SolutionsV_1 - ECE 3250 HOMEWORK V SOLUTIONS Fall 2009 1....

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Unformatted text preview: ECE 3250 HOMEWORK V SOLUTIONS Fall 2009 1. In each case, S ( x ) = Conv h ( x ). (a) S ( x ) = Conv h ( x ), so S ( x )( t ) = Z - h ( t- ) x ( ) d = Z - e- 3( t- ) u ( t- ) e- 5 u ( ) d = Z e- 3( t- ) u ( t- ) e- 5 d = e- 3 t R t e- 2 d if t if t < = (1 / 2)( e- 3 t- e- 5 t ) if t if t < . The equality on the third line holds because of the u ( ) on the second line. The equality on the fourth line holds because of the u ( t- ) on the third line. Note that we can write the final answer as S ( x )( t ) = (1 / 2)( e- 3 t- e- 5 t ) u ( t ) for all t R . (b) S ( x ) = Conv h ( x ), so S ( x )( t ) = Z - h ( t- ) x ( ) d = Z - e- 3( t- ) u ( t- ) e- 5 | | d = Z t- e- 3( t- ) e- 5 | | d = ( e- 3 t R t- e 8 d if t < e- 3 t R- e 8 d + e- 3 t R t e- 2 d if t = (1 / 8) e 5 t if t < (1 / 8) e- 3 t + (1 / 2) e- 3 t- (1 / 2) e- 5 t if t . The equality on the third line holds because of the u ( t- ) on the second line. Note that we can write the final answer as S ( x )( t ) = ((5 / 8) e- 3 t- (1 / 2) e- 5 t ) u ( t ) + (1 / 8) e 5 t u (- t ) for all t R . (c) Conv ( x ) = x , so Conv Shift 7 ( ) ( x ) = Shift 7 ( x ). Since the first term in h is Shift 7 of the impulse response of the system in (a), and since x is the same as in (a), the convolution of the first term in h with x is Shift 7 of the answer to (a). Accordingly, S ( x )( t ) = (1 / 2)( e- 3( t- 7)- e- 5( t- 7) ) u ( t- 7) + e- 5( t- 7) u ( t- 7) for all t R = (1 / 2)( e- 3( t- 7) + e- 5( t- 7) ) u ( t- 7) for all t R . 1 2 (d) S ( x ) = Conv h ( x ), so S ( x )( t ) = Z - h ( t- ) x ( ) d = Z - e- 3( t- ) u ( t- ) u (- ) d = Z- e- 3( t- ) u ( t- ) d = ( e- 3 t R- e 3 d if t e- 3 t R t- e 3 d if t < . = (1 / 3) e- 3 t if t 1 / 3 if t < . The equality on the third line holds because of the u (- ) on the second line. The equality on the fourth line holds because of the u ( t- ) on the third line. Note that we can write the final answer as (1 / 3) e- 3 t u ( t ) + 1 / 3 u (- t ) for all t R . 2. (a) S ( u ) = Conv h ( u ), so S ( u )( t ) = Z - h ( t- ) u ( ) d = Z - 7( e- 3( t- )- e- 5( t- ) u ( t- ) u ( ) d = Z 7( e- 3( t- )- e- 5( t- ) ) u ( t- ) d = 7 e- 3 t R t e 3 d- 7 e- 5 t R t e 5 d if t if t < = (7 / 3)(1- e- 3 t )- (7 / 5)(1- e- 5 t ) if t if t < The equality on the third line holds because of the u ( ) on the second line. The) on the second line....
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This note was uploaded on 03/26/2010 for the course ECE 3250 at Cornell University (Engineering School).

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SolutionsV_1 - ECE 3250 HOMEWORK V SOLUTIONS Fall 2009 1....

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