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ps2 - EE 429 Reading assignment Ch 3 PROBLEM SET 2 DUE 18...

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EE 429 PROBLEM SET 2 DUE: 18 Feb 2008 Reading assignment: Ch 3 Problem 6: (20 points) The one-sided Z transform of a sequence y ( k ) is defined as Z{ y ( k ) } = k =0 y ( k ) z - k , where y ( k ) = 0 for k < 0. Suppose that the sequences e ( k ), f ( k ) and g ( k ) have Z transforms E ( z ), F ( z ) and G ( z ) respectively. Show that 1. (5 points) Z{ e ( k + n ) } = z n E ( z ) - z n e (0) - z n - 1 e (1) · · · - ze ( n - 1) 2. (5 points) Z{ e ( k - n ) } = z - n E ( z ) 3. (5 points) lim k →∞ e ( k ) = lim z 1 ( z - 1) E ( z ) provided that the left-side limit exists. 4. (5 points) lim k 0 e ( k ) = lim z →∞ E ( z ). Problem 7: (20 points) Calculate the Z transform of the following sequences: 1. (5 points) δ ( k ) = 1 k = 0 0 k = 0 2. (5 points) u o ( k ) = 1 k 0 0 k < 0 3. (5 points) a k u o ( k ) 4. (5 points) ka k u o ( k ) The relationship k =0 a k = 1 1 - a , for | a | < 1, may be of use. Problem 8: (20 points) 1. (6 points) If applicable, use the initial and final value theorems to calculate e (0) and lim k →∞ e ( k ) for each E ( z ) below: (a) (2 points) E ( z ) = 0 . 5 z ( z 2 - 1 . 6 z + 0 . 6) (b) (2 points) E ( z ) = 12 z 2 - 18 z 2 z 2 - 5 z + 2 (c) (2 points) E ( z ) = 18 z 9 z 2 - 6 z + 1 2. (12 points) For each E ( z
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