HW3Soln - ECE 301: Homework 3 Solutions Landis Huffman...

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Unformatted text preview: ECE 301: Homework 3 Solutions Landis Huffman Purdue University, Summer 2009 Due July 9, 2009 in class 1. Chapter 3 Book problems 3.3, 3.8, 3.22(b), 3.30, 3.37(a) 3.3 Solution T = 6, w = 2 π/ 6 = π/ 3. The signal can be written as a sum of complex exponentials using Euler’s formula x ( t ) = 2 + 1 2 ( e j 2 w t + e- j 2 w t ) + 4 1 2 j ( e j 5 w t − e- j 5 w t ) = 2 + 1 2 e j 2 w t + 1 2 e- j 2 w t − 2 je j 5 w t + 2 je- j 5 w t The non-zero Fourier series coefficients of x ( t ) are: a = 2 , a 2 = a- 2 = 1 2 , a 5 = a *- 5 = − 2 j. 3.8 Solution Since x ( t ) is real and odd (clue 1), its Fourier series coefficients a k are purely imaginary and odd (See Table 3.1). Therefore, a k = − a- k and a = 0. Also, since it is given that a k = 0 for | k | > 1, the only unknown Fourier series coefficients are a 1 and a- 1 . Using Parseval’s relation, 1 T integraldisplay <T> | x ( t ) | 2 dt = ∞ summationdisplay k =-∞ | a k | 2 , for the given signal we have 1 2 integraldisplay 2 | x ( t ) | 2 dt = 1 summationdisplay k =- 1 | a k | 2 . 1 Using the information given in clue 4 along with the above equation, | a 1 | 2 + | a- 1 | 2 = 1 ⇒ 2 | a 1 | 2 = 1 . Therefore, a 1 = − a- 1 = 1 √ 2 j or a 1 = − a- 1 = − 1 √ 2 j . The two possible signals which satisfy the given information are: x 1 ( t ) = 1 √ 2 j e jπt − 1 √ 2 j e- jπt = √ 2 sin( πt ) . and x 2 ( t ) = − 1 √ 2 j e jπt + 1 √ 2 j e- jπt = − √ 2 sin( πt ) . 3.22(b) Solution x ( t ) periodic with period 2 and x ( t ) = e- t for − 1 < t < 1 a k = 1 2 integraldisplay 1- 1 e- t e- jkπt dt = 1 2 integraldisplay 1- 1 e (- 1- jkπ ) t dt = 1 2 bracketleftbigg 1 − 1 − jkπ e (- 1- jkπ ) t bracketrightbigg 1- 1 = 1 2(1 + jkπ ) [ e (1+ jkπ ) − e (- 1- jkπ ) ] = 1 2(1 + jkπ ) [ e ( − 1) k − e- 1 ( − 1) k ] = ( − 1) k 2(1 + jkπ ) [ e − e- 1 ] ....
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This note was uploaded on 01/10/2010 for the course CDFS 21000 taught by Professor Alisone.baroody during the Summer '09 term at Vincennes.

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HW3Soln - ECE 301: Homework 3 Solutions Landis Huffman...

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