HW6sol_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 6 Solutions Prof. Bresler, Prof. Jones October 8, 2008 Problem 1 (30 points) Determine the two-sided z -transform of each of the following sequences, if it exists. Include with your answer the region of convergence in the z-plane. Also specify, in each case, if the Discrete Time Fourier Transform of the sequence exists. (a) x [ n ] = 0 . 5 n u [ n ] - 2 n u [ - n - 1] (b) x [ n ] = 2 n u [ n ] + 0 . 5 n u [ - n - 1] (c) x [ n ] = 2 n u [ n ] + 2(3 n ) u [ - n - 1] (d) x [ n ] = 2(0 . 5 n ) u [ n + 2] + 0 . 8 n u [ n ] (e) x [ n ] = 0 . 5 n u [ n ] - 0 . 8 n u [ - n ] (f) x [ n ] = 2 n ( u [ n ] - u [ n - 10]) Solution: We begin by deriving some elementary z-transforms: w [ n ] = a n u [ n ] W ( z ) = X n = -∞ w [ n ] z - n = X n =0 a n z - n = 1 1 - a z = z z - a , ROC: a z < 1 ⇔ | z | > | a | Similarly, v [ n ] = b n u [ - n - 1] V ( z ) = X n = -∞ v [ n ] z - n = - 1 X n = -∞ b n z - n = X m =1 b - m z m = z b 1 - z b = - z z - b , ROC: z b < 1 ⇔ | z | < | b | 1
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Also recall the shift property of z-transforms y [ n ] = x [ n - n 0 ] Y ( z ) = z - n 0 X ( z ) With those preliminaries, and using the linearity of z-transforms: (a) x [ n ] = 0 . 5 n u [ n ] - 2 n u [ - n - 1] X ( z ) = z z - 0 . 5 + z z - 2 , ROC: 0 . 5 < | z | < 2 (b) x [ n ] = 2 n u [ n ] + 0 . 5 n u [ - n - 1] X ( z ) = z z - 2 - z z - 0 . 5 , ROC: { 2 < | z |} ∩ {| z | < 0 . 5 } = i.e. , the z-transform of the sequence does not exist. (c) x [ n ] = 2 n u [ n ] + 2(3 n ) u [ - n - 1] X ( z ) = z z - 2 - 2 z z - 3 , ROC: 2 < | z | < 3 (d) x [ n ] = 2(0 . 5 n ) u [ n + 2] + 0 . 8 n u [ n ] x [ n ] = 8(0 . 5 n +2 ) u [ n + 2] + 0 . 8 n u [ n ] X ( z ) = 8 z · z 2 z - 0 . 5 - z z - 0 . 8 , ROC: { 0 . 5 < | z |} ∩ { 0 . 8 < | z |} = 0 . 8 < | z | (e) x [ n ] = 0 . 5 n u [ n ] - 0 . 8 n u [ - n ] x [ n ] = 0 . 5 n u [ n ] - (0 . 8)0 . 8 n - 1 u [ - ( n - 1) - 1] X ( z ) = z z - 0 . 5 + 0 . 8 z - 0 . 8 , ROC: 0 . 5 < | z | < 0 . 8 (f) x [ n ] = 2 n ( u [ n ] -
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HW6sol_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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