Quiz5sol_FA08

Quiz5sol_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 Quiz 5 Solutions Thursday, October 30, 2008 Student Name: Section: D. Jones (1pm), Y. Bresler (3pm) Problem 1 (25 points) A LSI system is described by the dierence equation y [ n ] = x [ n-1] + x [ n ] . a. (5 points) Compute and sketch the magnitude and phase of the frequency response of the system. b. (20 points) Compute the output of the system due to the following inputs: i. (10 points) x 1 [ n ] = cos ( 2 n + 6 ) ii. (10 points) x 2 [ n ] = e jn/ 5 Solution: a. H d ( ) = 1 + e-jw = 2 cos ( / 2) e-j/ 2 The frequency response of the system is plotted in gure 1. The magnitude and phase of H d ( ) are: | H d | = 2 cos ( / 2) H d ( ) =-/ 2 b. 1 Figure 1: Magnitude and phase of the frequency response i. x 1 [ n ] = cos ( 2 n + 6 ) y 1 [ n ] = | H d 2 | cos n 2 + 6 + H d 2 = 2 cos 2 n + 6- 4 = 2 cos 2 n- 12 Alternatively, y 1 [ n ] = x [ n-1] + x [ n ] = cos 2 ( n-1) + 6 + cos 2 n + 6 = -sin 2 n + 6 1 2 + cos 2 n + 6 1 2 2 = 2 cos 2 n- 12 ii. x 2 [ n ] = e jn 5 y 2 [ n ] = | H d ( 5 ) | e j ( n/ 5+ H d ( / 5)) = 2 cos 10 e j ( n 5- 10 ) Alternatively, y 2...
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Quiz5sol_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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