An Interactive Guide to the Fourier Transform

On thing that is often overlooked but should be emphasized is that the considered frequencies are fixed values while the phase shifts are continuous values. This creates tons of downstream problems

If your underlying signal is at frequency that is not a harmonic of the sampling length, then you get "ringing" and it's completely unclear how to deal with it (something something Bessel functions)

Actually using DFTs is a nightmare ..

- If I have several dominant frequencies (not multiples of the sampling rate) and I want to know them precisely, it's unclear how I can do that with an FFT

- If I know the frequency a priori and just want to know the phase shift.. also unclear

- If I have missing values.. how do i fill the gaps to distort the resulting spectrum as little as possible?

- If I have samples that are not equally spaced, how am I supposed to deal with that?

- If my measurements have errors, how do I propagate errors through the FFT to my results?

So outside of audio where you control the fixed sample rate and the frequencies are all much lower than the sample rate... it's really hard to use. I tried to use it for a research project and while the results looked cool.. I just wasn't able to backup my math in a convincing way (though it's been a few years so I should try again with ChatGPT's hand-holding)

I recommend people poke around this webpage to get a taste of what a complicated scary monster you're dealing with

https://ccrma.stanford.edu/~jos/sasp/sasp.html

This is the best content on this topic - a 2023 video by Reducible: https://www.youtube.com/watch?v=yYEMxqreA10

I have not read the whole article. But, what is shown at the beginning is not the Fourier Transform, it is the Discrete Fourier Transform (DFT).

Though the DFT can be implemented efficiently using the Fast Fourier Transform (FFT) algorithm, the DFT is far from being the best estimator for frequencies contained in a signal. Other estimators (like Maximum Likelihood [ML], [Root-]MUSIC, or ESPRIT) are in general far more accurate - at the cost of higher computational effort.

The absolute best teaching of the Fourier transform I've ever encountered is the extremely bizarre book "Who is Fourier?"

https://www.amazon.com/Who-Fourier-Mathematical-Transnationa...

If you're only interested in the gist of the concept and how it can be applied to compression, without the mathematical rigor, here is my go to: https://bertolami.com/index.php?engine=blog&content=posts&de...

Brilliant!

I would just suggest the author to replace the sentence “99% of the time, it refers to motion in one dimension” with “most of the time” since this is a mathematical article and there’s no need to use specific numbers when they don’t reflect actual data.

It's quite interesting that our ears implement a better-than-Fourier-like algorithm internally: https://arxiv.org/pdf/1208.4611

If you're dealing with computer graphics, audio, or data analysis, I highly recommend learning Fourier transforms, because they explain a whole lot of things that are otherwise mysterious.

My only quibble is that the article is about the discrete Fourier transform.

Will I ever be able to learn the Fourier transform?

If anyone wants to learn about the 2D DFT, the best explanation I've ever read was the relevant chapter in Digital Image Processing by Nick Efford.

If anyone wants to see my favorite application of the 2D DFT, I made a video of how the DFT is used to remove rainbows in manga on Kaleido 3 color eink on Kobo Colour:

https://youtu.be/Dw2HTJCGMhw?si=J6dUYOj2IRX1nPRF

My favorite application of the Fourier transform is converting convolution into pointwise multiplication. This is used to speed up multiple sequence alignment in bioinformatics.

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.

It is more about the duality between the amplitude and frequency spaces and conversion between them. A bit similar to Hadamard gate for transforming a quantum state from computational basis to diagonal basis.

Past comments: https://news.ycombinator.com/item?id=38652794

>The Fourier Transform is one of deepest insights ever made.

No, it is not. In fact it is quite a superficial example of a much deeper theory, behind functions, their approximations and their representations.