ECE633F09_HW8solutions

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Unformatted text preview: Page 1 This is the html version of the file http://www.ece.unh.edu/courses/ece633/pdf/ECE633F09_HW8solutions.pdf. Google automatically generates html versions of documents as we crawl the web. Page 1 ECE633 Signals and Systems I, Fall 2009 Homework 8 Solutions From: Linear Systems and Signals, 2 nd ed. B. P. Lathi, Oxford University Press, 2005. 6.1-1 For each of the periodic signals shown in Fig. P6.1-1, find the compact trigonometric Fourier series and sketch the amplitude and phase spectra. If either the sine of cosine terms are absent in the Fourier series, explain why. 1 C0 = T0 ) f t(dt T0 2 an = T0 f t( ) cos 2 nt T0 dt T0 2 bn = T0 f t( ) sin 2 nt T0 dt T0 http://74.125.93.132/search?q=cache:Mj4s0lPWpPAJ:www.ece.unh.edu/courses/ece633/pdf/ECE633F09_HW8solutions.pdf... 2/11/2010 11:43:48 AM Page 2 1 Page 2 ECE633 Signals and Systems I, Fall 2009 Homework 8 Solutions (a) 0 C = mean = 0 2 an = 4 2 nt 4 1 -1 cos nt 3 2 nt 12 dt - cos dt = sin 1 4 2 n 2 - n 2 1 nt - 2 sin n 2 -1 n 2 3 1 n 1 - sin an = sin n 2 1 an = 3sin n n - sin 3 n + sin 2 2 - 4sin 3 = 1 3sin n - sin 3 n 2 n 2 - 3sin n 2 n 2 n n = 4 sin 3 = 4 sin n n 2 2 integers bn= (even symmetry) 0 n 4 C n = an = sin n 2 f t( ) = n =1 = tan - 1 n nt 2 0 an = 0 180 an 0 a <0 n 4 n sin n 2 cos t = 4 cos - 1 cos 3 t + 1 cos 5 t - 1 cos 7 t + " 2 3 2 5 2 7 2 http://74.125.93.132/search?q=cache:Mj4s0lPWpPAJ:www.ece.unh.edu/courses/ece633/pdf/ECE633F09_HW8solutions.pdf... 2/11/2010 11:43:48 AM Page 3 Page 3 ECE633 Signals and Systems I, Fall 2009 Homework 8 Solutions 1 10 (b) C 0 = - dt = [t ] - = 1 10 5 1 2 an = 10 - cos nt 5 1 dt = sin n nt 5 - = 1 sin n n 5 - sin - n 5 = 2 sin n n 5 bn= (even symmetry) 0 2 n C n = an = sin n 5 1 f t( ) = + 5 n =1 = tan - 1 n n 5 cos 4 5 0 an = 0 180 an 0 a <0 n 2 n sin sin cos t nt 5 2 5 1 6 cos sin 2t + 1 sin 5 3 6 5 cos 3 5 cos 3t 5 1 =+ 2 5 5 + 1 sin 5 2 cos + 1 sin 4 4t + + 0 5 6t + " 5 3 Page 4 http://74.125.93.132/search?q=cache:Mj4s0lPWpPAJ:www.ece.unh.edu/courses/ece633/pdf/ECE633F09_HW8solutions.pdf... 2/11/2010 11:43:48 AM ...
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