0.14 Laplace properties and transforms

State space systems Page 1 / 1

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	$0^{-} t f$	$1 s F s$

Note: $I t$ is a step function.

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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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