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lecture17 - EE313 Linear Systems and Signals Fall 2010...

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EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Laplace Transform
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17 - 2 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
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17 - 3 Inverse (Bilateral) Transform Inverse (Bilateral) Transform 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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17 - 4 ( 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
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