Learning about the FT in engineering and how it can represent pretty much any repeating signal was mind blowing. It was the culmination of so much learning in mathematics that brought me to that wow moment.

Before computer was ever invented we learned to make Fourier analysis of complex circuitry by hand. Huge matrixes and sliderules. It was most horrible torture, but bloody nazies of Helsinki Polytechnic insisted we must learn to do this without errors and also fast.

When I first time used computer for this task around 1974 on clanky ASR-33, I felt totally abused and brain-raped.

I found the 3Blue1Brown explanation to be more intuitive to understand, but this article has more practical examples/details.

Here's the 3Blue1Brown explanation:

https://www.youtube.com/watch?v=spUNpyF58BY

https://www.3blue1brown.com/lessons/fourier-transforms

IANAMathematician, but I've tried to come up with a metaphor which I hope isn't too wrong:

1. You've start with a signal fluctuating going up and down, and it's on a strip of little LEDs labeled from -1 to +1.

2. You mount that strip to a motor, and spin it at a certain rate. After a while the afterimages make a blob-shape.

3. For each rotation rate, measure how much the shape appears off-center.

In this way you can figure out how much the underlying signal does (or doesn't) harmonize with a given rotation hertz.

While we are talking about fun facts of the Fourier transform, the transfer function of the DFT is a sinc. That's how OFDM works, it performs an FFT on parallel input symbol streams. If you look at the spectrum of your signal it's sinc (approximately because they are truncated) channels spaced at 1/symbol rate which don't exhibit (theoretically if they weren't truncated) any interchannel interference.

My favorite algorithm (just for the name) is [Lomb-Scargle Periodogram](https://docs.astropy.org/en/latest/timeseries/lombscargle.ht...) which is a method for performing a Fourier transform on irregularly sampled data.

Kind of a tangent but why are there so many articles and videos popularizing the Fourier transform and practically none for the Laplace transform? The first person to do a well done 3Blue1Brown style video that focuses on intuition and visualization would probably be an overnight sensation (well, among engineers at least).

For lossy compression, turns out a sinusoidal (typically DCT) composition maximizes energy compaction and compress ability. A proof that this is true for AR-processes was a key realization for me. That you get a nice and intuitive domain to work with (modify frequencies) is a nice bonus on top :)

There's a lot of great Fourier visualizations out there (3Blue1Brown has a series on it as well). A great intuition pump would be a game where the player has to recreate a waveform given to them, by toggling on/off switches for each fundamental frequency (and phase).

Has anyone ever seen something like that?

See also its windowed version, https://en.wikipedia.org/wiki/Short-time_Fourier_transform