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0.14 Laplace properties and transforms

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Properties of Laplace Transforms

Laplace properties

f t F s t 0 - ∞ f t s t
Property Time-domain Frequency-domain
Linearity a f 1 t b f 2 t a F 1 t b F 2 t
Shifting in s -domain s 0 t f t F s s 0
Time Scaling ( a 0 ) f a t 1 a F s a
Convolution (causal functions) f 1 t f 2 t F 1 s F 2 s
Differentiation in Time t 1 f t s F s f 0 -
Differentiation in Freq. t f t s 1 F s
Integration in Time 0 - t f 1 s F s

Unilateral laplace transforms

Note: I t is a step function.

Time-domain Frequency-domain
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
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Source:  OpenStax, State space systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10143/1.3
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