# HW14_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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1 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Homework 14 Due Wednesday, December 10, 2008 Prof. Bresler / Prof. Jones Problem 1 (25 points) Compute the following cyclic convolutions: (a) {x n } 5 n=0 = {1, 2, 3, 4, 5, 6} and {y n } 5 n=0 = {1, 0, 0, 1, 0, 0}. Compute z[n] = = 5 0 6 ] [ ] [ k k n y k x , n=0,1,2,3,4,5. (b) {x n } 8 n=0 ={1, 2, 3, 4, 5, 6,0,0,0}; {y n } 8 n=0 ={1,0,0,1,0, 0, 0,0,0} Compute p[n] = = 8 0 9 ] [ ] [ k k n y k x n=0,1,2,3,4,5,6,7,8 (c) {x n } 7 n=0 ={1, 2, 3, 4, 5, 6,0,0}; {y n } 7 n=0 ={1,0,0,1,0, 0, 0,0} Compute w[n] = = 7 0 8 ] [ ] [ k k n y k x n=0,1,2,3,4,5,6,7 (d) Compute z linear [n], the linear convolution of the sequences in (a). (e) What is the relationship between the linear convolution of x[n] and y[n] computed in (d) and the circular convolutions computed in (a), (b) and (c) ? Problem 2 (15 points) The nonzero part of the impulse response of a particular lowpass filter is given by h[n] 4 n=0 = {0.1, 0.5, 1, 0.5, 0.1} Your goal is to convolve h[n] with the following signal:

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HW14_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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