20092ee132A_1_hwk1

20092ee132A_1_hwk1 - = by showing that S f S f 1 2 = 4 Time...

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EE132A, Spring 2009 Prof. John Villasenor Communication Systems TA: Pooya Monajemi, Erica Han Handout# 2 Homework 1 Assigned: Monday, March 30, 2009 Due: Monday, April 6, 2009 Reading Assignments: Fundamentals of Communication Systems , Chapter 2 (2.2, 2.3, 2.5) Note: It is useful to note that ) ( 2 1 ) 2 ( 1 ) ( )} ( { f f j f U t u F δ π + = = 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. 2. Convolve ) ( t u e at with ) ( t u e at 3. Shifted sinusoids. Given: ( ) s t f t s t f t c c 1 2 2 2 2 ( ) sin( ) ( ) cos = = π π π Show that s t s t 1 2 ( ) ( )
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Unformatted text preview: = by showing that S f S f 1 2 ( ) ( ) = . 4. Time Limited Sine. Given: ≤ ≤ = otherwise-for ) 2 sin( ) ( 2 T 2 T t t f t s c Express S f ( ) as a linear combination of sincs. 5. Find ∫ ∞ ∞ − − dt t u e t ) ( 2 using Parseval’s theorem. Hint: Use the fact that 2 2 ) ( ) ( t u e t u e t t − − = . 6 Use the time shift property to find the Fourier transform of: s t T s t T ( ) ( ) + − From this result, find the Fourier transform of ds dt ....
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