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Properties of the Laplace Transform

Module by: Melissa Selik, Richard Baraniuk

Summary: A table of signals, Laplace transforms, and regions of convergence for several properties.

Property Signal Laplace Transform Region of Convergence
Linearity αx1t+βx2t α x1 t β x2 t αX1s+βX2s α X1 s β X2 s At least ROC1⋂ROC2 ROC1 ROC2
Time Shifting xt-τ x t τ ⅇ-sτXs s τ X s ROCROC
Frequency Shifting (modulation) ⅇηtxt η t x t Xs-η X s η Shifted ROCROC ( s-η s η must be in the region of convergence)
Time Scaling xαt x α t 1-|α|Xs-α 1 α X s α Scaled ROCROC ( s-α s α must be in the region of convergence)
Conjugation xt¯ x t Xs¯¯ X s ROC ROC
Convolution x1t*x2t x1 t x2 t X1tX2t X1 t X2 t At least ROC1⋂ROC2 ROC1 ROC2
Time Differentiation ddtxt t x t sXs s X s At least ROCROC
Frequency Differentiation -txt t x t ddsXs s X s ROC ROC
Integration in Time ∫-∞txτdτ τ t x τ 1-sXs 1 s X s At least ROC⋂ℜs>0 ROC s 0

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