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# HW7sol_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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Unformatted text preview: UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Homework 7 Solution Prof. Bresler, Prof. Jones Wednesday, October 16, 2008 Problem 1 . Given that X ( z ) = 3 . 5- 7 . 5 z- 1- 19 z- 2 1- 3 z- 1- 10 z- 2 + 24 z- 3 . (a) (10 points) Find and sketch all possible ROC’s for X ( z ). Determine whether the system is left-sided/right-sided/2-sided in each case. When the system is left-sided or right-sided, is it causal or noncausal? X ( z ) = z ( z- (15 / 14- j √ 839 / 14))( z- (15 / 14 + j √ 839 / 14)) (4- z )(2- z )( z + 3) (1) X ( z ) has poles at z = 2 ,- 3 , 4 . Therefore, there are four possible ROCs: i. | z | < 2, left-sided, anticausal. ii. 2 < | z | < 3, 2-sided, iii. 3 < | z | < 4, 2-sided, iv. | z | > 4, right-sided, causal. (b) (5 points) For each ROC, state whether the system is BIBO stable. Give reasoning for full credit. The system is BIBO stable if the ROC contains unit circle. Therefore, i. | z | < 2, stable ii. 2 < | z | < 3, unstable iii. 3 < | z | < 4, unstable iv. | z | > 4, unstable (c) (5 points) Find the Inverse Z-transform x [ n ] for each ROC. X ( z ) = 2 1- 2 z- 1 + 1 1 + 3 z- 1 + . 5 1- 4 z- 1 i. | z | < 2 x [ n ] = (- 2(2) n- (- 3) n- . 5(4) n ) u [- ( n + 1)] 1 ii. 2 < | z | < 3 x [ n ] = 2(2) n u [ n ] + (- (- 3) n- . 5(4) n ) u [- ( n + 1)] iii. 3 < | z | < 4 x [ n ] = (2(2) n + (- 3) n ) u [ n ]- . 5(4) n u [- ( n + 1)] iv. | z | > 4 x [ n ] = (2(2) n + (- 3) n + 0 . 5(4) n ) u [ n ] Problem 2 . A sequence x [ n ] = 4(2)- n u [- n ]- 9(- 3)- n u [ n- 1] is the input to an LSI system with the transfer function H ( z ). It produces a two-sided output sequence with z-transform of the form: Y ( z ) = 2 ( z + 1 / 3)( z- 1) ....
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HW7sol_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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