EL611 Laplace Soln

# EL611 Laplace Soln - EL611 Laplace Fall 2008 1 State...

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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.

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ROC 1 } 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 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
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EL611 Laplace Soln - EL611 Laplace Fall 2008 1 State...

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