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Unformatted text preview: ECE 6010 Lecture 8 – Random Processes in Linear Systems Continuous time systems Recall: A signal X t through a linear system produces an output Y t = Z ∞∞ h ( t, s ) X s ds. • The system is causal if h ( t, s ) = 0 for t < s . In this case, Y t = Z t∞ h ( t, s ) X s ds. • The system is time invariant if h ( t, s ) = h ( t s, 0) 4 = h ( t s ) for all t and s . In this case, Y t is obtained by convolution: Y t = Z t ∞ h ( t s ) X s ds = h * X. In this case, we can do analysis using Fourier transforms: Y ( ω ) = H ( ω ) X ( ω ) Suppose the input function is a random process instead of a deterministic signal. How can we characterize the output function? Let Y t = Z ∞∞ h ( t, s ) X s ds The integral is to be interpreted in in the meansquare sense. Properties: • R ∞∞ h ( t, s ) X s ds exists ⇔ R ∞∞ R ∞∞ h ( t, s ) h ( t, q ) R X ( s, q ) dsdq < ∞ . • μ Y ( t ) = R ∞∞ h ( t, τ ) μ X ( τ ) dτ . That is, the mean of the output is the response of the mean of the input: E [ Y t ] = E Z ∞∞ h ( t, τ ) X τ d 1 t = Z ∞∞ h ( t, τ ) E [ X t ] dτ • R XY ( t, s ) = E [ Y t X s ] = R ∞∞ h ( t, τ ) R X ( τ, s ) dτ . That is, the correlation between the input and the output is the response to the autocorrelation with respect to the 1st input. • R Y ( t, s ) = E [ Y t Y s ] = R ∞∞ R ∞∞ h ( t, σ ) R X ( σ, τ ) h ( s, τ ) dσdτ ....
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This note was uploaded on 03/01/2012 for the course ECE 6010 taught by Professor Stites,m during the Spring '08 term at Utah State University.
 Spring '08
 Stites,M

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