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Unformatted text preview: EE 261 The Fourier Transform and its Applications Fall 2009 Problem Set Two Due Wednesday, October 7, 2009 1. (10 points) A famous sum You cannot go through life knowing about Fourier series and not know the application to evaluating a very famous sum. Let S ( t ) be the sawtooth function, that is S ( t ) = t for 0 t 1 and then periodized to have period 1. Show that the Fourier series for S ( t ) is 1 2- summationdisplay n = ,n negationslash =0 1 2 in e 2 int and use it to show that summationdisplay n =1 1 n 2 = 2 6 . 2. (30 points) Some practice with symmetry Considerations of symmetry arise frequently in Fourier analysis, e.g., evenness or oddness of a signal. It is helpful to introduce the reversed signal , defined by f ( t ) = f (- t ) . Thus f is even if and only if f = f f is odd if and only if f =- f For any signal, f + f is even and f- f is odd, and thus any signal can be decomposed into its even and odd parts: f = 1 2 ( f + f ) + 1 2 ( f- f ) = f even + f odd . We do not assume that f is real-valued. Suppose that a square integrable function f , defined on- < x < , is written as a sum of its even and odd parts as f ( x ) = f even ( x ) + f odd ( x ). (a) (10) Show that integraldisplay | f ( x ) | 2 dx = integraldisplay | f even ( x ) | 2 dx + integraldisplay | f odd ( x ) | 2 dx 1 (b) (10) Find an expression of a similar kind for integraldisplay f...
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