ECE633F09_HW9solutions

# ECE633F09_HW9solutions - Page 1 âˆ âˆ âˆ âˆ âˆ âˆ 11:43:27...

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Unformatted text preview: Page 1 âˆ« âˆ« âˆ« âˆ« âˆ« âˆ« 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 â€“ 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 Ï‰ 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 Ï‰ Ï‰ Ï‰ âˆ’ = = + + âˆ’ âˆ« âˆ« âˆ« ( ) ( ) ( ) 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 Ï‰ Ï‰ Ï‰ âˆ’ âˆ’ = + âˆ’ âˆ« âˆ« âˆ« ( ) ( ) ( ) ( ) ( ) 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 Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ âˆ’ âˆ’ Ú„ Ú„ â½· â½¹ â½· â½¹ â½· â½¹ = + + âˆ’ + â¾‡ â¾‡ â¾‡ â¾‡ â¾‡ â¾‡ â½¸ â½º â½¸ â½º â½¸ â½º ( ) ( ) ( ) ( ) ( ) 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 Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ â½· â½¹ â½· â½¹ = + âˆ’ âˆ’ âˆ’ + âˆ’ â¾‡ â¾‡ â¾‡ â¾‡ â½¸ â½º â½¸ â½º ( ) 2 2 2 2 even 4 1 cos 8 odd n n A a n A n n n Ï€ Ï€ Ï€ á¿² â¾‡ = âˆ’ = â½· â½¹ á¿³ â½¸ â½º â¾‡ á¿´ b = ( By inspection ) Page 2 âˆ« âˆ« âˆ« âˆ« âˆ« âˆ« 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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## This note was uploaded on 02/11/2010 for the course ECE 633 taught by Professor Staff during the Fall '09 term at University of Texas.

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ECE633F09_HW9solutions - Page 1 âˆ âˆ âˆ âˆ âˆ âˆ 11:43:27...

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