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20072ee132A_1_hwk1_sol

20072ee132A_1_hwk1_sol - EE132A Spring 2007 Prof John...

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EE132A, Spring 2007 Prof. John Villasenor Communication Systems TA: Choo Chin (Jeffrey) Tan Handout# 4 Homework 1 Solution Note: Euler’s Formula cos sin cos 2 cos sin sin 2 j j j j j j e e e j e e e j j θ θ θ θ θ θ θ θ θ θ θ θ - - - + = + = - = - = 1. Evaluate the Fourier transform of the following functions of time: ) ( ) ( ) ( ) 30 2 cos( ) ( ) ( ) ( 3 2 ) ( 1 t u te t s t u t t s t u e t s at b at - + - = + = = ringoperator where ) ( t u is the unit step function. (a) Use direct evaluation (a>0) 2 1 1 ( ) 2 ( 2 ) 0 ( 2 ) 0 ( ) ( ) ( ) ( 2 ) 2 j ft at b j ft b a j f t b a j f t b S f s t e dt e u t e dt e e dt e e a j f e a j f π π π π π π - -∞ - + - -∞ - - + - - + - = = = = - + = + You can also use the Fourier Transform table in the text book: { } ( ) 1 1 1 ( ), 0 2 ( ) ( ) ( ( )) ( ) ( ) 2 at at b b at b b at e u t a a j f s t e u t e e u t e S f e F e u t a j f π π - - + - - - - - > + = = = = + .

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(b) Given ( ) 1 t δ , by using the integration property, 1 1 ( ) ( ) ( ) ( ) 2 2 t u t d U f f j f δ τ τ δ π -∞ = = + . Using the frequency shift property, 6 6 (2 ) (2 ) 2 2 6 6 2 6 6 2 1 ( ) cos(2 ) ( ) ( ) ( ) ( ) 6 2 2 2 1 1 1 1 1 1 ( ) ( ) ( ) 2 2 ( 1/ ) 2 2 2 ( 1/ ) 2 j j j t j t j t j t j j e e s t t u t e e u t e
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