Posted by rbanffy 9/4/2025
This is the "Fourier transform" that is most often used, in practice in computing.
A longer window length gives lower frequency resolution, but longer latency.
And the window of course creates a latency, which is sometimes relevant for realtime audio filtering by FFT.
This setup then allowed Fourier to take the limit of the Fourier series using a complex exponential expression. But keep in mind, this is all in the context of 19th century determinisitic thinking(see Fourier's analysis of the heat equation 1807, and Maxwell's treatise on 'Theory of Heat' c. 1870s), and it ran into real-world limits in the late 19th century, first with Poincare and later with sensitive dependence on initial conditions. . Poincare showed that just because you have a deterministic system, you don't automatically get predictability. Regardless this Fourier transform mathematical approach worked well in astronomy (at least at solar-system scale) because the underlying system really was at least quasiperiodic - essentially this led to prediction of new planets like Neptune.
But what if you apply Fourier transform analytics to data that is essentially chaotic? This applies to certain aspect of climate science too, eg, efforts to predict the next big El Nino based on the historical record - since the underlying system is significantly chaotic, not strictly harmonic, prediction is poor (tides in contrast are predictable as they are mostly harmonic). How to treat such systems is an ongoing question, but Fourier transforms aren't abandoned, more like modified.
Also, the time-energy quantum mechanics relation is interesting though not really a pure QM uncertainty principle, more like a classical example of a Fourier bandwidth relation, squeeze it on one axis and it spreads out on the other - a nice quote on this is "nothing that lives only for a while can be monochromatic in energy." Which sort of leads on to virtual particles and quantum tunneling. (which places a size limit of about 1 nm on chip circuitry).
The bottom line is that if you're applying elegant and complex mathematical treatments to real-world physical problems, don't forget that nature doesn't necessarily follow in the footsteps of your mathemetical model.
Was he a dwarf or a hunchback? I seem to recall a physical setback.
Underpins much of mathematics, science and engineering
They do a really good job at breaking down the fundamental knowledge needed to build an understanding.
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 guess what i want to know, in the examples it always shows like 3 or 4 constituents frequencies as output, but why not hundreds or a million? Is that decided upfront /paramtetizable?
The article isn't helpful, it just says something like "all possible frequencies".
Suppose you blinde, and are on one side of a black picket fence. Let's say the fence uprights are 1" wide, and 1" apart.
If you aim a light meter at the fence, it will sum up all the light of the visible slices of the background. (This meter reads its values out loud!)
Suppose the background is all white - the meter sees white slices, and get a 50% reading (because the fence stripes are black). It's as high as you can get.
Now move the picket fence half an inch to the left - still all white, still a 50% reading. Now you know the background is 100% white, no other colors.
But if when you move it, black stripes are exposed, then the meter reads 0%, and you know that the background is 50/50 white and black, in stripes.
Congratulations! You've just done a Fourier transform that can detect the difference between 0-frequency and a 1"-striped frequency pattern. In reality, you're going to continuously move that fence to the left, watching all the time, but this is simple.
But instead, imagine you got readings of 40% and 10%. What kind of pattern is that? Gray stripes? White stripes that aren't as wide as the fence posts? You'll have to build another fence, with another spacing - say 1/2" fence posts 1/2" apart. Repeat.
The bins do not actually measure a specific frequency, more like an average of the power around that frequency.
As you take more and more samples, the bin spacing gets finer and the range of frequencies going into the average becomes tighter (kind of). By collecting enough samples (at an appropriate rate), you can get as precise a measurement as you need around particular frequencies. And by employing other tricks (signal processing).
If you graph the magnitude of the DFT, signals that are a combination of power at just a few frequencies show just a few peaks, around the related bins. Eg a major chord would show 3 fundamental peaks corresponding to the 3 tones (then a bunch of harmonics)
https://en.wikipedia.org/wiki/Fourier_transform#/media/File:...
So you detect these peaks to find out what frequency components are present. (though peak detection itself is complicated)
But in the continuous Fourier transform, the output you get is a continuous function that's defined on the entire real line.
Until I read this article I didn't properly understand Fourier transforms (I didn't know how image compression bitmaps were derived), now it's opened a whole new world - toying with my own compression and anything that can be continuous represented as it's constituent parts.
I can use it for colour quantisation too possibly to determine main and averaged RGB constituents with respect to hue, allowing colour reduction akin to dithering, spreading the error over the wave instead and removing the less frequent elements.
It may not work but it'll be fun trying and learning!
