| Line 1: | Line 1: | ||
| − | |||
{|style="width:75%; background: none; text-align: center; border:1px solid gray;" align="center" | {|style="width:75%; background: none; text-align: center; border:1px solid gray;" align="center" | ||
| + | ! colspan="4" style="background: #bbb; font-size: 110%;" | Laplace Transform Pairs and Properties | ||
| + | |- | ||
|- | |- | ||
! colspan="4" align="left" style="background: #b79256; font-size: 120%;" | Laplace Transform Pairs | ! colspan="4" align="left" style="background: #b79256; font-size: 120%;" | Laplace Transform Pairs | ||
|- style="background: #e4bc7e; font-size: 110%;" align="center" | |- style="background: #e4bc7e; font-size: 110%;" align="center" | ||
| − | ! width="75px"| | + | ! width="75px"|Note !! Signal |
! width="170px"|Transform | ! width="170px"|Transform | ||
! width="170px"|ROC | ! width="170px"|ROC | ||
Revision as of 07:07, 30 October 2009
| Laplace Transform Pairs and Properties | |||
|---|---|---|---|
| Laplace Transform Pairs | |||
| Note | Signal | Transform | ROC |
| 1 | $ \,\!\delta(t) $ | $ 1 $ | $ All\,\, s $ |
| 2 | $ \,\! u(t) $ | $ \frac{1}{s} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ |
| 3 | $ \,\! -u(-t) $ | $ \frac{1}{s} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $ |
| 4 | $ \frac{t^{n-1}}{(n-1)!}u(t) $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ |
| 5 | $ -\frac{t^{n-1}}{(n-1)!}u(-t) $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $ |
| 6 | $ \,\!e^{-\alpha t}u(t) $ | $ \frac{1}{s+\alpha} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ |
| 7 | $ \,\! -e^{-\alpha t}u(-t) $ | $ \frac{1}{s+\alpha} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $ |
| 8 | $ \frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t) $ | $ \frac{1}{(s+\alpha )^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ |
| 9 | $ -\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t) $ | $ \frac{1}{(s+\alpha )^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $ |
| 10 | $ \,\!\delta (t - T) $ | $ \,\! e^{-sT} $ | $ All\,\, s $ |
| 11 | $ \,\![cos( \omega_0 t)]u(t) $ | $ \frac{s}{s^2+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ |
| 12 | $ \,\![sin( \omega_0 t)]u(t) $ | $ \frac{\omega_0}{s^2+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ |
| 13 | $ \,\![e^{-\alpha t}cos( \omega_0 t)]u(t) $ | $ \frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ |
| 14 | $ \,\![e^{-\alpha t}sin( \omega_0 t)]u(t) $ | $ \frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ |
| 15 | $ u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}} $ | $ \,\!s^{n} $ | $ All\,\, s $ |
| 16 | $ u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n\,\,times} $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ |
