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I think the best explanation and understanding I received about Fourier Transform was from studying linear algebra. The Fourier Transform equation essentially maps a signal from the time domain onto orthogonal complex sinusoidal basis functions through projection. And the article does not even mention this. =)
I didn’t have time to read the article, but it’s cool that the position operator and the momentum operator in quantum mechanics are fourier transforms of one another.
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.
If you like Fourier, you're going to love Laplace (or its discrete counterpart, the z transform). This took me down a very fascinating and intricate rabbit hole years ago, and is still one of my favorite hobbies. Application of Fourier, Laplace, and z transforms is (famously) useful in an incredibly wide variety of fields. I mostly use it for signal processing and analog electronics.
The Wikipedia FFT article (https://en.wikipedia.org/wiki/Fast_Fourier_transform) credits Gauss with originating the FFT idea later expanded on by others, and correctly describes Cooley and Tukey's work as a "rediscovery."
The FT, as are many other transforms, are 1-1, so, in theory, there's no information lost or gained. In many real world conditions, looking at a function in frequency space greatly reduces the problem. Why? Pet theory: because many functions that look complex are actually composed of simpler building in the transformed space.
Take the sound wave of a fly and it looks horribly complex. Pump it through the FT and you find a main driver of the wings beating at a single frequency. Take the sum of two sine waves and it looks a mess. Take the FT and you see the signal neatly broken into two peaks. Etc.
The use of the FT (or DCT or whatever) for JPEG, MP3 or the like, is basically exploiting this fact by noticing the signal response for human hearing and seeing it's not uniform, and so can be "compressed" by throwing away frequencies we don't care about.
The "magic" of the FT, and other transforms, isn't so much that it transforms the signal into a set of orthogonal basis but that many signals we care about are actually formed from a small set of these signals, allowing the FT and cousins to notice and separate them out more easily.
As everyone in this thread is sharing links, I'm gonna pitch in, too. This lecture by Dennis Freeman from MIT 6.003 "Signals and Systems" gives an intuitive explanation of the connections between the four popular Fourier transforms (the Fourier transform, the discrete Fourier transform, the Fourier series, and the discrete-time Fourier transform): https://ocw.mit.edu/courses/6-003-signals-and-systems-fall-2...
Shameless plug: If you are interested in Fourier Transform and signal processing you might enjoy my somewhat artistic 3D visualisation of the fourier transform as well as the fractional fourier transform [1] (Fractional fourier transform on the top face of the cube) And for short time fourier transform showing how a filter kernel is shiftes across the signal. [2] [1]: https://static.laszlokorte.de/frft-cube/ [2]: htt…