EL611LaplaceFall - EL611 Laplace Fall 2008 1. State whether...

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EL611 Laplace Fall 2008 1. State whether the Laplace transforms of the following functions exist and, where they do exist, find the Laplace transforms and the regions of convergence. a) ) ( 5 ) ( 2 ) ( 3 t U e t U e t f t t - + = - - ) 3 )( 1 ( 13 3 1 5 3 2 ) ( + + - - = + - + = s s s s s s F ; ROC 1 } Re{ 3 - < < - s b) ) ( 5 ) ( 4 ) ( 2 t U e t U e t f t t - + = ) ( s F Does not exist because the causal part has ROC 2 } Re{ s , and the anticausal part has ROC 1 } Re{ < s . Thus, there are no common values of s where both terms converge. c) ) ( ) ( 7 3 t U e t t f t = 4 ) 7 ( ! 3 ) ( - = s s F ; ROC 7 } Re{ s 2. For the Laplace transform 3 2 3 ) 5 . 0 )( 1 ( 5 2 3 ) ( + - + + = s s s s s s F a) Find the Partial Fraction expansion of ) ( s F . This is the same function as in Homework 2, except that it is a function of s instead of z . Using the exact same solution as in solution 2, we get 3 2 3 2 3 ) 5 . 0 ( 12 / 19 ) 5 . 0 ( 9 / 22 5 . 0 27 / 1 1 27 / 80 ) 5 . 0 )( 1 ( 5 2 3 ) ( + + + - + + + - = + - + + = s s s s s s s s s s F b) Find all possible inverse transforms ) ( t f . Since there are poles at 1 = s and 5 . 0 - = s , there are three possible ROC’s, with the following inverses. ROC 1 } Re{ s :
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) ( 24 / 19 ) ( 9 / 22 ) ( 27 / 1 ) ( 27 / 80 ) ( 5 . 0 2 5 . 0 5 . 0 t U e t t U e t t U e t U e t f t t t t - - - + - + = ROC 1 } Re{ 5 . 0 < < - s : ) ( 24 / 19 ) ( 9 / 22 ) ( 27 / 1 ) ( 27 / 80 ) ( 5 . 0 2 5 . 0 5 . 0 t U e t t U e t t U e t U e t f t t t t - - - + - + - - = ROC 5 . 0 } Re{ < s : ) ( 24 / 19 ) ( 9 / 22 ) ( 27 / 1 ) ( 27 / 80 ) ( 5 . 0 2 5 . 0 5 . 0 t U e t t U e t t U e t U e t f t t t t - - - + - - - - = - - - 3. Determine whether the following LTI systems are BIBO stable. Do it two ways, first by calculating - dt t h ) ( , and then by checking the region of convergence of ) ( s H .
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This note was uploaded on 03/04/2010 for the course EE ee taught by Professor Ee during the Spring '10 term at Istanbul Technical University.

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EL611LaplaceFall - EL611 Laplace Fall 2008 1. State whether...

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