lecture2

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/12/2010 for the course ECE 382 taught by Professor Staff during the Fall '08 term at Purdue.

Page1 / 10

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online