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.
All these transforms are switching to an eigenbasis of some differential operator (that usually corresponds to a differential equation of interest). Spherical harmonics, Bessel and Henkel functions, which are the radial versions of sines/cosines and complex exponential, respectively, and on and on.
The next big jumps were to collections of functions not parameterized by subsets of R^n. Wavelets use a tree shapes parameter space.
There’s a whole, interesting area of overcomplete basis sets that I have been meaning to look into where you give up your basis functions being orthogonal and all those nice properties in exchange for having multiple options for adapting better to different signal characteristics.
I don’t think these transforms are going to be relevant to understanding neural nets, though. They are, by their nature, doing something with nonlinear structures in high dimensions which are not smoothly extended across their domain, which is the opposite problem all our current approaches to functional analysis deal with.
You may well be right about neural networks. Sometimes models that seem nonlinear turns linear if those nonlinearities are pushed into the basis functions, so one can still hope.
For GPT like models, I see sentences as trajectories in the embedded space. These trajectories look quite complicated and no obvious from their geometrical stand point. My hope is that if we get the coordinate system right, we may see something more intelligible going on.
This is just a hope, a mental bias. I do not have any solid argument for why it should be as I describe.
I’d argue that most if not all of the math that I learned in school could be distilled down to analyzing problems in the correct coordinate system or domain! The actual manipulation isn’t that esoteric once you get in the right paradigm. And those professors never explained things at that kind of higher theoretical level, all I remember was the nitty gritty of implementation. What a shame. I’m sure there’s higher levels of mathematics that go beyond my simplistic understanding, but I’d argue it’s enough to get one through the full sequence of undergraduate level (electrical) engineering, physics, and calculus.
It’s kind of intriguing that predicting the future state of any quantum system becomes almost trivial—assuming you can diagonalize the Hamiltonian. But good luck with that in general. (In other words, a “simple” reference frame always exists via unitary conjugation, but finding it is very difficult.)
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.
There is a saying about Gauss: when another mathematician came to show him a new result, Gauss would remark that he had already worked on it, open a drawer in his desk, and pull out a pile of papers on the same topic.
One of the things I admire about many top mathematicians today like Terence Tao is that they are clearly excellent mentors to a long list of smart graduate students and are able to advance mathematics through their students as well as on their own. You can imagine a half-formed thought Terence Tao has while driving to work becoming a whole dissertation or series of papers if he throws it to the right person to work on.
In contrast, Gauss disliked teaching and also tended to hoard those good ideas until he could go through all the details and publish them in the way he wanted. Which is a little silly, as after a while he was already widely considered the best mathematician in the world and had no need to prove anything to anyone - why not share those half-finished good ideas like Fast Fourier Transforms and let others work on them! One of the best mathematicians who ever lived, but definitely not my favorite role model for how to work.
> There is a saying about Gauss: when another mathematician came to show him a new result, Gauss would remark that he had already worked on it, open a drawer in his desk, and pull out a pile of papers on the same topic.
As if phd students need more imposter syndrom to deal with. Ona serious side, I wonder what conditions allow such minds to grow. I guess a big part is genetics, but I am curious if the "epi" is relevant and how much.
When I interned at Chevron someone said they (or some other oil company) were using Fourier transforms in the 1950's for seismic analysis but kept it a secret for obvious reasons. I think you couldn't (can't?) patent math equations.
Gauss's notes and margins is riddled with proofs he didn't bother to publish - he was wild.
Not sure if true, but allegedy he insisted his son not go into maths, as he would simply end up in his father's shadow as he deemed it utterly Impossible to surpass his brilliance in maths :'D
> as he would simply end up in his father's shadow as he deemed it utterly Impossible to surpass his brilliance in maths
Definitely true but also bad parenting. Gauss was somewhat of a freak of nature when it came to math. Him and Euler are two of the most unreasonably productive mathematicians of all time.
I only have 5 kids, and I am also not nearly as productive as Gauss but to a certain degree, it feels to me like responsibility kind of tries to force me to be more effective.
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.
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 :-)
Signals can be approximately frequency and time bandlimited, though, meaning the set of values such that the absolute value exceeds any epsilon is compact in both domains. A Gaussian function is one example.
Another example: ears are excellent at breaking down the frequency of sounds, but are imprecise about where the sound is coming from; whereas eyes are excellent at telling you where light is coming from, but imprecise about how its frequencies break down.
For those who don't get this comment, the Heisenberg uncertainty principle applies to any two quantities that are connected in QM via a Fourier transform. Such as position and momentum, or time and energy. It is really a mathematical theorem that there is a lower bound on the variance of a function times the variance of its Fourier transform.
That lower bound is the uncertainty principle, and that lower bound is hit by normal distributions.
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.
It's effectively the underpinning of all modern lossy compression algorithms. The DCT which underlies codecs like Jpeg, h264, mp3, is really just a modified FFT.
Inter/intra-prediction is more important than the DCT. H264 and later use simpler degenerate forms of it because that's good enough and they can define it with bitwise accuracy.
>Once you start looking at the world through the lens of frequency domain a lot of neat tricks become simple.
Not the first time I've heard this on HN. I remember a user commenting once that it was one of the few perspective shifts in his life that completely turned things upside down professionally.
It is, I've done it live on a laptop and via the front camera of a phone. I actually wrote this thing twice, once in Swift a few years back, and then again in Python more recently because I wanted to remember the details of how to do it. Since a few people seem surprised this is feasible maybe it's worth posting the code somewhere.
It is, but there's a lot of noise on top of it (in fact, the noise is kind of necessary to avoid it being 'flattened out' and disappearing). The fact that it covers a lot pixels and is relatively low bandwidth is what allows for this kind of magic trick.
Given how much of the talk is about the original paper the title references, and how the Fourier transform turns out to be unreasonably effective at allowing communication over noisy channels, I'd say it's a reasonable reference.
It's a play on the famous essay 1960 "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".
I agree this is getting old after 75 years. Not least because it seems slightly manipulative to disguise a declarative claim ("The Fourier transform is unreasonably effective."), which could be false, as a noun phrase ("The unreasonable effectiveness of the Fourier transform"), which doesn't look like a thing that can be wrong.
Also how most of the articles with this kind of title (those posted on HN at least) are about computation/logical processes, which are by definition, reasonable.
FTs are actually very reasonable, in the sense that they are a easy to reason about conceptually and in practice.
There's another title referenced in that link which is equally asinine: "Eugene Wigner's original discussion, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". "
Like, wtf?
Mathematics is the language of science, science would not compound or be explainable, communicable, or model-able in code without mathematics.
It's actually both plainly obvious for mathematics then to be extremely effective (which it is) and also be evidently reasonable as to why, ergo it is not unreasonably effective.
Also the slides are just FTs 101 the same material as in any basic course.
Hi, original presenter here :) The beginning is FTs 101. The end gets more application-centric around OFDM and is why it feels 'unreasonably effective' to me. If it feels obvious, there's a couple of slides at the end that are food for thought jumping off points. And if that's obvious to you too, let's collab on building an open source LTE modem!
So, biology and medicine are not sciences? Or are only sciences to the extent they can be mathematically described?
The scientific method and models are much more than math. Equating the reality with the math has let to myriad misconceptions, like vanishing cats.
And silly is good for a title -- descriptive and enticing -- to serve the purpose of eliciting the attention without which the content would be pointless.
I find it hard to parse the middle of your post. Are you saying Wigner's article, which is what all the "unreasonable effectiveness" titles reference, is silly?
If that is what you are saying I suggest that you actually go back and read it. Or at least the Wiki article:
By means of contrast: I think it's clear that mathematics is, for example, not unreasonably effective in psychology. It's necessary and useful and effective at doing what it does, but not surprisingly so. Yet in the natural sciences it often has been. This is not a statement about mathematics but about the world.
(As Wittgenstein put it some decades earlier: "So too the fact that it can be described by Newtonian mechanics asserts nothing about the world; but this asserts something, namely, that it can be described in that particular way in which as a matter of fact it is described. The fact, too, that it can be described more simply by one system of mechanics than by another says something about the world.")
It is likewise unreasonable to look down on any kind of world model from the past. Remember that you, in 2026, are benefitting from millions of aggregate improvements to a world model that you've absorbed passively through participation in society, and not through original thought. You have a different vantage point on many things as a result of the shoulders of giants you get to stand on.
> FTs are actually very reasonable, in the sense that they are a easy to reason about conceptually and in practice.
ok but it's not the FTs that are unreasonable, it's the effectiveness
I think we all understand at this point that "unreasonable effectiveness" just means "surprisingly useful in ways we might not have immediately considered"
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).
Anybody who does anything in the real world with Fourier transforms uses the fast Fourier transform operating on windowed data. This eliminates all of that infinite support and infinite resolution of frequencies.
To be more precise, when working with sampled data with uniform sample rate you use the Discrete Time Fourier Transform (DTFT), not the Fourier Transform!. None the less, you still end up with an approximate spectrum which is the signal spectrum convolved with the window function's spectrum.
In my view the Fourier Transform is still useful in the real world. For example you can use it to analytically derive the spectrum of a given window.
But I think the parent is hinting at wavelet basis.
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)
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.
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.
Pretty sure in the USA you can patent mathematics if it is an integral part of the realisation of a physical system.* There is a book "Math you Can't Use" that discusses this.
> Pretty sure in the USA you can patent mathematics if it is an integral part of the realisation of a physical system.
Yes, that's true. In that example, you're not patenting mathematics, you're patenting a specific application, which can be patented. In my reading I see that mathematics per se is an abstract intellectual concept, thus not patentable (reference: https://ghbintellect.com/can-you-patent-a-formula/).
There is plenty of case law in modern times where the distinction between an abstract mathematical idea, and an application of that idea, were the issues under discussion.
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.
All these transforms are switching to an eigenbasis of some differential operator (that usually corresponds to a differential equation of interest). Spherical harmonics, Bessel and Henkel functions, which are the radial versions of sines/cosines and complex exponential, respectively, and on and on.
The next big jumps were to collections of functions not parameterized by subsets of R^n. Wavelets use a tree shapes parameter space.
There’s a whole, interesting area of overcomplete basis sets that I have been meaning to look into where you give up your basis functions being orthogonal and all those nice properties in exchange for having multiple options for adapting better to different signal characteristics.
I don’t think these transforms are going to be relevant to understanding neural nets, though. They are, by their nature, doing something with nonlinear structures in high dimensions which are not smoothly extended across their domain, which is the opposite problem all our current approaches to functional analysis deal with.
You may well be right about neural networks. Sometimes models that seem nonlinear turns linear if those nonlinearities are pushed into the basis functions, so one can still hope.
For GPT like models, I see sentences as trajectories in the embedded space. These trajectories look quite complicated and no obvious from their geometrical stand point. My hope is that if we get the coordinate system right, we may see something more intelligible going on.
This is just a hope, a mental bias. I do not have any solid argument for why it should be as I describe.
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Note that I'm not great at math so it's possible I've entirely misunderstood you.
Here's an example of directly leveraging a transform to optimize the training process. ( https://arxiv.org/abs/2410.21265 )
And here are two examples that apply geometry to neural nets more generally. ( https://arxiv.org/abs/2506.13018 ) ( https://arxiv.org/abs/2309.16512 )
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I feel like this is the way we should have learned Fourier and Laplace transforms in my DSP class. Not just blindly applying formulas and equations.
I’d argue that most if not all of the math that I learned in school could be distilled down to analyzing problems in the correct coordinate system or domain! The actual manipulation isn’t that esoteric once you get in the right paradigm. And those professors never explained things at that kind of higher theoretical level, all I remember was the nitty gritty of implementation. What a shame. I’m sure there’s higher levels of mathematics that go beyond my simplistic understanding, but I’d argue it’s enough to get one through the full sequence of undergraduate level (electrical) engineering, physics, and calculus.
It’s kind of intriguing that predicting the future state of any quantum system becomes almost trivial—assuming you can diagonalize the Hamiltonian. But good luck with that in general. (In other words, a “simple” reference frame always exists via unitary conjugation, but finding it is very difficult.)
Indeed.
It's disconcerting at times, the scope of finite and infinite dimensional linear algebra, especially when done on a convenient basis.
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
There is a saying about Gauss: when another mathematician came to show him a new result, Gauss would remark that he had already worked on it, open a drawer in his desk, and pull out a pile of papers on the same topic.
One of the things I admire about many top mathematicians today like Terence Tao is that they are clearly excellent mentors to a long list of smart graduate students and are able to advance mathematics through their students as well as on their own. You can imagine a half-formed thought Terence Tao has while driving to work becoming a whole dissertation or series of papers if he throws it to the right person to work on.
In contrast, Gauss disliked teaching and also tended to hoard those good ideas until he could go through all the details and publish them in the way he wanted. Which is a little silly, as after a while he was already widely considered the best mathematician in the world and had no need to prove anything to anyone - why not share those half-finished good ideas like Fast Fourier Transforms and let others work on them! One of the best mathematicians who ever lived, but definitely not my favorite role model for how to work.
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> There is a saying about Gauss: when another mathematician came to show him a new result, Gauss would remark that he had already worked on it, open a drawer in his desk, and pull out a pile of papers on the same topic.
As if phd students need more imposter syndrom to deal with. Ona serious side, I wonder what conditions allow such minds to grow. I guess a big part is genetics, but I am curious if the "epi" is relevant and how much.
1 reply →
When I interned at Chevron someone said they (or some other oil company) were using Fourier transforms in the 1950's for seismic analysis but kept it a secret for obvious reasons. I think you couldn't (can't?) patent math equations.
Gauss's notes and margins is riddled with proofs he didn't bother to publish - he was wild.
Not sure if true, but allegedy he insisted his son not go into maths, as he would simply end up in his father's shadow as he deemed it utterly Impossible to surpass his brilliance in maths :'D
> as he would simply end up in his father's shadow as he deemed it utterly Impossible to surpass his brilliance in maths
Definitely true but also bad parenting. Gauss was somewhat of a freak of nature when it came to math. Him and Euler are two of the most unreasonably productive mathematicians of all time.
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Gauss is gonna Gauss.
How was Gauss so productive with 6 children?
Probably sexual division of labor.
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That's 30 more fingers he could count with.
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He wasn't as productive as he could have been. This seems like Chuck Norris and Jeff Dean territory after all.
I'd hazard by not dealing with them until they'd been schooled enough to operate as computers and GI (general intelligence) assistants.
There's a spread of farmers, railroad and telegraph directors, high level practical infomation management skills in the children.
I only have 5 kids, and I am also not nearly as productive as Gauss but to a certain degree, it feels to me like responsibility kind of tries to force me to be more effective.
no TV
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Chutzpah.
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.
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 :-)
Signals can be approximately frequency and time bandlimited, though, meaning the set of values such that the absolute value exceeds any epsilon is compact in both domains. A Gaussian function is one example.
Another example: ears are excellent at breaking down the frequency of sounds, but are imprecise about where the sound is coming from; whereas eyes are excellent at telling you where light is coming from, but imprecise about how its frequencies break down.
that's mostly due to light waves being FAR shorter and many orders of magnitude more "sensors"
Ears are essentially 2 "pixels" of sound sensing; and for that limitation they are ABSOLUTELY AMAZING at pointing out the sound source.
It’s literally the Heisenberg uncertainty principle, applied to signal processing.
For those who don't get this comment, the Heisenberg uncertainty principle applies to any two quantities that are connected in QM via a Fourier transform. Such as position and momentum, or time and energy. It is really a mathematical theorem that there is a lower bound on the variance of a function times the variance of its Fourier transform.
That lower bound is the uncertainty principle, and that lower bound is hit by normal distributions.
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> I was minimizing time so spent while she was minimizing number of washes
I'm probably just slow, but I'm not following. Do you mean because you went fast, you had to run another cycle to clean everything properly?
If you haven't already, you should watch the Technology Connections series on dishwashers.
https://www.youtube.com/watch?v=jHP942Livy0
Since I’m rushing to load it as fast as possible the packing is not as good as hers so some dishes are left out. Overall this leads to more loads.
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The self loading dishwasher would be the greatest marriage saving invention since car navigation systems.
you just need to go "if you want it loaded your way, you do it" and all is solved
And if loading dishwasher is on top of your marital issues you're probably in very happy marriage.
The constant small degree of conflict and strife is key to happiness, people can't be permanently happy, they just find ways to sabotage when they do
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.
It's effectively the underpinning of all modern lossy compression algorithms. The DCT which underlies codecs like Jpeg, h264, mp3, is really just a modified FFT.
Inter/intra-prediction is more important than the DCT. H264 and later use simpler degenerate forms of it because that's good enough and they can define it with bitwise accuracy.
>Once you start looking at the world through the lens of frequency domain a lot of neat tricks become simple.
Not the first time I've heard this on HN. I remember a user commenting once that it was one of the few perspective shifts in his life that completely turned things upside down professionally.
There is also a loose analogy with finance: act (trade) when prices cross a certain threshold, not after a specific time.
I don't think pulsing skin (due to blood flow) is visible from a webcam though.
Plenty of sources suggest it is:
https://github.com/giladoved/webcam-heart-rate-monitor
https://medium.com/dev-genius/remote-heart-rate-detection-us...
The Reddit comments on that second one have examples of people doing it with low quality webcams: https://www.reddit.com/r/programming/comments/llnv93/remote_...
It's honestly amazing that this is doable.
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MIT was able to reconstruct voice by filming a bag of chips on a 60FPS camera. I would hesitate to say how much information can leak through.
https://news.mit.edu/2014/algorithm-recovers-speech-from-vib...
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Sure it is. Smart watches even do it using the simplest possible “camera” (an LED).
It totally is. Look for motion-magnification in the literature for the start of the field, and then remote PPG for more recent work.
You will be surprised of The Unreasonable Effectiveness of opencv.calcOpticalFlowPyrLK
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It is, I've done it live on a laptop and via the front camera of a phone. I actually wrote this thing twice, once in Swift a few years back, and then again in Python more recently because I wanted to remember the details of how to do it. Since a few people seem surprised this is feasible maybe it's worth posting the code somewhere.
It is, but there's a lot of noise on top of it (in fact, the noise is kind of necessary to avoid it being 'flattened out' and disappearing). The fact that it covers a lot pixels and is relatively low bandwidth is what allows for this kind of magic trick.
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I have seen apps that use the principle for HRV. Finger pushed on phone cam.
You can do it with infrared and webcams see some of it, but I'm not sure if they're sensitive enough for that.
[dead]
I would heartily recommend Sebastian Lague's latest video, which covers this in a very approachable way: https://www.youtube.com/watch?v=08mmKNLQVHU
Okay, who's gonna write the story
> The unreasonable effectiveness of The Unreasonable Effectiveness title?
Unreasonable effectiveness is all you need.
"Unreasonable effectiveness is all you need" considered harmful
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I did some analysis on top title patterns. Both of these make the list pretty handily: https://projects.peercy.net/projects/hn-patterns/index.html
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Someone will end up writing "Scholastic Parrots"...
I lol’d
Given how much of the talk is about the original paper the title references, and how the Fourier transform turns out to be unreasonably effective at allowing communication over noisy channels, I'd say it's a reasonable reference.
It's a play on the famous essay 1960 "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".
I agree this is getting old after 75 years. Not least because it seems slightly manipulative to disguise a declarative claim ("The Fourier transform is unreasonably effective."), which could be false, as a noun phrase ("The unreasonable effectiveness of the Fourier transform"), which doesn't look like a thing that can be wrong.
Also how most of the articles with this kind of title (those posted on HN at least) are about computation/logical processes, which are by definition, reasonable.
The Antipode of unreasonable effectiveness ness
Agreed, these kind of titles are very silly.
FTs are actually very reasonable, in the sense that they are a easy to reason about conceptually and in practice.
There's another title referenced in that link which is equally asinine: "Eugene Wigner's original discussion, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". "
Like, wtf?
Mathematics is the language of science, science would not compound or be explainable, communicable, or model-able in code without mathematics.
It's actually both plainly obvious for mathematics then to be extremely effective (which it is) and also be evidently reasonable as to why, ergo it is not unreasonably effective.
Also the slides are just FTs 101 the same material as in any basic course.
Hi, original presenter here :) The beginning is FTs 101. The end gets more application-centric around OFDM and is why it feels 'unreasonably effective' to me. If it feels obvious, there's a couple of slides at the end that are food for thought jumping off points. And if that's obvious to you too, let's collab on building an open source LTE modem!
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> Mathematics is the language of science
So, biology and medicine are not sciences? Or are only sciences to the extent they can be mathematically described?
The scientific method and models are much more than math. Equating the reality with the math has let to myriad misconceptions, like vanishing cats.
And silly is good for a title -- descriptive and enticing -- to serve the purpose of eliciting the attention without which the content would be pointless.
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I find it hard to parse the middle of your post. Are you saying Wigner's article, which is what all the "unreasonable effectiveness" titles reference, is silly?
If that is what you are saying I suggest that you actually go back and read it. Or at least the Wiki article:
https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...
By means of contrast: I think it's clear that mathematics is, for example, not unreasonably effective in psychology. It's necessary and useful and effective at doing what it does, but not surprisingly so. Yet in the natural sciences it often has been. This is not a statement about mathematics but about the world.
(As Wittgenstein put it some decades earlier: "So too the fact that it can be described by Newtonian mechanics asserts nothing about the world; but this asserts something, namely, that it can be described in that particular way in which as a matter of fact it is described. The fact, too, that it can be described more simply by one system of mechanics than by another says something about the world.")
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It is likewise unreasonable to look down on any kind of world model from the past. Remember that you, in 2026, are benefitting from millions of aggregate improvements to a world model that you've absorbed passively through participation in society, and not through original thought. You have a different vantage point on many things as a result of the shoulders of giants you get to stand on.
It is pretty funny to flippantly call an influential paper by someone who received a Nobel Prize in Physics 'asinine'.
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> FTs are actually very reasonable, in the sense that they are a easy to reason about conceptually and in practice.
ok but it's not the FTs that are unreasonable, it's the effectiveness
I think we all understand at this point that "unreasonable effectiveness" just means "surprisingly useful in ways we might not have immediately considered"
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How about The unreasonable effectiveness title considered harmful?
The unreasonable effectiveness of considering something harmful.
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Why "The \"Unreasonable Effectiveness\" Title Considered Harmful" Matters
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"The unreasonable effectiveness" is all you need
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The Unreasonable Effectiveness of LLMs.
Ironically a very relevant and accurate title.
Not at all relevant in those instance because the blog post was not LLM generated slop.
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).
I dont like the Fourier Transform. It is infinite which makes it coarse and rough and it it gets everywhere.
Anybody who does anything in the real world with Fourier transforms uses the fast Fourier transform operating on windowed data. This eliminates all of that infinite support and infinite resolution of frequencies.
To be more precise, when working with sampled data with uniform sample rate you use the Discrete Time Fourier Transform (DTFT), not the Fourier Transform!. None the less, you still end up with an approximate spectrum which is the signal spectrum convolved with the window function's spectrum.
In my view the Fourier Transform is still useful in the real world. For example you can use it to analytically derive the spectrum of a given window.
But I think the parent is hinting at wavelet basis.
Yes but they commonly dont end up as FT/FFT. For example wavelets and DCT.
Not sure if an oblaque tomb raider reference or math metaphor.
It's both a math metaphor and a Star Wars prequel reference.
not like here.
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)
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
Moreover, The Unreasonable Effectiveness of Linear, Orthogonal Change of Basis.
You are spot on.
More here https://news.ycombinator.com/item?id=46553398
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.
And even non cycling functions (e.g. a gaussian)
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.
Pretty sure in the USA you can patent mathematics if it is an integral part of the realisation of a physical system.* There is a book "Math you Can't Use" that discusses this.
* not a legal definition, IANAL.
I would think you could only patent a particular usage of it.
> Pretty sure in the USA you can patent mathematics if it is an integral part of the realisation of a physical system.
Yes, that's true. In that example, you're not patenting mathematics, you're patenting a specific application, which can be patented. In my reading I see that mathematics per se is an abstract intellectual concept, thus not patentable (reference: https://ghbintellect.com/can-you-patent-a-formula/).
There is plenty of case law in modern times where the distinction between an abstract mathematical idea, and an application of that idea, were the issues under discussion.
An obligatory XKCD reference: https://xkcd.com/435/
And IANAL also.