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Unformatted text preview: UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Quiz 4 Solutions Thursday, October 16, 2008 Student Name: Section: D. Jones (1pm), Y. Bresler (3pm) Problem 1 (10 points) Determine the twosided z transform for each of the following sequences. Specify the poles, zeros, and the region of convergence of the ztransform in each case. (a) (5 points) x 1 [ n ] = ( 1 2 )  n  (b) (5 points) x 2 [ n ] = ( 1 2 )  5( n 2)  Solution: (a) x 1 [ n ] = 1 2  n  = 1 2 u [ n ] + 2 n u [ n 1] ⇒ X 1 ( z ) = z z 1 2 z z 2 = ( 3 2 ) z ( z 2)( z 1 2 ) , ROC: 1 2 <  z  < 2 X 1 ( z ) has a zero at z = 0 and poles at z = 2 and z = 0 . 5. (b) Define ˜ x 2 [ n ] = 1 2  5 n  = 1 32 u [ n ] + 32 n u [ n 1] ⇒ ˜ X 1 ( z ) = z z 1 32 z z 32 = ( 32 + 1 32 ) z ( z 32)( z 1 32 ) , ROC: 1 32 <  z  < 32 Now x 2 [ n ] 4 = ˜ x 2 [ n 2] ⇒ X 2 ( z ) = z 2 ˜ X 2 ( z ) = ( 32 + 1 32 ) z ( z 32)( z 1 32 ) , ROC: 1 32 <  z  < 32 1 We can also derive the same result directly from the definition of ztransform: x 2 [ n ] = 1 2  5( n 2)  ⇒ X 2 ( z ) = ∞ X n =∞ 1 2  5( n 2)  z n = z 2 ∞ X n =∞ 1 2 5  n 2  z ( n 2) = z 2 ∞ X m =∞ 1 32  m  z m = z 2 " ∞ X m =0 1 32 m z m + 1 X m =∞ 32 m z m # = z 2 " ∞ X m =0 1 32 z m + ∞ X m =1 z 32 m # = z 2 " 1 1 1 32 z + z 32 1 z 32 # , 1 32  z  < 1;  z  32 < 1 = z 2 " z z 1 32 + z 32 z # , ROC: 1 32 <  z  < 32 = ( 32 + 1 32 ) z ( z 32)( z 1 32 ) , ROC: 1 32 <  z  < 32 Hence, X 2 ( z ) has no zeros and poles at z = 0, z = 32 and z = 1 32 . 2 Problem 2 (10 points) Let the algebraic form of a system transfer function H ( z ) be: H ( z ) = z 1 1 3 2 z 1 z 2 (a) Determine all possible ROCs for H ( z ). (b) For each possible choice of ROC, specify if the system has a rightsided, leftsided, or two sided unit pulse response, and if it is BIBO stable. ( Note: You are not required to compute the unit pulse response. ) Solution: H ( z ) = z ( z 2)( z + 1 2 ) H ( z ) has poles at z = 2 and z = 1 2 . Therefore H ( z ) has three possible ROCs ROC h [ n ] BIBO stable  z  < 1 2 Leftsided...
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 Fall '09
 Digital Signal Processing, Signal Processing, Orders of magnitude, difference equation

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