I struggled for a long time to understand the Fourier transform, using visual materials like 3b1b [1] and betterexplained [2].
Eventually it clicked when considering the Discrete Fourier transform [3], which is just an orthogonal matrix you multiply onto a vector. All the stuff about the inverse FFT, Plancherel theorem and Parseval's theorem come for free: they just say that the matrix is orthonormal.
Maybe this only works if you've already invested in understanding linear algebra. But once I had this discrete understanding, back porting it to the continuous Fourier transform was easy enough.
Maybe I'm weird, but this was a case where just looking at the equations was much easier than all the animations of circles and stuff.
[1] https://www.youtube.com/watch?v=spUNpyF58BY
[2] https://betterexplained.com/articles/an-interactive-guide-to...
[3] https://en.wikipedia.org/wiki/Discrete_Fourier_transform
That's a really interesting case, and my gut feeling is that you are definitely very weird lol, not that that's a bad thing. How do you end up understanding linear algebra but turning to youtube videos to understand the fourier transform? My gut feeling is that 99% of people who understand linear algebra would learn fourier transforms through the same mechanism they used to learn advanced math (textbooks, university courses, etc) where the default way of introducing the subject would be something like 'just an orthogonal matrix you multiply onto a vector,' an otherwise unintuitive explanation that leaves a void that said youtube videos are trying to fill. For example the 3b1b video has to gently re-introduce why complex numbers are useful - different target audience you know?
People who are good at algebra aren’t necessarily good at calculus/analysis and vice versa. It sounds perfectly reasonable that someone might struggle with the continuous FT (integrals, brrh!) but grok the discrete version perfectly well (just a sum of basis functions).
Heh, my college signals courses first taught continuous time systems then focused on discrete time in a follow-up course. They assumed you knew enough calculus at the start that Laplace transforms wouldn't be a huge hurdle, even if you hadn't seen them before. Discrete time / Z-transform was treated as more "advanced".
But I agree, I've done enough integrals for a lifetime!
Computer scientists and programmers live in an inherently discrete world where there’s algebra everywhere you look but very little calculus outside certain niches. I’m reminded of the Feynman and the Connection Machine story [1] where he ended up analyzing some complex binary circuits in terms of differential equations because as a physicist he lived and breathed the continuum – unlike his computer engineer coworkers!
[1] https://longnow.org/essays/richard-feynman-connection-machin...
As a computer scientist I never actually had a calculus class where we used fourier series.
I just ran into them often enough, through stuff like the fast fourier transform for computing convolutions, that I thought I should understand it better.
So I googled the topic, and stuff like 3b1b is what came up, and what everyone said were the most intuitive explanations.
I eventually did a course on binary functional analysis, and it thought the discrete boolean fourier transform.
This is how people who lived before you did it. It's math. You can just read what they had to say instead of pretending a YouTube video or comic is actual hard won knowledge.
Nobody has anything more to say about Fourier series than what Walter Rudin figured out long ago. They can be defined for any locally compact abelian group. They are just trying to teach themselves about what is established theory.
Uhh... there is a lot more to say about Fourier series than "just that they're defined for any locally compact abelian group". Give me a break.
Do you know understand how quoting works? You are supposed to use the words the person actually typed.
Feel free to cite a textbook on Fourier analysis proving a result not contained in Rudin's text. Uhh.. here's your break. Put up, or shut up.
https://link.springer.com/journal/41/articles
As far as I know Rudin focuses on Fourier analysis on locally compact Abelian groups. There's been plenty of attention paid to doing Fourier analysis on other objects, especially compact Lie groups or symmetric spaces.
Of course there are people who would say that this isn't Fourier analysis anymore but the same ideas are still at play.
I felt like it made total sense when I understood the idea of a basis, plus
"sinusoids are a basis for (a class of functions)"
Everything else basically follows from that. The Fourier transform itself integrates (in one notation) e^{-ikx} against f(x). Well, the integral is a giant dot product, e^{-ikx} is the "transpose" of e^{ikx}, one of the basis vectors, so this amounts to saying f_i = <e_i, f> for a basis element e.
Right, they are very simple and elegant in algebra.
Visualizing them with nested circles flying around and drawing pictures, definitely makes them seem more weird.
The circles never really bothered me, in the end one circle just gives you the value/time of a partial sine wave. Instead of nesting them you could just put one partial into each row and add them up down below.
My experience has been the same: all these intuitive and visual explanations of math/physics stuff on YouTube/else just makes me "feel" like I learnt it, but then as soon as I need to use it in practice, I realize the only thing that actually gets the job done is some solid math equations, no questions asked. I still do all the proofs for math subjects in order to know I can derive them.
It's unfortunate, and I wish just watching such beautiful visuals would magically instill the idea in my brain, but it just feels like intellectual dopamine for me.
I don't think it's that as much as...
You have to do something, apply the knowledge. That's the piece that learning solely from nice visuals misses.
It just so happens that actually solving problems often involves using equations. But I don't think that the essential ingredient.
I agree. I much prefer the clean math language as well. I guess some people can process abstraction better than others. Fourier series are just a topic in approximation theory, which is a rich area.
You are not weird; you are just differently wired.
I find beauty in the fact that a bunch of circles are spinning on a stick (axis) with increasing frequencies, and if sum up their "tracks", you end up approximating shapes.
To your point, I happened across this video recently and it was a nice break from animated circles-within-circles: https://www.youtube.com/watch?v=QmgJmh2I3Fw
I was about to comment (as a joke) "it's just a change of basis, what's so hard to understand"
It's the signals & systems version of "monoid in the category of endofunctors"
It's sad to see that the educational system stayed generalized, and never customized to each person's mental capabilities, I always find neat articles discussing different topics from a totally different perspective. If only these existed back then when I was a studen.