lecture2

lecture2 - Lec 2. Laplace Transform Laplace transform:...

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1 Lec 2. Laplace Transform • Laplace transform: definition and properties • Use Laplace transform to solve ODE • Partial Fraction Expansion • Reading: Chap. 2 Definition of Laplace Transform f ( t ): a function of time t with f ( t )=0 for t <0 (i.e., causal signal) F ( s ): a function of the complex variable s Laplace Transform F ( s ) = L [ f ( t )] = i 0 f ( t ) e st dt For a given f ( t ), its Laplace transform F ( s ) may not be defined for every s . Re [ s ] Im [ s ] σ Abcissa of convergence : smallest σ such that for all s with Re[ s ]> σ , the integral in F ( s ) converges
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2 Example f ( t ) = 0 t < 0 t 0 f ( t ) Laplace transform: F ( s ) = i 0 f ( t ) e st dt e at = 1 s + a Re [ s ] = i 0 e ( s + a ) t dt Im [ s ] = e at 1( t ) Other Useful Examples F ( s ) = ω s 2 + ω 2 f ( t ) = cos( ωt ) 1( t ) F ( s ) = s s 2 + ω 2 All have abcissa of convergence zero f ( t ) = sin( ωt ) 1( t ) 0 1 F ( s ) = 1 s f ( t ) = 1( t ) = Unit step function t < 0 t 0 f ( t ) = 0 t < 0 F ( s ) = 1 s 2 Unit ramp function t 0 t
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3 Unit Impulse Function δ ( t ) 1 h h h 0 1 δ ( t ) δ ( t ) = d dt 1( t ) 1( t ) = i t −∞ δ ( s ) ds i −∞ f ( t ) δ ( t ) dt = f (0) Important property: L [ δ ( t )] = 1 Inverse Laplace Transform f ( t ): a function of time t with f ( t )=0 for t <0 Laplace Transform F ( s ) = L [ f ( t )] = i 0 f ( t ) e st dt Inverse Laplace Transform Inverse Laplace transform: f ( t ) = L 1 [ F ( s )] = 1 2 πj
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This note was uploaded on 10/12/2010 for the course ECE 382 taught by Professor Staff during the Fall '08 term at Purdue.

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lecture2 - Lec 2. Laplace Transform Laplace transform:...

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