HWA_ece220_2007_solution

HWA_ece220_2007_solution - Homework 10 Solution(ECE220 Fall...

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Unformatted text preview: Homework 10 Solution (ECE220 - Fall 2007) 1. (30 points) An LTI system has input x [ n ] = [ a- n + a- 3 n ] u [ n ] and output y [ n ] = [ a- n + a- 4 n ] u [ n ] (a) (15 points) Determine the frequency response of the system. Solution: X ( e jω ) = 1 1- a- 1 e- jω + 1 1- a- 3 e- jω = 1- a- 1 e- jω + 1- a- 3 e- jω (1- a- 1 e- jω )(1- a- 3 e- jω ) = 2- ( a- 1 + a- 3 ) e- jω (1- a- 1 e- jω )(1- a- 3 e- jω ) Y ( e jω ) = 1 1- a- 1 e- jω + 1 1- a- 4 e- jω = 1- a- 1 e- jω + 1- a- 4 e- jω (1- a- 1 e- jω )(1- a- 4 e- jω ) = 2- ( a- 1 + a- 4 ) e- jω (1- a- 1 e- jω )(1- a- 4 e- jω ) H ( e jω ) = Y ( e jω ) X ( e jω ) = (2- ( a- 1 + a- 4 ) e- jω )(1- a- 3 e- jω ) (1- a- 4 e- jω )(2- ( a- 1 + a- 3 ) e- jω ) = 2- ( a- 1 + 2 a- 3 + a- 4 ) e- jω + ( a- 4 + a- 7 ) e- j 2 ω 2- ( a- 1 + a- 3 + 2 a- 4 ) e- jω + ( a- 5 + a- 7 ) e- j 2 ω (b) (15 points) Use section 5.8 from the book to provide the difference equation that corre- sponds to this system....
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HWA_ece220_2007_solution - Homework 10 Solution(ECE220 Fall...

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