> Joseph Fourier (1768–1830) was attracted by the problem of heat diffusion because he wanted to find the ideal (soil) depth to build his cellar so that the wine remained stored at the perfect temperature in the course of a year. He then attempted to understand how the heat would spread across the surface (for a rich Fourier’s biography, we refer to https://mathshistory.st-andrews.ac.uk/Biographies/Fourier/)
But the biography says nothing about wine or cellars. I think the "wine cellar problem" is a kind of textbook application of his work, but I couldn't find any evidence that this was Fourier's motivating problem.
2/10.
[1]. The Fourier Transform and Its Applications. (Ronald Bracewell)
However saying it is "just" curve fitting with sinusoids fails to mention that, among an infinite number of basis functions, there are some with useful properties, and sinusoids are one such: they are eigenvectors of shift-invariant linear systems (and hence are also eigenvectors of derivative operators).
But then the statements about the discontinuous "vibrations". E.g. in the case of the 1 Hz cycle over half the window the author states that:
> Yet the FFT of this data is also very complex. Again, there are many harmonics with energy. They indicate that the signal contains vibrations at 0.5Hz, 1.0Hz, 1.5Hz, etc. But the time signal clearly shows that the 'vibration' was only at 1Hz, and only for the first second.
The implication that there is a vibration only at 1Hz is plain wrong. To have a vibration abruptly stop, you need many frequencies (in general the shorter a feature in the time domain the more frequency components you need in the frequency domain). If we compare for example a sine wave with a square wave at the same frequency, the square wave will have many more frequency components in the Fourier domain (it's a sinc envelope of delta functions spaced at the frequency of the wave in fact). That's essentially what is done in the example the sine wave is multiplied by a square wave with half the frequency (similar things apply to the other examples). Saying only the fundamental frequency matters is just wrong.
This is also not just a "feature of the fitting to sines", it's fundamental and has real world implications. The reason why we e.g. see ringing on an oscilloscope trace of a square wave input is because the underlying analog system has a finite bandwidth, so we "cut-off"/attenuate higher frequency components, so the square wave does not have enough of those higher frequencies (which are irrelevant according to the author) to represent the full square wave.
FFT is simply an algorithm to efficiently compute the DFT. The fact that the article makes no mention of DFT vs fourier series vs DTFT is going to end up creating more confusion that it solves. For some reason introductory tutorials always start with the DFT (usually mistakenly using FFT and DFT interchangeably), even though to me the continuous fourier transform is far easier to conceptually understand. Going from continuous fourier transform to the DTFT, is just applying the FT to a dirac-combed (sampled) function. Then from DTFT to DFT you introduce periodic boundary condition. Fourier series is just applying FT to a function that happens to already be periodic, resulting in a finite set of discrete frequencies.
There is a connection between fourier series and DFT in that if the fourier series is computed for the periodic resummation of a signal, and then the DFT is computed for the original signal (which implicitly involves applying a periodic boundary condition), the DFT is just the periodic resummation of the fourier series.
I spent ages meditating on this image https://en.wikipedia.org/wiki/Discrete_Fourier_transform?#/m... before everything finally clicked, it's a shame that introductions never once mention DTFT
An interesting fact (somewhat related to your mentioning of the DTFT) is that one can consider the DFT as a filter with a sinc transfer function. That's essentially how you can understand the spectrum of an OFDM signal. You perform a block based FFT on your input bit/symbol stream, so you have waves at different carriers. However, because the stream is timevarying you essentially get sinc shaped spectra spaced at the symbol rate (excluding cycling prefixes etc.). So your OFDM spectrum is composed of many sincs spaced at fb, which is very squarish which is one of the reasons why OFDM is so advantageous.
Sadly that wasn’t my linear systems class, which omitted this in both the lectures and textbook.
I love the visualizations on that page. There were some other cool interactive visualizatiosn on bl.ocks.org, but sadly, that site has be shattered. This is the closest I could find:
* https://observablehq.com/@drio/visualizing-the-fourier-serie...
* https://blocks.roadtolarissa.com/denisemauldin/b0424b73fae8c...
Frequentist vs Bayesian get debated constantly. I liked this video about the difference:
Also how do we construct those orthogonal basis functions for any downstream task is an interesting research question!
https://youtu.be/Dw2HTJCGMhw?si=Qhgtz5i75v8LwTyi
Learning about Fourier is really interesting in image processing, I'm glad I found a good textbook explaining it.
Its not that unintuitive to imagine that if all of your signals are pulses, something like the wavelet transform might do a better job at giving you meaningful insights into a signal than the fourier transform might.
Reality is often different from ideal and not that linear. So basic wave-forms often aren't really sinus. But people usually only know sinus, so they'll use this hammer on every nail. Some guys into electrical engineering maybe know about rectangles, but there's, not yet, enough deeper understanding out there for playing with the mathematical tools correctly.
Actual engineers:
- use sinusoids because LTI systems respond, uh... linearly to them
- use square waves for digital logic
- use triangle waves for modulation
- use wavelets for compression or time/frequency localization
- use Hilbert transforms (and actually know what "orthonormal" means)
- use STFT, CWT, FFTs... and know exactly why Fourier works and when it breaksGreat stuff. Thank you for linking this.
Which is an absolutely subjective choice in an of itself and immediately breaks the notion that curve-fitting done that way is going to be telling you some absolute truth about the function.
For example, you might want, at each point of the non-linear curve being fitted, throw a line perpendicular to its tangent, compute the distance to the linear fit, and sum those distances over all points of the non-linear curve.
About as intuitively correct (if not more) as the "fit" proposed, yet yields a very different result.
Statistics are by definition subjective unless you use a specifically demonstrated property of the particular way you decide to project your data to the simple-minded underlying statistical model.
There was another comment that referred to why we use this orthonormal basis versus another, and I think to appreciate the full reason of why this was done in the first place is important. But this presentation is a very good introduction for someone with my particular training.
For example, the Fourier transform of Ly is. F[L(2)y] = L(2)˜y(k), where y(k) is the factor from L(2)y's distribution.
maybe this is the way mechanical engineers look at it, but leaving out stfts and spectrograms seems super weird to me.
That is not even wrong. A Fourier transform is a basis expansion. In particular, the full expansion is exact (not just an approximation). Of course, truncated expansions are approximations.
The actually interesting part: Why is this basis expansion so much more useful than, e.g. expanding into some eigenfunctions, Hermite polynomials, etc.? The decomposition into (complex) exponentials converts between addition and multiplication, i. e. sin(x+y), cos(x+y) you get from multiplying sin(x), cos(x), sin(y) and cos(y). This in turn has important implications such as turning derivatives into multipliers. More generally you can consider nonlinear Fourier transforms with different groups and generators other than exponentials.
TLDR: It is a transform. What you are transforming between is what makes it so useful.