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**Unformatted text preview: **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 (0 , ) for which H d ( ) = 0, what is ? (d) Given that H d (0) = 1, give an expression for H d ( ). (e) What is the difference equation for this filter in part (d)?...

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