HW12s09_soln

# HW12s09_soln - ECE-2025 Homework#12 Solutions Spring-2009...

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Unformatted text preview: ECE-2025 Homework #12 Solutions Spring-2009 Problem 12.1: Use the known Fourier transform pairs in Tables 11-2 and 11-3: (a) ] 2 [ 4 13 ] 2 [ 13 ) 19 ( cos 13 ) ( 38 38 2 19 19 2 t j t j t j t j e e e e t t x π π π π π − − + + = + = = , then )] 38 ( ) 38 ( ) ( 2 [ 2 13 )] 38 ( 2 ) ( 4 ) 38 ( 2 [ 4 13 ) ( π ω δ π ω δ δ π π ω πδ πδ π ω πδ ω + + − + = + + + − = t t j X ; (b) ) 1 ( 2 ) 1 ( ) 1 ( 2 ) 1 ( ) ( ) 1 ( 5 5 5 − − − = − − − = − − − − t u e e t t u e t t x t t δ δ then ] 5 2 1 [ ) ( 5 ω ω j e e j X jt + − = − − ; (c) If ] ) 2 / 1 ( ) 2 / 1 ( 17 sin ) 2 / 1 ( [ 2 1 2 ) 2 / 1 ( 17 sin ) 2 / 1 ( 2 1 ) ( − − − − = − − − − = t t t t t t t x π π δ π π π δ , then )]} 17 ( ) 17 ( [ 1 { 2 1 ) ( 2 / π ω π ω ω ω − − + − = − u u e j X j which is an ideal HPF. (d) )] 1 12 ( ) 1 12 ( [ )] 13 ( ) 11 ( [ ) ( − − − + − = − − − = t u t u t u t u t x , then 2 / sin ) ( 12 ω ω ω ω j e j X − = . Problem 12.2: Use the known Fourier transform pairs in Tables 11-2 and 11-3: (a) If ] 2 [ 4 1 ) 11 ( cos ) ( 22 22 2 ω ω ω ω j j e e j X − + + = = then )] 22 ( ) ( 2 ) 22 ( [ 4 1 ) ( − + + + = t t t t x δ δ δ . Notice the similarity between the answer here and that for 12.1(a) above (a duality property); (b) If ω ω ω ω j j j j X + + − = + − = 4 14 1 4 10 ) ( then ) ( 14 ) ( ) ( 4 t u e t t x t − + − = δ ; (c) If )] 37 ( ) 37 ( [ 13 ) ( π ω δ π ω δ ω + − − = j j X then t t x π π 37 sin 13 ) ( − = ; (d) If 10 / )] 47 ( ) 47 ( [ ) ( ω π ω π ω ω j e u u j X − + − − = , then ) 10 / 1 ( ) 10 / 1 ( 47 sin ) ( − − − = t t t x π π . Problem 12.3: A continuous-time LTI system has an impulse response: ) 2 ( )] 2 ( sin[ 7 ) ( − − = t t t h CO π ω , and the input to the system is: t k j jk t k j k e e e j t x π π π 6 . 2 / 6 ....
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## This note was uploaded on 01/27/2011 for the course ECE 2025 taught by Professor Juang during the Spring '08 term at Georgia Tech.

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HW12s09_soln - ECE-2025 Homework#12 Solutions Spring-2009...

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