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Unformatted text preview: EECS 20N: Structure and Interpretation of Signals and Systems Department of EECS University of California Berkeley Problem Set 5 Issued: 16 November 2010 Due: 30 November 2010, 6pm The creative mind It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clari ed and exhausted a subject, then I turn away from it, in order to go into darkness again. The neversatis ed man is so strange; if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others. Carl Friedrich Gauss 1 Policy Statement We encourage you to collaborate, but only in a group of up to ve current EECS 20N students. On the solution document that you turn in for grading, you must write the names of your collaborators below your own; each teammate must submit for our evaluation a distinct, selfprepared solution document containing original contributions to the collaborative e ort. Please write neatly and legibly, because if we can't read it, we can't grade it. Unless we explicitly state otherwise, you will receive full credit only if you explain your work succinctly, but clearly and convincingly. Typically, we evaluate your solutions for only a subset of the assigned problems. A priori, you do not know which subset we will grade. It is to your advantage to make a bona de e ort at tackling every assigned problem. If you are asked to provide a \sketch," it refers to a handdrawn sketch, well labeled to indicate all the salient featuresnot a plot generated by a computing device. Overview This problem set reviews the DiscreteTime Fourier Series (DFS), ContinuousTime Fourier Series (CFS) and the DiscreteTime Fourier Transform (DTFT) after we cover it. Reading Chapter 7 and Chapter 10 of the book of Lee and Varaiya. 2 HW5.1 (LTI Processing of Periodic Signals) Consider a periodic, discretetime sig nal x : Z ! R having the discrete Fourier series (DFS) expansion x ( n ) = X k = h p i X k e ik! n ; where ! denotes the fundamental frequency of the signal; if p is the period of x , then ! = 2 =p . Suppose x is the input signal applied to a linear timeinvariant (LTI) system charac terized by the impulse response h : Z ! R and corresponding frequency response H , where H ( ! ) = 1 X n = 1 h ( n ) e i!n ; 8 !: Let y be the corresponding output signal. (a) Prove that the output signal y is periodic; that is, show that if x ( n + p ) = x ( n ), then y ( n + p ) = y ( n )....
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 Spring '11
 BABAK
 Fourier Series, HPI, Carl Friedrich Gauss, sion tht

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