An Interactive Guide to the Fourier Transform

I've decided math isn't my thing. The first part of the article I couldn't stop thinking "how the hell would you construct a banana filter?" And the entire smoothie metaphor seemed to describe nothing at all.

Then there was something about circles and why do some people call them some other silly thing?

So far, so utterly meaningless, as far as I could tell. just seemed like meaningless babble to make even a kindergartner feel comfortable with the article, but it didn't seem to have communicated much of anything, really.

Then there were circles. Some of them were moving, one of them had a sinus wave next to it and some balls were tracing both in sync, indicating which part of the sinus wave equalled which part of the circle I guess?

I understood none of it.

I asked chat gpt to explain to me, i think it has read this article cause it used the smoothie analogy as well. I still don't understand what that analogy is meant to mean.

Then finally I found this: If someone plays a piano chord, you hear one sound. But that sound is actually made of multiple notes (multiple frequencies).

The Fourier Transform is the tool that figures out:

which notes (frequencies) are present, and how loud each one is

That, finally, makes sense.

I wonder if my approach would help with your understanding?

https://dsego.github.io/demystifying-fourier/

The piano analogy is incomplete. First, of all, a piano constructs sounds by combining multiple string sounds in a unique manner. But the idea behind transforms (Fourier being a particular case) is that you can take a function (“sound”) that isn’t necessarily produced by combining components and you can still decompose it into a sum of components. This decomposition is not unique in the general case as there are many different transforms yielding different results. However, from the mathematical (and i believe, quantum mechanical) standpoint, there is full equivalence between the original function and its transforms.

The other important point is that Fourier doesn’t really give you frequency and loudness. It gives you complex numbers that can be used to estimate the loudness of different frequencies. But the complex nature of the transform is somewhat more complex than that (accidental pun).

A fun fact. The Heisenberg uncertainty principle can be viewed as the direct consequence of the nature of the Fourier transform. In other words, it is not an unexplained natural wonder but rather a mathematical inevitability. I only wish we could say the same about the rest of quantum theory!

All analogies are incomplete. It's kinda inherent in the definition of the word.

But it is a lovely, real-world and commonly understood example of how harmonics can work, and thus a nice baby-step into the idea of spectral analysis.

Yes, I could understand almost all of this actually! Thanks for explaining Fourier so well!

I really don't have any mathematics in my background, so you lost me towards the very end when the actual math came in, but I can't fault your Fourier explanation for not also explaining imaginary numbers: even I can see they're out of scope for this post!