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{|
 
{|
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|-
 
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! colspan="4" style="background:  #e4bc7e; font-size: 110%;" | Laplace Transform Pairs and Properties
 
 
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|-
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! colspan="4" style="background: #eee;" | Laplace Transform Pairs ||    ||
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! colspan="4" style="background: #e4bc7e; font-size: 110%;" | Laplace Transform Pairs and Properties
 
|-
 
|-
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| align=center" style="padding-right: 1em;" |  notes || Signal || Laplace Transform || ROC
+
! colspan="4" style="background: #eee;" | Laplace Transform Pairs
 +
!
 +
!
 
|-
 
|-
−
| 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>
+
| style="padding-right: 1em;" | notes
 +
| Signal
 +
| Laplace Transform
 +
| ROC
 
|-
 
|-
−
| 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="left" style="padding-right: 1em;" | unit impulse/Dirac delta
−
|-
+
| <math>\,\!\delta(t)</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>
+
| <span class="texhtml">1</span>  
−
|-
+
| <math>\text{All}\, s \in {\mathbb C}</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>\text{All}\,\, s\in {\mathbb C}</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>
+
 
|-
 
|-
 +
| 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>\text{All}\,\, s\in {\mathbb C}</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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[[ECE301|Go to the ECE 301 homepage]]
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[[Category:Formulas]]
 
[[Category:Formulas]]

Revision as of 15:51, 5 April 2010

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} $ $ \text{All}\,\, s\in {\mathbb C} $
$ \,\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 $

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