ECE130A-F10-HW05(2)

ECE130A-F10-HW05(2) - ECE 130A Signal Analysis and...

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ECE 130A: Signal Analysis and Processing Prof. Michael Liebling Fall 2009 TAs: Jingning Han, Hari Sivakumar Distribution: Monday, 1 November 2010 Electrical & Computer Engineering Dept. Homework Due: Tuesday, 9 November 2010 at 6:00 p.m. University of California, Santa Barbara Homework 5 Problem 1: Fourier Transform Computation (15 Points) Compute the Fourier transform of each of the following signals: (a) x ( t ) = ( 1 - t 2 , 0 < t < 1 0, otherwise. (b) x ( t ) = e -| t | cos(2 t ) (c) x ( t ) = sin(2 π t ) 2 π t sin[ π ( t - 1)] π ( t - 1) Problem 2: Inverse Fourier Transform Computation (15 Points) Compute the inverse Fourier trans- form of each of the following signals: (a) X (j ω ) = - πδ ( ω ) + πδ ( ω - 2 π ) + πδ ( ω + 2 π ) (b) X (j ω ) = 1, 0 ω 1 - 2, - 2 ω < 0 0, otherwise. (c) X (j ω ) = [ u ( ω + 1) - u ( ω - 1)]e j( - ω /2 + π ) Problem 3: System Frequency Response (25 Points) An LTI system has impulse response h ( t ) = u ( t - 2) - u ( t + 1) where the unit of time is seconds, that is, [ t ] = s. (a) Determine the magnitude and phase of the frequency response H (j2 π f ) as a function of f , with [ f ] = s - 1 = Hz.
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