11.2 Properties of the laplace transform

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A table of Laplace Transform properties.

Table of Laplace Transform properties.

Table of laplace transform properties
Property	Signal	Laplace Transform	Region of Convergence
Linearity	$α x_{1} t β x_{2} t$	$α X_{1} s β X_{2} s$	At least ${ROC}_{1} {ROC}_{2}$
Time Shifting	$x t τ$	$s τ X s$	$ROC$
Frequency Shifting (modulation)	$η t x t$	$X s η$	Shifted $ROC$ ( $s η$ must be in the region of convergence)
Time Scaling	$x α t$	$1 α X s α$	Scaled $ROC$ ( $s α$ must be in the region of convergence)
Conjugation	$x t$	$X s$	$ROC$
Convolution	$x_{1} t x_{2} t$	$X_{1} t X_{2} t$	At least ${ROC}_{1} {ROC}_{2}$
Time Differentiation	$t x t$	$s X s$	At least $ROC$
Frequency Differentiation	$t x t$	$s X s$	$ROC$
Integration in Time	$τ$ ∞ t x τ	$1 s X s$	At least $ROC s 0$

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Source: OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15

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