- People go all dopey eyed about "frequency space", that's a red herring. The take away should be that a problem centric coordinate system is enormously helpful.
After all, what Copernicus showed is that the mind bogglingly complicated motion of planets become a whole lot simpler if you change the coordinate system.
Ptolemaic model of epicycles were an adhoc form of Fourier analysis - decomposing periodic motions over circles over circles.
Back to frequencies, there is nothing obviously frequency like in real space Laplace transforms *. The real insight is that differentiation and integration operations become simple if the coordinates used are exponential functions because exponential functions remain (scaled) exponential when passed through such operations.
For digital signals what helps is Walsh-Hadamard basis. They are not like frequencies. They are not at all like the square wave analogue of sinusoidal waves. People call them sequency space as a well justified pun.
My suspicion is that we are in Ptolemaic state as far as GPT like models are concerned. We will eventually understand them better once we figure out what's the better coordinate system to think about their dynamics in.
* There is a connection though, through the exponential form of complex numbers, or more prosaically, when multiplying rotation matrices the angles combine additively. So angles and logarithms have a certain unity, or character.
by derektank
5 subcomments
- My favorite story about the Fourier Transform is that Carl Friedrich Gauss stumbled upon the algorithm for the Fast Fourier Algorthim over a century before Cooley and Tukey’s publication in 1965 (which itself revolutionized digital signal processing).[1] He was apparently studying the motion of the asteroids Pallas and Juno and wrote the algorithm down in his notes but it never made it into public knowledge.
[1] https://www.cis.rit.edu/class/simg716/Gauss_History_FFT.pdf
- A signal cannot be both time and frequency band limited. Many years ago I was amazed when I read that this fact I learned in my undergraduate is equivalent to the Uncertainty Principle!
On a more mundane note: my wife and I always argue whose method of loading the dishwasher is better: she goes slow and meticulously while I do it fast. It occurred to me we were optimizing for frequency and time domains, respectively, ie I was minimizing time so spent while she was minimizing number of washes :-)
- Once you start looking at the world through the lens of frequency domain a lot of neat tricks become simple. I have some demo code that uses fourier transform on webcam video to read a heartrate off a person's face, basically looking for what frequency holds peak energy.
- I would heartily recommend Sebastian Lague's latest video, which covers this in a very approachable way: https://www.youtube.com/watch?v=08mmKNLQVHU
- My biggest missing feature for Grafana is that I want a Fourier transform that can identify epicycles in spikes of traffic. Like the first Monday of the month, or noon on tuesdays.
I had a couple charts that showed a trend line of the last n days until someone in OPs noticed that three charts were fully half of our daily burn rate for Grafana. Oops. So I started showing a -7 days line instead, which helped me but confused everyone else.
- If you are from ML/Data science world, the analogy that finally unlocked FFT for me is feature size reduction using Principal Component Analysis. In both cases, you project data to a new "better" co-ordinate system ("time to frequency domain"), filter out the basis vectors that have low variance ("ignore high-frequency waves"), and project data back to real space from those truncated dimension ("Ifft: inverse transform to time domain").
Of course some differences exist (e.g. basis vectors are fixed in FFT, unlike PCA).
- Okay, who's gonna write the story
> The unreasonable effectiveness of The Unreasonable Effectiveness title?
by petermcneeley
3 subcomments
- I dont like the Fourier Transform. It is infinite which makes it coarse and rough and it it gets everywhere.
by RossBencina
1 subcomments
- Moreover, The Unreasonable Effectiveness of Linear, Orthogonal Change of Basis.
- This talk was given at crowd supply’s 2025 teardown convention which after going for the first time last year I highly recommend it to anyone interested in hardware development. Met a lot of super cool people and managed to get my ticket price back 4x in the amount of free dev boards I got lol
- Learning about the Fourier Transform in my Signals and Systems class was mind opening. The idea you can represent any cycling function with sinusoidal functions would not only never occur to me but I would have said it wasn't possible.
- So he explains OFDM in a way that implicitly does Amplitude shift keying.
I guess if you want to use different modulations you treat the complex number corresponding to the subcarrier as an IQ point in quadrature. So you take the same symbols, but read them off in the frequency domain instead of the time domain.
And I guess this works out quite equivalently to normally modulating these symbols at properly offset frequencies (just by the superposition principle)
- At the time of his death by a Roman soldier the ancient mathematician Archimedes is said to yell: Don't disturb my circles, while he was calculating on sand. Much later, a few years ago, one of his handbooks, an overwritten palimpsest, was found to contain elements of modern calculus. If both these concepts where saved and spread through the middle ages, human civilisation might have been developed 1000 years earlier.
- Too bad -- the article doesn't mention Gauss. The Fourier transform is best presented to students in its original mathematical form, then coded in the FFT form. It also serves as a practical introduction to complex numbers.
As to the listed patent, it moves uncomfortably close to being a patent on mathematics, which isn't permitted. But I wouldn't be surprised to see many outstanding patents that have this hidden property.
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