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ECE535_10S_hw2soln

# ECE535_10S_hw2soln - ECE 535 DISCRETE TIME SYSTEMS HOMEWORK...

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ECE 535 DISCRETE TIME SYSTEMS SPRING 2010 HOMEWORK ASSIGNMENT #2 Solutions 1. (a) Poles are at e - (1 . 6)(0 . 2) = e - 0 . 32 = 0 . 73 e - (4 . 2)(0 . 2) = e - 0 . 84 = 0 . 43 The first pole lies outside [0 . 4 , 0 . 7]; therefore, FALSE. (b) FALSE. We need the system to be stable to have the bilinear transformation be stable, since poles from the left half of the s -plane map to poles outside the unit circle in the z -plane. (c) TRUE. The DC gain of the ZOH-equivalent has to equal the DC gain of the original continuous-time transfer function, so that it doesn’t depend on T . (d) FALSE. For instance a 2-pole, no-zero H ( s ) gets converted to a 2-pole, 1-zero b H ( z ) under the ZOH-equivalent. (e) TRUE. A pole at s = s 0 maps to z = exp ( s 0 T ), and exp ( s 0 T ) 1 as T 0. (f) FALSE. If H ( s ) is strictly proper, then its frequency response approaches zero as ω approaches infinity. This means that the frequency response of the bilinear transformation b H ( z ) approches zero as ωT approaches π , so that b H ( z ) must have a zero at z = - 1. This is independent of T , so in some cases some of the zeros

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ECE535_10S_hw2soln - ECE 535 DISCRETE TIME SYSTEMS HOMEWORK...

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