HW2Ans_2005

# Digital Signal Processing: A Practical Approach (2nd Edition)

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Homework #Set 2 z-Transforms 1. (a) ], 1 [ ] 2 [ ) 1 ( ) 1 ( ] 1 [ ] 2 [ ) 1 ( ] [ 2 2 + = + = n u n u n u n u n x n n 1 z : ROC , 1 ] 2 [ ) 1 ( 1 2 2 > + ⎯→ z z n u Z n Q , 1 z : ROC , 1 ] 1 [ 1 1 > z z n u Z , 1 1 ) ( 1 1 1 2 + + = z z z z z X ROC: 1 > z . (b) ], 1 [ ) 3 ( ] 3 [ ) 2 ( ] [ + = n u n u n x n n , 3 1 1 ) 3 1 ( 2 1 1 ) 2 1 ( ) 3 ( ) 2 ( ) 3 ( ) 2 ( ) ( 3 3 3 00 1 1 3 3 1 z z z z z z z z z X nn n n n n n n n n n n + + + = + = + = ∑∑ = = = + + = ROC: |(-1/2)z| < 1 and |(-1/3)z| < 1 |z| < 2 (c) ], 2 [ 3 ] 2 [ 2 ] [ + + = n n n x δ , 3 2 ) ( 2 2 z z z X + = ROC: all , except 0, . (d) ], 1 [ ) 1 ( ) 2 . 0 ( ] [ = n u n n x n let ], [ ) 2 . 0 ( ] [ n u n y n = from (a), 2 . 0 ) ( + = z z z Y 2 ) 2 . 0 ( 2 . 0 ) ( ] [ ] [ ) 2 . 0 ( + = = z z dz z dY z n ny n u n Z n , , ) 2 . 0 ( 04 . 0 ) 2 . 0 ( 2 . 0 ) 2 . 0 ( ) ( ] 1 [ ) 1 ( ) 2 . 0 )( 2 . 0 ( ] 1 [ ) 1 ( ) 2 . 0 ( ] [ 2 2 1 1 + = + = = = z z z z z X n u n n u n n x Z n n ROC: 2 . 0 > z (e) ], 2 [ ) 2 ( ) 2 ( ] 2 [ ) 2 ( ] [ 2 2 = = n u n u n x n n , , ) 2 ( 4 ) 2 ( 4 ) ( 2 + = + = z z z z z z X ROC: 2 > z . (f) ] [ ) ( ] [ , 1 0 , ] [ N n u N n n nu N n N N n n n x = = , where N = 7.

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2 1 1 1 ) 1 ( 1 1 ] [ = ⎯→ = z z z dz d z n nu Z , ROC: 1 > z , ) 1 ( ) ( 2 1 1 1 = z z z z X N ROC: 1 > z . , ) 1 ( ) ( 2 1 8 1 = z z z z X ROC: 1 > z .
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HW2Ans_2005 - Homework#Set 2 z-Transforms 1(a x[n =(1 n u[n...

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