Connexions

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Properties of the Laplace Transform

By Melissa Selik, Richard Baraniuk

Property Signal Laplace Transform Region of Convergence
Linearity α x1 t+β x2 t α x1 t β x2 t α X1 s+β X2 s α X1 s β X2 s At least ROC1ROC2 ROC1 ROC2
Time Shifting xtτ x t τ e(sτ)Xs s τ X s ROCROC
Frequency Shifting (modulation) eηtxt η t x t Xsη X s η Shifted ROCROC ( sη s η must be in the region of convergence)
Time Scaling xαt x α t (1|α|)Xsα 1 α X s α Scaled ROCROC ( sα s α must be in the region of convergence)
Conjugation xt¯ x t Xs¯¯ X s ROC ROC
Convolution x1 t* x2 t x1 t x2 t X1 t X2 t X1 t X2 t At least ROC1ROC2 ROC1 ROC2
Time Differentiation d d t xt t x t sXs s X s At least ROCROC
Frequency Differentiation (t)xt t x t d d s Xs s X s ROC ROC
Integration in Time txτd τ τ t x τ (1s)Xs 1 s X s At least ROC(s>0) ROC s 0
Table 1: Table of Laplace Transform Properties