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# HW5 - n ≤ N-1 elsewhere(b t n = n ≤ n ≤ N-1 2 N-n N...

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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 310 Digital Signal Processing Problem Set 5 Fall 2003 Due: Wednesday, October 1, 2003 Problem 5.1 Find the two-sided z-transforms and the associated ROCs for the following sequences: (a) x ( n ) = ( 1 3 ) | n - 1 | (b) x ( n ) = ( - 1) n ( 1 2 ) n u ( n ) (c) x ( n ) = ( 1 2 ) n [ u ( n + 5) - u ( n - 26)] (d) x ( n ) = ( 1 4 ) n cos( ω 0 n + θ ) u ( n ) Problem 5.2 Determine the region of convergence of Y ( z ) where (a) Y ( z ) = X 1 ( z ) + X 2 ( z ), X 1 ( z ) = z z +1 , | z | > 1 X 2 ( z ) = z - 2 z +1 , | z | > 1 (b) Y ( z ) = H ( z ) X ( z ), H ( z ) = 1 z 2 +7 z +10 , | z | > 5 x ( n ) = δ ( n ) + 2 δ ( n - 1) (c) Y ( z ) = z - 2 X ( z )

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x ( n ) = 2 n u ( - n - 1) Problem 5.3 Find the z-transforms and the associated ROCs for the following sequences: (a) r
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Unformatted text preview: n ≤ N-1 elsewhere (b) t ( n ) = n ≤ n ≤ N-1 2 N-n N ≤ n ≤ 2 N elsewhere Hint: To solve part (b), use the convolution and shifting properties of the z-transform. Problem 5.4 Find the inverse z-transform of each of the following z-transforms: (a) X ( z ) = 1-az-1 z-1-a , | z | > | 1 /a | (b) X ( z ) = z 3-2 z z-2 , left-sided sequence Problem 5.5 Evaluate the inverse z-transform of G ( z ) = z 2 ( z +2) 2 ( z 2 +9) for all possible ROCs. 2...
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HW5 - n ≤ N-1 elsewhere(b t n = n ≤ n ≤ N-1 2 N-n N...

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