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ee313_laplace_transform

# ee313_laplace_transform - EE 313 Linear Systems and Signals...

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Laplace Transform EE 313 Linear Systems and Signals Fall 2008 Prof. Adnan Kavak Dept. of Electrical and Computer Engineering The University of Texas at Austin Courtesy of Prof. Brian Evans

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Forward Laplace Transform Decompose a signal f ( t ) into complex sinusoids of the form e s t where s is complex: s = σ + j 2 π f Forward (bilateral) Laplace transform f ( t ): complex-valued function of a real variable t F ( s ): complex-valued function of a complex variable s Bilateral means that the extent of f ( t ) can be infinite in both the positive t and negative t direction (a.k.a. two-sided) ( 29 ( 29 - - = dt e t f s F t s
Inverse (Bilateral) Transform Inverse (Bilateral) Transform a is a contour integral which represents integration over a complex region– recall that s is complex c is a real constant chosen to ensure convergence of the integral Notation F ( s ) = L { f ( t )} variable t implied for L f ( t ) = L -1 { F ( s )} variable s implied for L -1 ( 29 ( 29 ds e s F j t f t s j c j c 2 1 + - = π ( 29 ( 29 s F t f L →

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( 29 ( 29 ( 29 ( 29 s F t f s F t f L L 2 2 1 1 and → → ( 29 ( 29 ( 29 ( 29 ? 2 2 1 1 2 2 1 1 s F a s F a t f a t f a L + → + ( 29 ( 29 { } ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 s F a s F a dt e t f a dt e t f a dt e t f a t f a t f a t f a L st st st 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 + = + = + = + - - - - - - L F (s) f ( t ) Laplace Transform Properties Linear or nonlinear? Linear operator
Laplace Transform Properties Time-varying or time-invariant? This is an odd question to ask because the output is in a different domain than the input. ( 29 ( 29 s F t f L 1 1 → ( 29 { } ( 29 - - - = - dt e t t f t t f L st 0 1 0 1 : change not do limits , , Let 0 0 du dt t u t t t u = + = - = ( 29 { } ( 29 ( 29 ( 29 - - - - - + - = = = - ) ( 1 1 1 0 1 0 0 0 s F e du e u f e du e u f t t f L st su st t u s

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Example ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 a s e a s e a s dt e dt e e dt e t u e s F t u e t f t a s t t a s t a s st at st at at + + + - = + - = = = = = + - + - + - - - - - - - 1 1 lim 1 ) ( ) ( term? this to happens What 0 0 0 o o o o o o o o o ( 29 n oscillatio t as ion amplificat or n attenuatio , number complex a For t j t t j t z e e e e α z β α β α - - + - - = = + =
Convergence

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ee313_laplace_transform - EE 313 Linear Systems and Signals...

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