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 + + = + = = , then )] 38 ( ) 38 ( ) ( 2 [ 2 13 )] 38 ( 2 ) ( 4 ) 38 ( 2 [ 4 13 ) ( + + + = + + + = 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 then ] 5 2 1 [ ) ( 5 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 , then )]} 17 ( ) 17 ( [ 1 { 2 1 ) ( 2 / + = 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 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 j j e e j X + + = = then )] 22 ( ) ( 2 ) 22 ( [ 4 1 ) ( + + + = t t t t x . Notice the similarity between the answer here and that for 12.1(a) above (a duality property); (b) If j j j j X + + = + = 4 14 1 4 10 ) ( then ) ( 14 ) ( ) ( 4 t u e t t x t + = ; (c) If )] 37 ( ) 37 ( [ 13 ) ( + = j j X then t t x 37 sin 13 ) ( = ; (d) If 10 / )] 47 ( ) 47 ( [ ) ( j e u u j X + = , then ) 10 / 1 ( ) 10 / 1 ( 47 sin ) ( = t t t x . Problem 12.3: A continuous-time LTI system has an impulse response: ) 2 ( )] 2 ( sin[ 7 ) ( = t t t h CO , and the input to the system is: t k j jk t k j k e e e j t x 6 . 2 / 6 ....
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HW12s09_soln - ECE-2025 Homework #12 Solutions Spring-2009...

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