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| − | = | + | = Table of Laplace Transform Pairs = |
| − | {{: | + | {|style="width:75%; background: none; text-align: center; border:1px solid gray;" align="center" |
| + | |- | ||
| + | ! colspan="4" align="left" style="background: #b79256; font-size: 120%;" | Laplace Transform Pairs | ||
| + | |- style="background: #e4bc7e; font-size: 110%;" align="center" | ||
| + | ! width="75px"|Transform Pair !! Signal | ||
| + | ! width="170px"|Transform | ||
| + | ! width="170px"|ROC | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;"|1 || <math>\,\!\delta(t)</math> || <math>1</math> || <math>All\,\, s</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;"|2 || <math>\,\! u(t)</math> || <math>\frac{1}{s}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> | ||
| + | |- | ||
| + | |align="right" style="padding-right: 1em;"| 3 || <math>\,\! -u(-t)</math> || <math>\frac{1}{s}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;"|4 || <math>\frac{t^{n-1}}{(n-1)!}u(t)</math> || <math>\frac{1}{s^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;"|5 || <math>-\frac{t^{n-1}}{(n-1)!}u(-t)</math> || <math>\frac{1}{s^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;"|6 || <math>\,\!e^{-\alpha t}u(t)</math> || <math>\frac{1}{s+\alpha}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;"|7 || <math>\,\! -e^{-\alpha t}u(-t)</math> || <math>\frac{1}{s+\alpha}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;"|8 || <math>\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t)</math> || <math>\frac{1}{(s+\alpha )^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;"|9 || <math>-\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t)</math> || <math>\frac{1}{(s+\alpha )^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;"|10 || <math>\,\!\delta (t - T)</math> || <math>\,\! e^{-sT}</math> || <math>All\,\, s</math> | ||
| + | |- | ||
| + | |align="right" style="padding-right: 1em;"| 11 || <math>\,\![cos( \omega_0 t)]u(t)</math> || <math>\frac{s}{s^2+\omega_0^{2}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;"|12 || <math>\,\![sin( \omega_0 t)]u(t)</math> || <math>\frac{\omega_0}{s^2+\omega_0^{2}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> | ||
| + | |- | ||
| + | |align="right" style="padding-right: 1em;"| 13 || <math>\,\![e^{-\alpha t}cos( \omega_0 t)]u(t)</math> || <math>\frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math> | ||
| + | |- | ||
| + | |align="right" style="padding-right: 1em;"| 14 || <math>\,\![e^{-\alpha t}sin( \omega_0 t)]u(t)</math> || <math>\frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;"|15 || <math>u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}}</math> || <math>\,\!s^{n}</math> || <math>All\,\, s</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;"|16 || <math>u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n\,\,times}</math> || <math>\frac{1}{s^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> | ||
| + | |} | ||
| + | |||
| + | |||
| + | |||
| + | |||
*[[Laplace Pairs 1_ECE301Fall2008mboutin| (1)]]{{:Laplace Pairs 1_ECE301Fall2008mboutin}} | *[[Laplace Pairs 1_ECE301Fall2008mboutin| (1)]]{{:Laplace Pairs 1_ECE301Fall2008mboutin}} | ||
*[[Laplace Pairs 2_ECE301Fall2008mboutin| (2)]]{{:Laplace Pairs 2_ECE301Fall2008mboutin}} | *[[Laplace Pairs 2_ECE301Fall2008mboutin| (2)]]{{:Laplace Pairs 2_ECE301Fall2008mboutin}} | ||
Revision as of 04:39, 27 October 2009
Table of Laplace Transform Pairs
| Laplace Transform Pairs | |||
|---|---|---|---|
| Transform Pair | 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 $ |
- (1)$ \delta(t) \leftrightarrow 1 $, for all s
- (2)$ \ u(t) \leftrightarrow \frac{1}{s} $, for Re{s} > 0
- (3)$ \ -u(-t) \leftrightarrow \frac{1}{s} $, for Re{s} < 0
- (4)$ \frac{t^{n - 1}}{(n - 1)!}u(t) \leftrightarrow \frac{1}{s^{n}} $, for Re{s} > 0
- (5)$ - \frac{t^{n - 1}}{(n - 1)!}u(-t) \leftrightarrow \frac{1}{s^{n}} $, for Re{s} < 0
- (6)$ \ e^{\alpha t }u(t) \leftrightarrow \frac{1}{s + \alpha} $, for Re{s} > $ \ - \alpha $
- (7)$ \ -e^{\alpha t }u(-t) \leftrightarrow \frac{1}{s + \alpha} $, for Re{s} < $ \ - \alpha $
