HW3 - sociated ROCs. (a) z 2 + z z 2 + z-2 ; | z |...

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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 310 Digital Signal Processing Problem Set 3 Fall 2003 Due: Wednesday, September 17, 2003 Problem 3.1 Using the zero-input/zero-state approach, find the total response of the system y ( n ) - 1 2 y ( n - 1) = x ( n ) + x ( n - 1) for the input x ( n ) = 2 - n u ( n ) and the initial condition y ( - 1) = 2. Problem 3.2 Find the one-sided z-transforms and the corresponding ROCs for the following sequences. The unit step is denoted by u ( n ). (a) x ( n ) = ( - 1) n ( 1 2 ) n u ( n ) (b) x ( n ) = ( 1 2 ) n [ u ( n ) - u ( n - 26)] (c) x ( n ) = (1 + e jπ/ 3 ) n u ( n ) (d) x ( n ) = cos( π 4 n + θ ) u ( n ) (e) x ( n ) = ( 1 2 ) n cos( π 3 n ) u ( n ) (f) x ( n ) = n ( 1 4 ) n u ( n ) (g) x ( n ) = n 2 u ( n ) (h) x ( n ) = n ( - 1) n u ( n ) Problem 3.3 Find the inverse z-transform of each of the following 1-sided z-transforms with the as-
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Unformatted text preview: sociated ROCs. (a) z 2 + z z 2 + z-2 ; | z | > 2 (b) z z 2 +9 ; | z | > 3 (c) z z 2-9 5 z-2 5 ; | z | > 2 (d) z ( z-1)( z-4) 2 ; | z | > 4 (e) z z 2 +2 z +2 ; | z | > 2 (f) 1-1 2 z-1 1+ 3 4 z-1 + 1 8 z-2 ; | z | > 1 2 Problem 3.4 Use the 1-sided z-transform to solve (a) y ( n ) + y ( n-1) + 1 4 y ( n-2) = ( 1 3 ) n , n , y (-1) = 1 , y (-2) = 2 . (b) y ( n + 2)-y ( n + 1) + y ( n ) = 4 , n , y (0) = 0 , y (1) = 6 . Problem 3.5 A causal LSI system is described by the D.E. y ( n ) + 1 3 y ( n-1) = 2 x ( n ) + 3 x ( n-2) Determine the unit pulse response of the system. 2...
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HW3 - sociated ROCs. (a) z 2 + z z 2 + z-2 ; | z |...

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