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**Unformatted text preview: **ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 22 Today: Discrete-Time (1) DTFT (2) Power Spectral Density (3) Examples • HW 10 due Tue Dec 9 at 5pm. Application Assignment 6 due Thu Dec 11 at 5pm. • Discussion items? 0.1 Cts-time WSS Process/LTI Filtering Examples 1. (From Miller & Childers, p448) The input to a linear filter is a random process with the following autocorrelation function: R X ( τ ) = Aω π sin( ω τ ) ω τ . The filter impulse response is of the same form and is h ( t ) = ω 1 π sin( ω 1 τ ) ω 1 τ . Determine the autocorrelation function of the filter output for ω ≥ ω 1 and for ω < ω 1 . 2. (from A. Yagle) A WSS r.p. X ( t ) with PSD S X ( f ) = 6 (2 πf ) 2 +16 is input into a LTI system with impulse response h ( t ) = δ ( t ) + e- 3 t u ( t ) where δ ( t ) is an impulse function and u ( t ) is a unit step function. Compute the PSD S Y ( f ) and variance Var Y [ Y ( t )] of the output Y ( t ) 1 Discrete-Time R.P. Spectral Analysis Traditionally, the discrete time case is not taught in a random pro- cess course. But real digital signals are everywhere, and so I imagine that most of the time that this analysis is necessary is with discrete- time random processes. For example, image processing, audio pro- cessing, video processing, all are mostly digital. Traditionally, com-cessing, video processing, all are mostly digital....

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