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Unformatted text preview: 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 . )( 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 . ( 12 / 19 ) 5 . ( 9 / 22 5 . 27 / 1 1 27 / 80 ) 5 . )( 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 . = s , there are three possible ROCs, with the following inverses. ROC 1 } Re{ s : ) ( 24 / 19 ) ( 9 / 22 ) ( 27 / 1 ) ( 27 / 80 ) ( 5 . 2 5 . 5 . t U e t t U e t t U e t U e t f t t t t + + = ROC 1 } Re{ 5 . < < s : ) ( 24 / 19 ) ( 9 / 22 ) ( 27 / 1 ) ( 27 / 80 ) ( 5 . 2 5 . 5 . t U e t t U e t t U e t U e t f t t t t + + = ROC 5 . } Re{ < s : ) ( 24 / 19 ) ( 9 / 22 ) ( 27 / 1 ) ( 27 / 80 ) ( 5 . 2 5 . 5 ....
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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.
 Spring '10
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