02Averages - Averages To describe a random process...

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Averages To describe a random process (phenomenon), { } xt () , we need to compute the average properties: {}{ } { } pxt ( ) ,( ) , ( ) ) , ( ) , ( ) , 11 21 2 3 ⋅⋅⋅⋅⋅ Stationarity {} { } pxt xt ( ) ) , () ) , , , + + + ττ τ 2 Normal Distribution px e x = 1 2 2 2 2 πσ σ { } , ( ) 8 Ensemble Averages µµ τ τ xx N i i N xx N i i N i tx p x d x N R xtxt dxtdxt N ( ) l im () ( ) ) ) l ) == = =++ + = −∞ →∞ = = 1 1 1 1 τ + Ergodicity + µ x = 0 R T d t xx T T ( ) lim ( ) ( ) τ = 1 0 τ + Temporal average Finite record length T ± ()( ) R T d t xx T τ τ τ τ = + 1 0 Sampling ht TN h = = ; ± ), , , , , , Rr h Nr xnhx n rh r m xx n = += = 1 0123 1
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MAE 591 RANDOM DATA Example of Nonstationary Process )) ( ( )) 4 ( ( )) 3 ( ( )) 2 ( ( )) 1 ( ( t x p x p x p x p x p = = = = ) ( 1 t x ) ( 2 t x 2 1 -1 1 2 1 -1 1 ) ( x p x 2 1 2 1 t t The first order probability is time invariant.
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02Averages - Averages To describe a random process...

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