HW11s09_soln

HW11s09_soln - ECE-2025 Homework #11 Solutions Spring-2009...

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Unformatted text preview: ECE-2025 Homework #11 Solutions Spring-2009 Problem 11.1: If ) ( t x has the Fourier transform ) ( ω j X then ) ( d t t x − has the Fourier transform ) ( ω ω j X e d t j − (a) If ) ( ) ( t t x δ = then 1 ) ( = ω j X . So if ) 1 ( ) ( 2 ) 1 ( ) ( − + + + = t t t t x δ δ δ then ω ω ω ω cos 2 2 2 ) ( + = + + = − j j e e j X ; (b) If t t t x b π ω / sin ) ( = then ) ( ) ( ) ( b b u u j X ω ω ω ω ω − − + = So if 2 ) 2 ( 100 sin ) ( − − = π π t t x then )] 100 ( ) 100 ( [ ) ( 2 π ω π ω ω ω − − + = − u u e j X j ; (c) If ) ( ) ( t u e t x t − = then ω ω j j X + = 1 1 ) ( So if )] 4 ( [ ) ( )] 4 ( ) ( [ ) ( ) 4 ( 4 − − = − − = − − − − − t u e e t u e t u t u e t x t t t then ω ω ω ω ω ω ω 4 4 4 4 1 1 1 1 1 1 1 ) ( j j e j e j j e e j j X − − − − + − + = + − + = . Problem 11.2: Use the known Fourier transform pairs in Tables 11-2 and 11-3: (a) If ) ( 6 9 6 2 9 ) ( 3 1 ω ω ω j j j X + = + = then ) ( 6 9 ) ( 3 1 t u e t x t − = ; (b) If ) ( 2 cos 2 2 ) ( ω ω ω ω j j e e j X − + + = + = then ) 1 ( ) 1 ( ) ( 2 ) ( − + + + = t t t t x δ δ δ ; (c) If ω ω ω j j j X + − + = 2 1 1 1 ) ( then ) ( ) ( ) ( 2 t u e t u e t x t t − − − = ; (d) If ) 100 ( ) 100 ( ) ( π ω δ π ω δ ω + − − = j j j X , then t t x π 100 sin ) ( − = . Problem 11.3: A continuous-time LTI system has a frequency response: ω ω ω ω 8 7 sin ) ( j e j H − = (a) The DC response is 7 | ] / / 7 sin [ ) ( = = = ω ω ω ω ω d d d d j H by using the l’Hopital’s rule of differentiation; (b) From the shape of the magnitude response, it is clearly an LPF, with the magnitude and phase plots shown below (in the next page), and with zeros at 7 / π ω k ± = for any positive integer k ; (c) Using the delay property from Problem 11-1 and from the third row of Table 11-2, we have T=14 and )] 15 ( ) 1 ( [ 2 1 )] 7 8 ( ) 7 8 ( [ 2 1 ) ( − − − = − − − + − = t u t u t u t u t h clearly we have a rectangular pulse function with a height of 1/2, a width of 14 (similar to a length-14 digital filter), starting at...
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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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HW11s09_soln - ECE-2025 Homework #11 Solutions Spring-2009...

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