20095ee102_1_EE102_ HW7_Solution

# 20095ee102_1_EE102_ HW7_Solution - , 1 ) ( ) ( ) 2 0 , 1, 1...

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EE102 HW 7 Fall 2009 1. Given the following formula for the Fourier Transform and the Inverse Fourier Transform: 2 2 ( ) ( ) ( ) ( ) j ft t j ft f F f f t e dt f t F f e df π - =-∞ =-∞ = = Find the Fourier Transform of the following functions: | | | | 2 2 2 2 | | 2 2 2 2 2 2 2 2 2 2 2 2 2 | | | | 1 1 ) ( ) (1 | |) 1 1 2 1 2 1 2 1 4 1 1 2(1 4 ) | | 2 1 2 2 1 2 (1 4 ) 2 2(1 4 ) 4 ( ) 1 4 (1 4 ) (1 4 ) sin( ) ) ( ) ( ) t t t t t a f t e t e j f j f f j d j d f t e df j f df j f f f F f f f f t b f t e t x t e X - - - - - = + + = + - + - - = + - + - = + = + + + = = 2 2 2 2 1/2 1/2 1 1 2 2 2 1/2 1/2 1 1 | | | | 2 2 ( ) 1 4 1 1 1, sin( ) 2 2 ( ) ( ) 0 2 1 ( ) ( ) ( ) tan (2 ) 1 4 1 ( ) [tan (2 ) tan (2 )] ) ( ) ( ) ( ) ( ) f f f u u f t t f f f f t x t X f t else F f X u X f u du du u u F f f f c f t e t F f e t e δ + + - - =-∞ = - - - - - - = + - ≤ = = = - = = + = + - - = = 1 t t dt =-∞ = 2. Find the Inverse Fourier Transform of the following functions:

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2 1 1 1 1 | | 2 2 sin( ) ) ( ) 1, 1/ 2 1/ 2 sin( ) ( ) ( ) 0, 1 , 1 0 ( ) ( ) ( ) 1 , 1 0 1 1 ) ( ) ( ) 1 4 2 t f a F f f t f F f f t f else t t f t f t f d t t b F f f t e f τ π =-∞ - = - ≤ ≤ = = + - ≤ ≤ = - = - - ≤ ≤ = = + 3. What is the Fourier Transform of the following function:
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Unformatted text preview: , 1 ) ( ) ( ) 2 0 , 1, 1 1 ) ( ) ( ) ( ) 2 2 0 , kt e t a f t F f k j f t t b f t F f f j f t δ- ≥ = ↔ = + < ≥ = ↔ = + < What you would get for part b, if you consider f(t) as the integral of ( ) t ? 4. Show that the Fourier Transform of ( ) cos(2 ) f t f t = is [ ] 1 ( ) ( ) ( ) 2 F f f f f f =-+-2 2 1 1 1 1 ( ) cos(2 ) ( ) ( ) ( ) 2 2 2 2 j f t j f t f t f t e e F f f f f f-= = + ↔ =-+ +...
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## This note was uploaded on 01/16/2012 for the course EE 102 taught by Professor Levan during the Spring '08 term at UCLA.

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20095ee102_1_EE102_ HW7_Solution - , 1 ) ( ) ( ) 2 0 , 1, 1...

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