ECE130A-F10-HW04(2)

# ECE130A-F10-HW04(2) - where H(j ω = Z ∞-∞ e-j ω t h...

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ECE 130A: Signal Analysis and Processing Prof. Michael Liebling Fall 2009 TAs: Jingning Han, Hari Sivakumar Distribution: Wednesday, 27 October 2010 Homework Due: Tuesday, 2 November 2010 at 6:00 p.m. University of California, Santa Barbara Homework 4 Problem 1: Fourier Series and LTI systems (40 points) Let x ( t ) be a periodic function with fundamental pulsation ω 0 , whose Fourier series representation is: x ( t ) = X k =-∞ a k e j k ω 0 t , and h ( t ) the impulse response of a stable LTI system. Show that the output of the system, y ( t ) = h * x ( t ), has the following Fourier series representation y ( t ) = X k =-∞ b k e j k ω 0 t , with b k = a k H (j k ω
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Unformatted text preview: ), where H (j ω ) = Z ∞-∞ e-j ω t h ( t )d t . Problem 2: Using the System Frequency Response to Compute System Response to a Periodic Signal (60 Points) Consider a continuous-time LTI system whose impulse response is h ( t ) = sin(5 t ) π t . (a) Calculate the frequency response H (j ω ) = Z ∞-∞ h ( t )e-j ω t d t . (b) If the input to this system is x ( t ) = sin ± 2 π 8 t ¶ + cos(2 π t ), determine the corresponding output y ( t ). Hint: Decompose x ( t ) as a Fourier series and determine the response of the individual components....
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## This note was uploaded on 12/06/2010 for the course ECE 130A taught by Professor Madhow during the Fall '07 term at UCSB.

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