Understanding Properties of the Laplace Transform using Wolfram Language 🐺
In this blog post I’ll be sharing my experience in comprehending the Laplace transform using Mathematica.
When I first used theLaplaceTransformfunction on Mathematica, I wasn’t able to get the transforms of Signals which spread to the negative side of the time domain. Digging into the Wolfram Language, I found out thatLaplaceTransform on it was defined for theUnilateral Laplace Transform instead. So to get the transforms of non causal signals I manually defined the Bilateral Laplace Transform asLT(wanted a shorter function name!)
So, here’s the Bilateral Laplace Transform in action! Let’s compute the Laplace Transform of and
Now everything is set to understand laplace transform properties. I’ll be noting the properties in the order as discussed inour lecture EN1060 .
-
If x(t) is a finite duration signal, then the ROC is the entire s-plane -
If x(t) is a right sided signal, and some
such that
is in ROC, then all values for which
are in ROC
(simply what this means is in the s-plane, ROC is right sided) so you should get that if the time domain signal is causal, then the ROC will be right sided. See the following code example to see this in action.
(to be updated with examples, and to be extended with Z transform.. Open for suggestions and comments! )
Figure 1: © Credit: Prof. Dennis Freeman (MIT 6.003)
Figure 2: © Credit: Prof. Dennis Freeman (MIT 6.003)
For further reading:
https://dsp.stackexchange.com/questions/53875/can-a-fourier-transform-exist-even-if-the-j-omega-axis-is-not-in-the-region-of
#link("https://github.com/rangarodrigo/EN1060Lectures/blob/master/i%20Signals%20and%20Systems%20Laplace%20Transforms.pdf")[https://github.com/rangarodrigo/EN1060Lectures/blob/master/i Signals and Systems Laplace Transforms.pdf]http://www.ece.uvic.ca/~imanmoaz/teaching_files/Laplace.pdf
http://www.ee.cityu.edu.hk/~hcso/EE3210_9.pdfhttp://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node4.html