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Unformatted text preview: MAE 591 RANDOM DATA Stationary Process 1. Stationary in the strict sense: strongly stationary A process is stationary if its statistics are not affected by a shift at time scale, i.e. ), ( ), ( > + ε ε t x t x , have the same statistics for every, e.g. )) ( ( )) ( ( ε + = t x p t x p . 2. Stationary in the wide sense: weakly stationary (stationarity of order two) A process is stationary in the wide sense if ) ( t x mean: )] ( [ ) ( t x E t x x = = µ µ = constant autocorrelation (even function) : 1 2 2 1 ), ( ) ( )] ( ) ( [ ) , ( t t R R t x t x E t t R xx xx xx − = − = = + = τ τ τ τ 3. Jointly stationary in the wide sense: Two processes x and are jointly stationary in the wide sense if each of them is a stationary process in the wide sense and ) ( t ) ( t y ) ( ) ( )] ( ) ( [ ) , ( 2 1 τ τ τ − = = + = yx xy xy R R t y t x E t t R . Temporal (Time) Averages Temporal mean (static, time invariant component): ∫ − ∞ → >= =< 2 2 ) ( 1 lim T T T dt t x T x x Temporal mean square (measure of data intensity):...
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- Spring '09