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The unreasonable effectiveness of the Fourier transform
- emil-lp 9mo agoOkay, who's gonna write the story > The unreasonable effectiveness of The Unreasonable Effectiveness title?
- kingkongjaffa 9mo agoAgreed, 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.
- jwise0 9mo agoHi, 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!
- slwvx 9mo agoIf one wants to contribute to an open-source LTE modem, the best place to start may be OpenLTE: https://en.wikipedia.org/wiki/OpenLTE https://en.wikipedia.org/wiki/OpenLTE The core of any LTE modem is software, even if it is written for DSPs or other low-level software.
- metalliqaz 9mo ago> 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"
- IAmBroom 9mo agoThe HN audience is filled with the world's densest collection of literalists and nitpickers: software engineers. Metaphorical language compels them to microrebuttals.
- redhed 9mo agoIt is pretty funny to flippantly call an influential paper by someone who received a Nobel Prize in Physics 'asinine'.
- groby_b 9mo agoI mean... this one's actually a pretty good paper, but we also had Linus Pauling pontificate on Vitamin C, so maybe we should cool it with the appeals to Nobel authority alone.
- srean 9mo agoHe did have a very long life, so there's that. It's not easy to separate cause and effect from direct and strong correlations that we experience. The job of a scientist is not to give up on a hunch with a flippant "correlation is not causation" but pursue such hunches to prove it this way or that (that is, prove it or disprove it). It's human to lean a certain way about what could be true.
- Certhas 9mo agoI 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_of_Mathematics_in_the_Natural_Sciences 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.")
- kingkongjaffa 9mo agoYeah it's silly, I don't mean it in any mean spirited way. > Wigner's first example is the law of gravitation formulated by Isaac Newton. Originally used to model freely falling bodies on the surface of the Earth, this law was extended based on what Wigner terms "very scanty observations"[3] to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations." So despite 'very scant observations' they yielded a very effective model. Okay fine. But deciding they should be 'unreasonably' so is just a pithy turn of phrase. That mathematics can model science so well, is reductive and reduces to the core philosophy of mathematics question of whether it is invented or discovered. https://royalinstitutephilosophy.org/article/mathematics-discovery-or-invention/ https://royalinstitutephilosophy.org/article/mathematics-dis... Something can be effective, and can be unreasonably so if it's somehow unexpected, but I basically disagree that FTs or mathematics in general are unreasonably so since we have so much prior information to expect that these techniques actually are effective, almost obviously so.
- Certhas 9mo agoI am not discussing the FT case. But as regards Wigner's article, the core thing he points out is that while we are used to the effectiveness of maths, centuries after Newton, there in fact is not any prior grounds to expect this effectiveness. And no, this is unrelated to whether math is invented or discovered. If anything this is related to the extreme success of reductionism in physics. As a general point of reflection: If an influential article by a smart person seems silly to you, it's good practice to entertain the question if you missed something, and to ask what others are seeing in it that you're missing.
- staticshock 9mo agoIt 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.
- w10-1 9mo ago> 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.
- hogehoge51 9mo agoThey are still capable of being described with math, we are just not capable of doing the math, or probably better put is there is a diminishing return of doing the formalisation of those systems as our cognitive abilities are limited and couldn't reason about those models. It leaves them using very approximate models based on human language descriptions that can be reasoned about. Which means, the language of some fields can’t be math. However, I don’t think the original presenter was asserting those fields aren’t science, that’s an unreasonable interpretation. More so , ideally they would be use math as it is a language that would help prevent the silly argument “so, Y is not X?, or is Y only X provided Y is in the subset of X that excludes Z? “ (Even in Engineering, we hit this cognitive limit, and all sorts of silliness emerges about why things are or are not formalised)
- cubefox 9mo agoIt'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.
- flufluflufluffy 9mo agoAlso 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.
- jmyeet 9mo agoHow about The unreasonable effectiveness title considered harmful?
- bitwize 9mo agoComing up next on Hackernews: Why "The \"Unreasonable Effectiveness\" Title Considered Harmful" Matters The Unreasonable Effectiveness of "\"Why \\\"The \\\\\\\"Unreasonable Effectiveness\\\\\\\" Title Considered Harmful\\\" Matters\" Considered Harmful"
- BrokenCogs 9mo ago"The unreasonable effectiveness" is all you need
- deleted 9mo ago[deleted]
- seba_dos1 9mo agoThe unreasonable effectiveness of considering something harmful.
- pizza 9mo agoLies, Damned lies, and Unreasonable Effectiveness
- rbonvall 9mo agoLies, Damned lies, and Unreasonable Effectiveness For Fun and Profit
- flufluflufluffy 9mo agoLies, Damned Lies, and Unreasonable Effectiveness: How Lies in Titles are Damn Near Unreasonably Effective
- deleted 9mo ago
- threethirtytwo 9mo agoThe Unreasonable Effectiveness of LLMs. Ironically a very relevant and accurate title.
- tverbeure 9mo agoNot at all relevant in those instance because the blog post was not LLM generated slop.
- observationist 9mo agoUnreasonable effectiveness is all you need.
- chaboud 9mo agoSomeone will end up writing "Scholastic Parrots"...
- temp123789246 9mo agoI lol’d
- nextaccountic 9mo ago"Unreasonable effectiveness is all you need" considered harmful
- Nevermark 9mo agoDon’t let unreasonable effectiveness become the enemy of reasonable effectiveness.
- AlienRobot 9mo agoA closer look at unreasonable effectivenessness.
- cbracketdash 9mo agoA rigorous study of the "unreasonable effectiveness" method.
- Qem 9mo agoWhy Johnny can't be unreasonably effective?
- alberto_ol 9mo ago
- Yodan2025 9mo agoThe Antipode of unreasonable effectiveness ness
- SAI_Peregrinus 9mo agoGiven 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.
- rcarmo 9mo agoI would heartily recommend Sebastian Lague's latest video, which covers this in a very approachable way: https://www.youtube.com/watch?v=08mmKNLQVHU https://www.youtube.com/watch?v=08mmKNLQVHU
- derektank 9mo agoMy 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 https://www.cis.rit.edu/class/simg716/Gauss_History_FFT.pdf
- pastrami_panda 9mo agoGauss'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
- cogman10 9mo ago> 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.
- ggm 9mo agoBut what he deemed being posited as true, was this really bad parenting? It could be to head off competition or it could be brutal realism to head off future depression. Nepotism existed since time immemorial but for a mathematical genius, what was the nepotistic deliverable for the child? A sinecure placement at university?
- refulgentis 9mo agoIt’s unusual to tell others not to do something because you’re projecting they’re secretly doing it to compete with you, or that they’ll be depressed when they don’t do what you did. Doubly so when the rationale is “I’m so fucking awesome” Triply so when it’s something you’re passionate about, presumably inherently. Quadruply so when it’s your child. Its tough as a kid hearing your parents come up with elongated excuses why you can’t dream and work towards a future. When you let people find their own way, you might even learn something from it (ex. 70 yo Gauss learns he didn’t need to tie his mental state to his work because his son doesn’t suddenly become depressed from not matching dads output) Re: second half, sounds about right, confused at relevancy though (is the idea the child would only do it to pursue nepotistic spoils and an additional reason is the spoils aren’t even good?)
- petermcneeley 9mo agoI dont like the Fourier Transform. It is infinite which makes it coarse and rough and it it gets everywhere.
- ted_dunning 9mo agoAnybody 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.
- petermcneeley 9mo agoYes but they commonly dont end up as FT/FFT. For example wavelets and DCT.
- RossBencina 9mo agoTo 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.
- globalnode 9mo agonot like here.
- monuszero 9mo agoNot sure if an oblaque tomb raider reference or math metaphor.
- andrewflnr 9mo agoIt's both a math metaphor and a Star Wars prequel reference.
- hedgehog 9mo agoOnce 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.
- cogman10 9mo agoIt'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.
- astrange 9mo agoInter/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.
- Chrupiter 9mo agoI don't think pulsing skin (due to blood flow) is visible from a webcam though.
- moralestapia 9mo agoYou can do it with infrared and webcams see some of it, but I'm not sure if they're sensitive enough for that.
- javawizard 9mo agoPlenty of sources suggest it is: https://github.com/giladoved/webcam-heart-rate-monitor https://github.com/giladoved/webcam-heart-rate-monitor https://medium.com/dev-genius/remote-heart-rate-detection-using-webcam-and-50-lines-of-code-2326f6431149 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_heart_rate_detection_using_webcam_and_50/ https://www.reddit.com/r/programming/comments/llnv93/remote_... It's honestly amazing that this is doable.
- lutusp 9mo agoToo 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.
- RossBencina 9mo agoPretty 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.
- wtetzner 9mo agoI would think you could only patent a particular usage of it.
- lutusp 9mo ago> 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/ 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/ https://xkcd.com/435/ And IANAL also.
- RossBencina 9mo agoMoreover, The Unreasonable Effectiveness of Linear, Orthogonal Change of Basis.
- srean 9mo agoYou are spot on. More here https://news.ycombinator.com/item?id=46553398 https://news.ycombinator.com/item?id=46553398
- Jun8 9mo agoA 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 :-)
- hahahahhaah 9mo agoThe self loading dishwasher would be the greatest marriage saving invention since car navigation systems.
- PunchyHamster 9mo agoyou 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
- tsoukase 9mo agoOne of the things that I feel blessed of in my marriage is that my wife has the same way of loading the dishwasher like me. Anyway, fellow husbands think about the dozens of other conflicts that you are might avoiding.
- bschwindHN 9mo ago> 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 https://www.youtube.com/watch?v=jHP942Livy0
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- shihab 9mo agoIf 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).
- fracus 9mo agoLearning 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.
- bobbylarrybobby 9mo agoAnd even non cycling functions (e.g. a gaussian)
- alden5 9mo agoThis 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
- rocqua 9mo agoSo 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)
- srean 9mo agoPeople 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.
- alexlesuper 9mo agoI 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.
- patentatt 9mo agoI’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.
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- tsoukase 9mo agoAt 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.
- srean 9mo agoYes! Archimedes was probably alien intelligence compared to his peers. His method of exhaustion does qualify as integral calculus and it's not limited to that overwritten document. It appears many times in his work
- hinkley 9mo agoMy 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.
- physicsguy 9mo agoThat wouldn't really work well because the sparsity of the periodic spikes wouldn't fit the assumption that the signal has a frequency component 'everywhere', even though it's periodic. You can see that mathematically - if you take the Fourier transform of an impulse signal you get a smeared result in frequency space. You'd probably want to use a tool like calculating the cepstrum rather than fourier transform. Cepstral methods are commonly used in mechanical analysis to detect periodic impacts like where a gear tooth gets damaged.