Laplace Transform

# Laplace Transform - Laplace Transform Laplace Transform(LT...

This preview shows pages 1–3. Sign up to view the full content.

1 Laplace Transform Laplace Transform (LT) = 0 ) ( ) ( dt e t x s X st In the Fourier analysis, we focus on ω j s = . In the Laplace Transform we let + In the Laplace Transform, we let σ j s = . Fourier analysis is more focused on steady-state response LT focus on transient response, so we use the one-sided LT. Region of convergence (ROC) Given a signal, the set of all complex member s for which the integral 0 ) ( dt e t x st exists is called the region of convergence (ROC) of the LT X(s) of x(t). Note ROC depends on the signal x(t).

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

View Full Document
2 Ex. 1 x(t)=u(t) = = 0 0 1 ) ( st st e s dt e s X the limit st t e lim exists iff Re(s)>0, in which case 0 lim = st t e and X 1 = denoted as 1 u(t and s s ) ( denoted as s u(t) { } ROC : Re( ) 0 ss => Ex. 2: ) ( ) ( t u e t x bt = , b is real number. + = = ) ( 1 t b s st bt e d X + 0 0 ) ( b s dt e e s t b s t e ) ( lim + exist iff Re(s)> -b in which case 0 lim ) ( = + t b s t e b s s X + = 1 ) ( { } ROC : Re( ) s sb b s 1 u(t) e bt - + Properties of Laplace Transform Linearity : ) ( ) ( ) ( ) ( s bV s aX t bv t ax + + Right shift in time : ) ( ) ( ) ( s X e c t u c t x cs , for c > 0. Note u(t-c) is required to eliminate any non-zero values of x(t) for t<0. Time Scaling : ) ( 1 ) ( a s X a at x , for a > 0. Note there is no time reversal property in the one-sided LT. Differentiation in time : () (0 ) x ts X s x Differentiation in the time domain corresponds to multiplication by s in the LT domain. very useful in solving differential equations.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 9

Laplace Transform - Laplace Transform Laplace Transform(LT...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online