-order statistics) 8===TTTdttXTdxxxftXE )(21lim)()]([+==+=TTTdttXtXTdxdxxxfxxtXtXER )()(21lim);,()]()([)( 212121 Ergodic processes q Since we are averaging over absolute time, the statistical averages of all orders cannot depend on absolute time. q That is, ergodicity implies stationarity. q However, the converse is not true. 9

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