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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 ( ) = H ( ) X ( ) 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 mean-square sense. Properties: R - h ( t, s ) X s ds exists R - R - h ( t, s ) h ( t, q ) R X ( s, q ) dsdq < . Y ( t ) = R - h ( t, ) X ( ) d . That is, the mean of the output is the response of the mean of the input: E [ Y t ] = E Z - h ( t, ) X d 1 t = Z - h ( t, ) E [ X t ] d R XY ( t, s ) = E [ Y t X s ] = R - h ( t, ) R X ( , s ) d . 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, ) R X ( , ) h ( s, ) dd ....
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- Spring '08