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630F09_hw5

# 630F09_hw5 - ENEE 630 Fall 2009 Homework#5 Problem 1 A rst...

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ENEE 630 Fall 2009 Homework #5 Problem 1 A first order autoregressive (AR) process { u(n) } that is real-valued satisfies the real-valued difference equation u ( n ) + a 1 u ( n - 1) = v ( n ) where a 1 is a constant and { v(n) } is a white-noise process of varance σ 2 v . Such a process is also referred to as a first-order Markov process . (a) Suppose in practical implementation, the generation of the process { u(n) } starts at n=1 with initialization u(0)=0. Determine the mean of the actual { u(n) } process that we have obtained. Under what conditions E [ u ( n )] converges to a constant and what the constant is? (b) Now consider the case when { v(n) } has zero mean. Determine the variance of the actual { u(n) } process that we have obtained. Under what conditions V ar [ u ( n )] converges to a constant and what the constant is? (c) For the conditions specified in part (b), find the autocorrelation function of the AR process { u(n) } . Sketch this autocorrelation function for the two cases 0 < a 1 < 1 and - 1 < a 1 < 0.

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