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| + | ! colspan="4" style="background: #e4bc7e; font-size: 110%;" | Laplace Transform Pairs and Properties | ||
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| + | ! colspan="4" style="background: #eee;" | Laplace Transform Pairs || || | ||
| + | |- | ||
| + | | align=center" style="padding-right: 1em;" | notes || Signal || Laplace Transform || ROC | ||
| + | |- | ||
| + | | align="left" style="padding-right: 1em;" | unit impulse/Dirac delta || <math>\,\!\delta(t)</math> ||<math>1</math> || <math>\text{All}\, s \in {\mathbb C}</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | unit step function || <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;"| || <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;"| || <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;"| ||<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;"| || <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;"| || <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;"| || <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;"| || <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;"| || <math>\,\!\delta (t - T)</math> || <math>\,\! e^{-sT}</math> || <math>All\,\, s</math> | ||
| + | |- | ||
| + | |align="right" style="padding-right: 1em;"| || <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;"| || <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;"| || <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;"| || <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;"| || <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;"| || <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> | ||
| + | |- | ||
| + | |} | ||
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| + | |||
{|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" style="background: #bbb; font-size: 110%;" | Laplace Transform Pairs and Properties | ||
Revision as of 16:01, 2 November 2009
| Laplace Transform Pairs and Properties | |||||
|---|---|---|---|---|---|
| Laplace Transform Pairs | |||||
| notes | Signal | Laplace Transform | ROC | ||
| unit impulse/Dirac delta | $ \,\!\delta(t) $ | $ 1 $ | $ \text{All}\, s \in {\mathbb C} $ | ||
| unit step function | $ \,\! u(t) $ | $ \frac{1}{s} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | ||
| $ \,\! -u(-t) $ | $ \frac{1}{s} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $ | |||
| $ \frac{t^{n-1}}{(n-1)!}u(t) $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | |||
| $ -\frac{t^{n-1}}{(n-1)!}u(-t) $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $ | |||
| $ \,\!e^{-\alpha t}u(t) $ | $ \frac{1}{s+\alpha} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ | |||
| $ \,\! -e^{-\alpha t}u(-t) $ | $ \frac{1}{s+\alpha} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $ | |||
| $ \frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t) $ | $ \frac{1}{(s+\alpha )^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ | |||
| $ -\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t) $ | $ \frac{1}{(s+\alpha )^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $ | |||
| $ \,\!\delta (t - T) $ | $ \,\! e^{-sT} $ | $ All\,\, s $ | |||
| $ \,\![cos( \omega_0 t)]u(t) $ | $ \frac{s}{s^2+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | |||
| $ \,\![sin( \omega_0 t)]u(t) $ | $ \frac{\omega_0}{s^2+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | |||
| $ \,\![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 $ | |||
| $ \,\![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 $ | |||
| $ u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}} $ | $ \,\!s^{n} $ | $ All\,\, s $ | |||
| $ u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n\,\,times} $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | |||
| 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 $ |
