10Fall_hw08soln - ECE210 - Fall 2010 - Homework 08...

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Unformatted text preview: ECE210 - Fall 2010 - Homework 08 Solutions 1. Find the exponential and trigonometric Fourier series coefficients of the periodic signal f ( t ) = e- 2 t , t < 3 . The period is T = 3 . Solution: F n = 1 T Z T f ( t ) e- jn t dt = 1 3 Z 3 e- 2 t e- jn t dt = 1 3 Z 3 e- ( jn +2) t dt =- 1 j 3 n + 6 e- ( jn +2) t 3 =- 1 j 3 n + 6 ( e- j 3 n- 6- 1) Substituting for = 2 T = 2 3 F n = 1 j 2 n + 6 ( 1- e- jn 2 - 6 ) = 1 2 1- e- 6 3 + jn = 1 2 1- e- 6 q 9 + ( n ) 2 e- j arctan ( n 3 ) = 1 2 1- e- 6 9 + ( n ) 2 ! (3- jn ) Then a n = F n + F- n = 3 1- e- 6 9 + ( n ) 2 ! b n = j ( F n- F- n ) = ( 1- e- 6 ) n 9 + ( n ) 2 2. Problem 6.3 from text. Solution: There is not one unique approach to deriving this solution, but most approaches should follow one of these two methods: 1 2 2 e- jn-- jn 2- 1 2 e- jn 2 t (- jn 2 ) 2 2 = e- jn- jn 2- 1 2 e- jn (- jn 2 ) 2- 1 (- jn 2 ) 2 evaluate the right side at bounds = e- jn- jn 2 + 2(1...
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This note was uploaded on 03/05/2011 for the course ECE 210 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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10Fall_hw08soln - ECE210 - Fall 2010 - Homework 08...

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