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| + | = Some Laplace Transform Pairs = | ||
| + | {{:LaplaceTransforms_ECE301Fall2008mboutin}} | ||
| + | *[[Laplace Pairs 1_ECE301Fall2008mboutin| (1)]]{{:Laplace Pairs 1_ECE301Fall2008mboutin}} | ||
| + | *[[Laplace Pairs 2_ECE301Fall2008mboutin| (2)]]{{:Laplace Pairs 2_ECE301Fall2008mboutin}} | ||
| + | *[[Laplace Pairs 3_ECE301Fall2008mboutin| (3)]]{{:Laplace Pairs 3_ECE301Fall2008mboutin}} | ||
| + | *[[Laplace Pairs 4_ECE301Fall2008mboutin| (4)]]{{:Laplace Pairs 4_ECE301Fall2008mboutin}} | ||
| + | *[[Laplace Pairs 5_ECE301Fall2008mboutin| (5)]]{{:Laplace Pairs 5_ECE301Fall2008mboutin}} | ||
| + | *[[Laplace Pairs 6_ECE301Fall2008mboutin| (6)]]{{:Laplace Pairs 6_ECE301Fall2008mboutin}} | ||
| + | *[[Laplace Pairs 7_ECE301Fall2008mboutin| (7)]]{{:Laplace Pairs 7_ECE301Fall2008mboutin}} | ||
| − | + | ---- | |
| − | + | [[ MegaCollectiveTableTrial1|Back to Collective Table]] | |
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| − | [[ MegaCollectiveTableTrial1|Back to | + | |
Revision as of 04:37, 27 October 2009
Some 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 $
