Laplace Properties and Transforms 2.9 2001/01/28 2002/10/24 Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu Roy Ha rha@rice.edu Elizabeth Chan lizychan@rice.edu Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu laplace transform property properties Properties of Laplace Transforms

Laplace Properties ℒ f t F s t 0- f t s t

Property Time-domain Frequency-domain Linearity a f1 t b f2 t a F1 t b F2 t Shifting in s-domain s0 t f t F s s0 Time Scaling (a0) f a t 1 a F s a Convolution (causal functions) f1 t f2 t F1 s F2 s Differentiation in Time t1 f t s F s f 0- Differentiation in Freq. t f t s1 F s Integration in Time τ 0- t f τ 1 s F s

Unilateral Laplace Transforms Note: It is a step function.

Time-domain Frequency-domain δ t 1 I t 1 s t I t 1 s 2 t n I t n s n 1 a t I t 1 s a t a t I t 1 s a 2 a t I t s s 2 a 2 a t I t a s 2 a 2