Site Feedback

# Properties of the Laplace Transform

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 $ROC1∩ROC2 ROC1 ROC2$
Time Shifting $x⁢t−τ x t τ$ $e−(s⁢τ)⁢X⁢s s τ X s$ $ROCROC$
Frequency Shifting (modulation) $eη⁢t⁢x⁢t η t x t$ $X⁢s−η X s η$ Shifted $ROCROC$ ( $s−η s η$ must be in the region of convergence)
Time Scaling $x⁢α⁢t x α t$ $(1−|α|)⁢X⁢s−α 1 α X s α$ Scaled $ROCROC$ ( $s−α s α$ must be in the region of convergence)
Conjugation $x⁢t¯ x t$ $X⁢s¯¯ X s$ $ROC ROC$
Convolution $x1 ⁢t* x2 ⁢t x1 t x2 t$ $X1 ⁢t⁢ X2 ⁢t X1 t X2 t$ At least $ROC1∩ROC2 ROC1 ROC2$
Time Differentiation $d d t x⁢t t x t$ $s⁢X⁢s s X s$ At least $ROCROC$
Frequency Differentiation $(−t)⁢x⁢t t x t$ $d d s X⁢s s X s$ $ROC ROC$
Integration in Time $∫−∞tx⁢τd τ τ t x τ$ $(1−s)⁢X⁢s 1 s X s$ At least $ROC∩(ℜ⁢s>0) ROC s 0$
Table 1: Table of Laplace Transform Properties