ECE633F09_HW9solutions - Page 1 2/11/2010 11:43:27 AM

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Unformatted text preview: Page 1 2/11/2010 11:43:27 AM http://74.125.93.132/search?q=cache:dJaSjAkzai8J:www.ece.unh.edu/courses/ece633/pdf/ECE633F09_HW9solutions.pdf+*+s... This is the html version of the file http://www.ece.unh.edu/courses/ece633/pdf/ECE633F09_HW9solutions.pdf . Google automatically generates html versions of documents as we crawl the web. Page 1 ECE 633 Signals and Systems I , Fall 2009 Homework 9 Solutions From : Linear Systems and Signals , 2 nd ed . B . P . Lathi , Oxford University Press , 2005 . 6 . 1- 4 ( a ) Find the trigonometric Fourier series for the periodic signal ( ) g t in Fig . P 6 . 1- 4 . Take advantage of the symmetry . ( b ) Observe that ( ) g t is identical to ( ) x t in Fig . 6 . 3 a left- shifted by . 5 second . Use this fact to obtain the Fourier series for ( ) g t from the results in Example 6 . 2 . Verify that the Fourier series thus obtained is identical to that found in part ( a ) . ( c ) Show that , in general , a time shift of T seconds of a periodic signal does not affect the amplitude spectrum . However , the phase of the n th harmonic is increased or decreased n T depending on whether the signal is advanced or delayed by T seconds . ( a ) a = ( By inspection ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 2 cos 2 cos 2 cos n T a x t n t dt A At n t dt A At n t dt T T T = = + + ( ) ( ) ( ) 1 1 1 1 2 4 4 cos cos cos n A A A a n t dt t n t dt t n t dt T T T = + ( ) ( ) ( ) ( ) ( ) 1 1 2 2 2 2 1 1 sin cos sin cos sin 2 4 4 n n t n t t n t n t t n t A A A a T n T n n T n n = + + + ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 sin cos sin cos sin 1 1 2 2 2 n n n n n n a A A A n n n n n n n = + + ( ) 2 2 2 2 even 4 1 cos 8 odd n n A a n A n n n = = b = ( By inspection ) Page 2 2/11/2010 11:43:27 AM http://74.125.93.132/search?q=cache:dJaSjAkzai8J:www.ece.unh.edu/courses/ece633/pdf/ECE633F09_HW9solutions.pdf+*+s......
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ECE633F09_HW9solutions - Page 1 2/11/2010 11:43:27 AM

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