| Laplace Transform Properties | |||
|---|---|---|---|
| Property | Signal | Laplace Transform | ROC |
| Linearity | $ \,\! ax_1(t) + bx_2(t) $ | $ \,\! aX_1(s)+bX_2(s) $ | At least $ R_1 \cap R_2 $ |
| Time Shifting | $ \,\! x(t-t_0) $ | $ e^{-st_0}X(s) $ | R |
| Shifting in the s-Domain | $ e^{s_0 t}x(t) $ | $ \,\! X(s-s_0) $ | Shifted version of R (i.e., s is in the ROC if $ s - s_0 $ is in R) |
| Time scaling | $ \,\! x(at) $ | $ \frac{1}{|a|}X\Bigg( \frac{s}{a} \Bigg) $ | Scaled ROC (i.e., s is in the ROC if s/a is in R) |
| Conjugation | $ \,\! x^{*}(t) $ | $ \,\! X^{*}(s^{*}) $ | R |
| Convolution | $ \,\! x_1(t)*x_2(t) $ | $ \,\! X_1(s)X_2(s) $ | At least $ R_1 \cap R_2 $ |
| Differentiation in the Time Domain | $ \frac{d}{dt}x(t) $ | $ \,\! sX(s) $ | At least R |
| Differentiation in the s-Domain | $ \,\! -tx(t) $ | $ \frac{d}{ds}X(s) $ | R |
| Integration in the Time Domain | $ \int_{-\infty}^{t}x(\tau)\,d\tau $ | $ \frac{1}{s}X(s) $ | At least $ R \cap \lbrace \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 \rbrace $ |
$ \,\!\mbox{Initial- and Final-Value Theorem} $
$ \,\!\mbox{If } x(t) = 0 \mbox{ for } t < 0 \mbox{ and } x(t)\mbox{ contains} $
$ \,\! \mbox{no impulses or higher-order singularities at }t = 0\mbox{, then} $
| |||
