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HW3 - Spring 2006 EE113 HW 3 Solution Due Problem 1 10 d 1...

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Spring 2006 EE113 HW 3 Solution ( Due 5/9/06 ) Problem 1: ( ) ( ) ( ) ( ) 3 1 11 1 3 0 10 0 1 1 1 11 3 1 = - - - = - = = - + - - = - - = = Z Z n n n n Z Z dZ d Z Z dZ d Z Z n u n u n n S ( ) 7499 . 0 1 10 11 3 2 10 9 = - + - = = - - Z Z Z Z Z Problem 2: a. ( ) ( ) 4 - = n nu n x ( ) ( ) ( ) 2 1 1 4 1 4 1 3 4 1 1 - - - - - - - = - - = Z Z Z Z Z dZ d Z Z X , R.O.C: 1 > Z b. ( ) ( ) n u n n x n = ( ) ( ) = = - = = 0 0 1 n n n n Z n nZ Z X , but for any finite Z as n>Z, the sequence become unbounded. Therefore, the Z-transform does not exist. Problem 3: ( ) ( ) ( ) ( ) - -∞ = - - - -∞ = - - = - - = + + + - = - = 1 1 3 2 4 4 ...... 3 4 2 4 4 4 1 log n n n n n n Z n Z n Z Z Z Z Z X By inspection, ( ) ( ) 1 4 - - = - n u n n x n Problem 4: a. ( ) ( ) ( ) ( ) ( ) 2 2 1 , 0 2 4 1 1 2 1 = - = - = - - - + y y n y n y n y ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) n u n y Z Z Z Z Z Z Y Z Z Y Z Z Z Z Z Y Z Z Z Y Z Z Y n n - + = + - + - = - + + - = + + - = - + = + + - + + + + - - - - - + - + - - + - + - + 1 1 1 1 2 1 1 1 2 1 2 2 1 81 . 0 31 . 0 81 . 0 1 81 . 0 31 . 0 1 31 . 0 25 . 0 5 . 0 1 5 . 0 5 . 0
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