630_hw_3-1 - Advanced DSP HW#3 Problem Set ENEE 630...

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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 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) 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 white-noise process of zero mean and variance 0.5 (a) Write the Yule-Walker 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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630_hw_3-1 - Advanced DSP HW#3 Problem Set ENEE 630...

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