20092ee132A_1_hwk1_sol

# 20092ee132A_1_hwk1_sol - EE132A Spring 2009 Prof John...

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Unformatted text preview: EE132A, Spring 2009 Prof. John Villasenor Communication Systems TA: Pooya Monajemi , Erica Han 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 − + − = + = = o where ) ( t u is the unit step function. (a) Use direct evaluation (a>0) 2 1 1 ( ) 2 ( 2 ) ( 2 ) ( ) ( ) ( ) ( 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 ( ), 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 − − + − − − − − > ⇔ + = = ∴ = = + . (b) Given ( ) 1 t δ ⇔...
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20092ee132A_1_hwk1_sol - EE132A Spring 2009 Prof John...

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