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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 πin e 2 πint and use it to show that ∞ summationdisplay n =1 1 n 2 = π 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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- Fourier Series, 1 2 g, S. Engelberg