# 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 diﬀerence 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 ( π 2 n + π 6 ) ii. (10 points) x 2 [ n ] = e jnπ/ 5 Solution: a. H d ( ω ) = 1 + e-jw = 2 cos ( ω/ 2) e-jω/ 2 The frequency response of the system is plotted in ﬁgure 1. The magnitude and phase of H d ( ω ) are: | H d | = 2 cos ( ω/ 2) ∠ H d ( ω ) =-ω/ 2 b. 1 Figure 1: Magnitude and phase of the frequency response i. x 1 [ n ] = cos ( π 2 n + π 6 ) y 1 [ n ] = | H d ± π 2 ² | cos ± nπ 2 + π 6 + ∠ H d ± π 2 ²² = √ 2 cos ± π 2 n + π 6-π 4 ² = √ 2 cos ± π 2 n-π 12 ² Alternatively, y 1 [ n ] = x [ n-1] + x [ n ] = cos ± π 2 ( n-1) + π 6 ² + cos ± π 2 n + π 6 ² = ³-sin ± π 2 n + π 6 ² 1 √ 2 + cos ± π 2 n + π 6 ² 1 √ 2 ´ √ 2 = √ 2 cos ± π 2 n-π 12 ² ii. x 2 [ n ] = e jn π 5 y 2 [ n ] = | H d ( π 5 ) | e j ( nπ/ 5+ ∠ H d ( π/ 5)) = 2 cos ± π 10 ² e j ( n π 5-π 10 ) Alternatively, y 2...
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Quiz5sol_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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