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Unformatted text preview: UCSB Fall 2009 ECE 235: Problem Set 6 (and recap of random processes through linear systems) Assigned: Wednesday, November 25 Due: Thursday, December 3 (by noon, in course homework box) Reading: Chapters 7 and 8 (you only need to go over the highlights covered in class); also prior material on Markov processes Office hours for homework: For next week, I will hold office hours Wednesday December 2 (10-noon) instead of on Monday; Final exam and review: The final exam is on Friday, December 11, 4-7 pm in the regular classroom; I will hold special office hours for the final exam on Friday, December 4, 9-11 am. Random processes through linear systems Suppose that a random process X is the input to a linear time-invariant (LTI) system with impulse response h . Let Y denote the random process at the output of the system. That is, Y ( t ) = ( X * h )( t ) = integraldisplay ∞-∞ X ( s ) h ( t- s ) ds Fact: If X is Gaussian, then X and Y are jointly Gaussian random processes. Fact: If X is WSS, then X and Y are jointly WSS. The mean function of Y is a constant, given by μ Y = μ X integraldisplay h ( t ) dt The crosscorrelation function (and cross-spectrum) is given by R Y X ( τ ) = ( R X * h )( τ ) S Y X ( f ) = S X ( f ) H ( f ) The autocorrelation function is given by R Y ( τ ) = ( R Y X * h mf )( τ ) S Y ( f ) = S Y X ( f ) H * ( f ) = S X ( f ) | H ( f ) | 2 where h mf ( t ) = h (- t )....
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- Fall '08
- Signal Processing, LTI system theory