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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 Solution Prof. Bresler, Prof. Jones Wednesday, December 3, 2008 Problem 1 (30 points) 1. In this problem you will design a second-order digital filter using the Bilinear Transformation method that converts an analog filter H a ( a ) 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. The expression s = 2(1- z- 1 )(1+ z- 1 ) is equivalent to z = (2+ s ) / (2- s ) So the pole s =- 1+ j transforms to z = 2 + s 2- s = 2 + (- 1 + j ) 2- (- 1 + j ) = 1 + j 3- j = (1 + j )(3 + j ) (3- j )(3 + j ) = (3 + j 2 + 4 j ) (9- j 2 ) = 2 + 4 j 10 = 1 5 + j 2 5 and similarly the zero at s = j transforms to z = 2 + s 2- s = 2 + j 2- j = (2 + j )(2 + j ) (2- j )(2 + j ) = 3 + 4 j 5 = 3 5 + j 4 5 The transformation of the other pole and zero can be worked out similarly or you can note that (by the property of distributivity of conjugation over addition, multiplication and division of complex numbers) ((2 + s * ) / (2- s * )) = ((2 + s ) * / (2- s ) * ) = ((2 + s ) / (2- s )) * . So the pole at s =- 1- j is transformed to z = 1 5- j 2 5 and the zero at s =- j is transformed to z = 3 5- j 4 5 See figure 1. The zeros on the imaginary axis in the s plane were mapped to the unit circle in the z-plane. This agrees with the theory. (b) If the digital filter h d is causal, is it stable ? Why or why not ? Yes it is stable as the poles of the digital filter lie within the unit circle. Alternatively, note that the poles of the analog filter lie in the left-half-plane and therefore is stable, and since the BLT preserves stability, the digital filter is also stable. 1 Figure 1: Poles at (1/5) j(2/5) and zeros at (3/5) j(4/5) (c) Given that there is only one value of ω ∈ (0 ,π ) for which H d ( ω ) = 0 what is ω ?...

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