Laplace Transform

Laplace Transform - Laplace Transform Laplace Transform...

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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).
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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.
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Laplace Transform - Laplace Transform Laplace Transform...

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