ee313_laplace_transform

ee313_laplace_transform - Laplace Transform EE 313 Linear...

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Unformatted text preview: 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 Forward Laplace Transform Decompose a signal f ( t ) into complex sinusoids of the form e s t where s is complex: s = + j 2 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 ( 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 + - = ( 29 ( 29 s F t f L ( 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 1 1 : change not do limits , , Let du dt t u t t t u = + = - = ( 29 { } ( 29 ( 29 ( 29 ---- - +- = = =- ) ( 1 1 1 1 s F e du e u f e du e u f t t f L st su st t u s 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 o o o o o o o o o ( 29 n oscillatio t as...
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This note was uploaded on 02/10/2009 for the course EE 313 taught by Professor Cardwell during the Spring '07 term at University of Texas at Austin.

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ee313_laplace_transform - Laplace Transform EE 313 Linear...

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