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# HW13_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Homework 13 Prof. Bresler, Prof. Jones Wednesday, December 3, 2008 Problem 1 In this problem you will design a second-order digital filter using the Bilinear Transformation method that converts an analog filter H a ( s ) into a digital filter H d ( z ). The mapping you will use is s = 2 1 - z - 1 1 + z - 1 . The analog filter H a ( s ) has poles at - 1 + j and - 1 - j and zeros at + j and - j . (a) Draw a pole-zero diagram for the digital filter and specify the exact locations of the poles and zeros. (b) If the digital filter h d is causal, is it stable? Why or why not? (c) Given that there is only one value of ω 0 (0 , π ) for which H d ( ω 0 ) = 0, what is ω 0 ? (d) Given that H d (0) = 1, give an expression for H d ( ω ). (e) What is the difference equation for this filter in part (d)? (f) Use the matlab command freqz to plot the magnitude and phase of the frequency response of the filter. Check that it evaluates to zero at ω 0 (from part (c)).

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HW13_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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