Thursday, 20 September 2012

Trying to Understand Laplace('s Transform)

The Laplace transform is a commonly used tool in engineering and digital signal stuff e.g. filter design.  I think my first encounter with it was watching a Khanacademy video.  The title looked cool, I was procrastinating for an exam, and I raised myself the challenge of trying to understand it.

The khanacademy video is brilliant, and Khan is an excellent teacher.  For someone who's been exposed to a lot of calculus in school, given the basic transform rule, it's just a matter of integration (by parts usually) to derive any of the transforms and rules.  The process of doing it clicked straight away.

However, the only thing lacking was a more fundamental explanation of what the transform actually represents.  What is the mysterious "s-domain"?  It just comes across as some magical tool that somehow simplifies evil differential equations into pathetic algebraic ones. I like to have a deeper understanding of difficult concepts, because once they're broken down I can relate them to other concepts I already understand.  I also like to think, how did the inventor(s) come to invent this formula or process?  To me this makes it much more intuitive.

More recently I came across the Laplace transform and even weirder z-transform again in filter design.  So I decided to really hunt for a better explanation for the Laplace transforms.  I was met with many forum posts with indecipherable replies with mentions of "infinite vector space" and "complex frequency domain".  I'm sure they're perfectly valid explanations, but as a simple minded engineer I like things broken down into more English and relatable terms before I can go on to describe it mathematically.  There was really one video - an MIT lecture - that blew my mind and answered all the questions.

(There are two parts to the video, this is the first part)

This video connects Laplace transforms with the more straightforward power series.  When I questioned myself on why a power series should work I realised I didn't know, so I had to backtrack a little.   After quite a few hours though I am now satisfied with the explanations of power series, Laplace transforms and z-transforms I found.  I compiled it all into a single document which I'll use later to prompt my memory.  I however release it into the wild for similarly confused people to share!

Here it is.

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