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Unformatted text preview: Advanced DSP HW#3 Problem Set ENEE 630 Homework #3 Problem 1 A first order autoregressive (AR) process { u(n) } that is realvalued satisfies the realvalued difference equation u ( n ) + a 1 u ( n 1) = v ( n ) where a 1 is a constant and { v(n) } is a whitenoise process of varance σ 2 v . Such a process is also referred to as a firstorder Markov process . (a) Show that if { v(n) } has a nonzero mean, the AR process { u(n) } is nonstationary (assuming u(0) = 0). (b) For the case when { v(n) } has zero mean, and the constant a 1 satisfies the condition  a 1  < 1, show that the variance of { u(n) } equals V ar [ u ( n )] = σ 2 v 1 a 2 1 (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. Problem 2 Consider an autoregressive process { u(n) } of order 2, described by the difference equation u ( n ) = u ( n 1) . 5 u ( n 2) + v ( n ) where { v(n) } is a whitenoise process of zero mean and variance 0.5 (a) Write the YuleWalker equations for the process. (b) Solve these two equations for the autocorrelation function values r(1) and r(2). (c) Find the variance of { u(n) } . Problem 3 Consider an MA process { x(n) } of order 2 described by the difference equation x ( n ) = v ( n ) + 0 . 75 v ( n 1) + 0 . 25 v ( n 2) where { v(n) } is a zero mean white noise process of unit variance. The requirement is to approximate the process by an AR process { u(n) } of order M. Do this approximation forof order M....
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This note was uploaded on 10/31/2010 for the course EE 630 taught by Professor Wu during the Spring '10 term at Aarhus Universitet, Aarhus.
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