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Electrical Engineering 20N - Spring 2002 - Varaiya - Final

# Electrical Engineering 20N - Spring 2002 - Varaiya - Final...

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EECS 20. Final Exam May 22, 2002. Please use these sheets for your answer and your work. Use the backs if necessary. Write clearly and show your work for full credit. Please check that you have 12 numbered pages. Print your name below Name: Problem 1 (20): Problem 2 (15): Problem 3 (15): Problem 4 (20): Problem 5 (10): Problem 6 (10): Problem 7 (10): Total: 1

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1. 20 points. Consider a continuous-time signal x : Reals Reals de£ned by t Reals, x ( t ) = cos( πt/ 2) + cos( πt ) + cos(3 πt ) + cos(5 πt ) . (a) Obtain the Fourier series coef£cients of x ( t ) , i.e., £nd the coef£cients A 0 ,A 1 ,A 2 ,... and φ 1 2 ,... and w 0 such that x ( t ) = A 0 + X k =1 A k cos( kw 0 t + φ k ) . (b) Obtain the Fourier series expansion for x ( t ) , i.e., £nd the coef£cients X k for all k Integers such that x ( t ) = X k = -∞ X k e ikw 0 t . 2
(c) Consider a continuous-time LTI system Filter : [ Reals Reals ] [ Reals Reals ] with the following frequency response H( ω ) = ( e if 2 < | ω | < 4 radians per second 0 otherwise For the x ( t ) given above, £nd an expression for the output y ( t ) of the system, where

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Electrical Engineering 20N - Spring 2002 - Varaiya - Final...

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