Line 25: Line 25:
 
| 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>\,\!\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>\,\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>\, \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}\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>\, 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) = \frac{d^{n}\delta (t)}{dt^{n}}</math> || <math>\,\!s^{n}</math> || <math>All\,\, s</math>
Line 39: Line 39:
 
| 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;"| || <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>
 
|-
 
|-
−
|}
 
−
 
−
 
−
{|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
 
−
|- style="background: #e4bc7e; font-size: 110%;" align="center"
 
−
! width="75px"|Note !! 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>
 
 
|}
 
|}
  

Revision as of 16:06, 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} $ $ \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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BSEE 2004, current Ph.D. student researching signal and image processing.

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