Properties of the Laplace Transform 2.8 2001/06/20 2004/08/23 14:53:24 GMT-5 Melissa Selik mselik@alumni.rice.edu Richard G. Baraniuk richb@rice.edu Justin Romberg jrom@rice.edu Melissa Selik mselik@alumni.rice.edu Richard G. Baraniuk richb@rice.edu Ricardo Anthony Radaelli-Sanchez ricky@alumni.rice.edu Mariyah Poonawala mariyah@rice.edu Prashant Singh prash@ece.rice.edu continuous time laplace transform A table of signals, Laplace transforms, and regions of convergence for several properties. Property Signal Laplace Transform Region of Convergence Linearity α x1 t β x2 t α X1 s β X2 s At least ROC1 ROC2 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 x1 t x2 t X1 t X2 t At least ROC1 ROC2 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