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22SISO - MAE591 RANDOM DATA Single-input/single-output...

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MAE591 RANDOM DATA Single-input/single-output relationships (Random Vibration) Response of a linear time-invariant discrete dynamic system : x t ( ) h H f ( ) ( ) τ y t ( ) y t h x t d h x t d ( ) ( ) ( ) ( ) ( ) = = −∞ τ τ τ τ τ 0 τ For stationary processes: Mean value [ ] µ τ τ τ τ τ τ τ τ µ π τ y j f f x E y t E h x t d h E x t d E x t h d E x t h e d H = = = = = = = [ ( )] [ ( ) ( ) ] ( ) [ ( )] [ ( )] ( ) [ ( )] ( ) ( ) 0 0 0 2 0 0 0 τ τ Correlation functions R E x t y t E h x t x t d h E x t x t d h R d h R xy xx xx ( ) [ ( ) ( )] ( ) ( ) ( ) ( ) [ ( ) ( )] ( ) ( ) ( ) ( ) τ τ ξ τ ξ ξ ξ τ ξ ξ ξ τ ξ ξ τ = + = + = + = = 0 0 0 τ η Similarly, R E y t y t E h h x t x t d d h h E x t x t d d h h R d d h h R yy xx xx ( ) [ ( ) ( )] ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ( )] ( ) ( ) ( ) ( ) ( ) ( ) τ τ ξ η ξ τ η ξ η ξ η ξ τ η ξ η ξ η τ ξ η ξ τ τ τ = + = + = + = + =
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