Explanation: Top left is a Taylor series, which expresses an infinitely differentiable function as an infinite polynomial. Center left is a Fourier transform, which extracts from periodic function into the frequencies of the sines and cosines composing it. Bottom left is the Laplace transform, which does the same but for all exponentials (sines and cosines are actually exponentials, long story). It seems simpler than the Fourier transform, until you realize that the s is a complex number. In all of these the idea is to break down a function into its component parts, whether as powers of x, sines and cosines or complex exponentials.
Edit: I’ll try to explain if something is unclear, but… uh… better not get your hopes up.
It seems simpler than the Fourier transform, until you realize that the s is a complex number.
Oh, look at that hornet’s nest. I wonder what happens if I poke it
As someone who worked with system modelling, analysis and control for years… I do think the Laplace transform is easier to work with 🙈🏃♂️
It is, but conceptually it’s a lot weirder than the Fourier transform, whose idea at least is very straightforward. I mean, when doing Laplace transforms you do have to assume that int(e^tdt){0}{∞}=-1. I’d definitely rather use the Laplace transform, but you couldn’t pay me to explain how that shit actually works to an undergrad student.
Basically the assumption is that the signal x(t) is equal to 0 for all t < 0 and that the integral converges. And what is a bit counter-intuitive: Laplace transformations can be regarded as generalizations of Fourier transformations, since the variable s is not only imaginary but fully complex. But yeah… I would have to brush up on it again, before explaining it as well. It’s… been a while.
What kind of work do you do?
I’m in the process of wrapping up my degree and I work a lot with signals and controls. I agree that Laplace is much less of a headache than Fourier.
I was at the intersection between mechanical and electrical engineering as well as computer science. And worked in/with (electric) mobility, agriculture, medical/rehabilitation tech., solar energy, energy grid, construction and building tech. As well as some very limited stuff with economics. And I intentionally chose my study courses to be able to work in multiple areas and inter-disciplinary. My latest work is more on the business and management side of things and less technical, though.
What are you studying and what direction are you hoping to head in?
Can you elaborate on why without getting us all stung to death?
Basically two things: 1. Complicated operations in the time domain like convolutions become simple operations in the frequency domain. 2. It is way easier to handle complex numbers and do analysis with them than with explicit frequencies to the point where some things like stability and robustness can be judged by simple geometry (e.g. are the eigenvalues within the unit circle) or the sign of the imaginary part.
I understand some of these words
They should have cancelled this series instead of Rome and Firefly.
Taylor… You fear to go into those series. Euler delved too greedily and too deep. You know what they awoke in the depths of the number line…
Wat?
Added an explanation comment. Should’ve probably done that sooner.
Ah yes… it’s crystal clear now!
