| Line 3: | Line 3: | ||
Let the signal be: | Let the signal be: | ||
| − | <math>x(t) =e^ {-at} \mathit{u} (t)</math> | + | <math>x(t) =e^ {-at} \mathit{u} (t).</math> |
| − | + | Here is how to compute the Laplace Transform of <math>x(t)</math>: | |
| − | <math>X(s)= \int_{-\infty}^{\infty}x(t){e^{-st}}\, dt | + | <math> |
| − | + | \begin{align} | |
| − | + | X(s) &= \int_{-\infty}^{\infty}x(t){e^{-st}}\, dt, \\ | |
| − | + | &= \int_{-\infty}^{\infty}{e^{-at}}{e^{-st}}dt ,\text{ since }\mathit{u} (t)=1,\text{ for }t>0, \text{ else }\mathit{u} (t)=0, \\ | |
| − | + | &=\frac{1}{s+a}. ~^* | |
| + | \end{align} | ||
| + | </math> | ||
| + | Note: the last equality (with a *) is untrue. Please do not write this on the test or you will get points marked off. I really appreciate this mistake being on Rhea, please do not erase it --[[User:Mboutin|Mboutin]] 11:58, 21 November 2008 (UTC) | ||
| + | |||
* [[Homework _ECE301Fall2008mboutin#10 Daniel Morris: Properties of the Region of Convergence(ROC)]] | * [[Homework _ECE301Fall2008mboutin#10 Daniel Morris: Properties of the Region of Convergence(ROC)]] | ||
Revision as of 07:58, 21 November 2008
== Fundamentals of Laplace Transform ==
Let the signal be:
$ x(t) =e^ {-at} \mathit{u} (t). $ Here is how to compute the Laplace Transform of $ x(t) $:
$ \begin{align} X(s) &= \int_{-\infty}^{\infty}x(t){e^{-st}}\, dt, \\ &= \int_{-\infty}^{\infty}{e^{-at}}{e^{-st}}dt ,\text{ since }\mathit{u} (t)=1,\text{ for }t>0, \text{ else }\mathit{u} (t)=0, \\ &=\frac{1}{s+a}. ~^* \end{align} $
Note: the last equality (with a *) is untrue. Please do not write this on the test or you will get points marked off. I really appreciate this mistake being on Rhea, please do not erase it --Mboutin 11:58, 21 November 2008 (UTC)
