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# HW6_FA08 - transform Specify the region of convergence for...

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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Homework 6 Wednesday, October 1, 2008 Prof. Bresler, Prof. Jones Due by October 8, 2008 Problem 1 (30 points) Determine the two-sided z -transform of each of the following sequences, if it exists. Include with your answer the region of convergence in the z-plane. Also specify, in each case, if the Discrete Time Fourier Transform of the sequence exists. (a) x [ n ] = 0 . 5 n u [ n ] - 2 n u [ - n - 1] (b) x [ n ] = 2 n u [ n ] + 0 . 5 n u [ - n - 1] (c) x [ n ] = 2 n u [ n ] + 2(3 n ) u [ - n - 1] (d) x [ n ] = 2(0 . 5 n ) u [ n + 2] + 0 . 8 n u [ n ] (e) x [ n ] = 0 . 5 n u [ n ] - 0 . 8 n u [ - n ] (f) x [ n ] = 2 n ( u [ n ] - u [ n - 10]) Problem 2 (30 points) For each of the following z-transforms: Sketch the pole-zero plot and shade the region of convergence. Compute the inverse z-transform, corresponding to the ROC you determined. (a) X 1 ( z ) = z - 1 1 + 3 jz - 1 , ROC: | z | > 3 (b) X 2 ( z ) = 1 - 1 3 z - 1 1 + 3 4 z - 1 + 1 8 z - 2 , x 2 [ n ] is causal (c) X 3 ( z ) = 1 + z - 2 1 - 5 2 z - 1 + z - 2 , ROC contains the unit circle | z | = 1 1

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Problem 3 (20 points) Compute the convolution y [ n ] = x [ n ] * h [ n ] for the x [ n ] and h [ n ] given below, using the z-
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Unformatted text preview: transform. Specify the region of convergence for X ( z ), Y ( z ) and H ( z ). (a) x [ n ] = 1 2 u [-n ] and h [ n ] = ( 1 2 ) n u [ n ] (b) x [ n ] = u [ n-5] and h [ n ] = 2cos( n ) u [ n ] Problem 4 (20 points) When the input to a LTI system is x [ n ] = ± 1 4 ¶ n u [ n ]-2 n u [-n-1] the output is y [ n ] = 14 ± 1 4 ¶ n u [ n ]-14 ± 1 2 ¶ n u [-n-1] (a) Find the system function H ( z ) = Y ( z ) X ( z ) of the system. Determine the poles and zeros of H ( z ) and indicate the region of convergence. (b) Find the impulse response h [ n ] of the system. (c) Write a diﬀerence equation that is satisﬁed by the given input and output. Is the system causal ? Comment if this is consistent with the ROC in part (a). (d) Draw a system block diagram corresponding to the diﬀerence equation found in part (c). 2...
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HW6_FA08 - transform Specify the region of convergence for...

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