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Unformatted text preview: 6.003 Homework 9 Due at the beginning of lecture on Thursday, November 12, 2009 . Problems 1. Fourier Series Determine the Fourier series coefficients for each of the following periodic CT signals. x 1 ( t )= x 1 ( t + 10) t 1 1 10 x 2 ( t )= x 2 ( t + 10) t 1 2 10 x 3 ( t )= x 3 ( t + 10) t 1 − 1 1 2 3 10 x 4 ( t )= x 4 ( t + 10) t 1 1 2 3 10 2. Inverse Fourier series Determine the CT signals with the following Fourier series coefficients. Assume that the signals are periodic in T = 4. a. a k = î jk ;  k  < 3 otherwise b. b k = 1; k odd 0; k even 6.003 Homework 9 / Fall 2009 2 3. Periodic convolution One of the nice properties of the Laplace and Z transforms is that they map convolution in the time domain into multiplication in the transform domain (see Tables 9.1 and 10.1 on pages 691 and 775 respectively in Oppenheim and Willsky). In this problem, you will work with an analogous property of CT Fourier series. Consider the periodic signal f 1 ( t ) shown below: f 1 ( t )= f 1 ( t + 6) 1 t 0 1 − 1 T = 6 12 a) What is problematic with evaluating the convolution of f 1 ( t ) with itself? To circumvent the difficulties you found in part a), we define a new type of convolution for periodic signals, called periodic convolution and written...
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 Spring '11
 DennisM.Freeman
 Fourier Series, Fourier series coefficients

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