[HN Gopher] The unreasonable effectiveness of the Fourier transform
___________________________________________________________________
The unreasonable effectiveness of the Fourier transform
Author : voxadam
Score : 298 points
Date : 2026-01-08 19:00 UTC (1 days ago)
(HTM) web link (joshuawise.com)
(TXT) w3m dump (joshuawise.com)
| emil-lp wrote:
| Okay, who's gonna write the story
|
| > The unreasonable effectiveness of _The Unreasonable
| Effectiveness_ title?
| kingkongjaffa wrote:
| 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.
| jwise0 wrote:
| 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!
| slwvx wrote:
| If one wants to contribute to an open-source LTE modem, the
| best place to start may be 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 wrote:
| > 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 wrote:
| The HN audience is filled with the world's densest
| collection of literalists and nitpickers: software
| engineers.
|
| Metaphorical language compels them to microrebuttals.
| redhed wrote:
| It is pretty funny to flippantly call an influential paper by
| someone who received a Nobel Prize in Physics 'asinine'.
| groby_b wrote:
| I 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 wrote:
| He 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 wrote:
| 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.")
| kingkongjaffa wrote:
| Yeah 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-
| 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 wrote:
| I 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 wrote:
| 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.
| w10-1 wrote:
| > 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 wrote:
| They 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 wrote:
| 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.
| flufluflufluffy wrote:
| 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.
| jmyeet wrote:
| How about _The unreasonable effectiveness title considered
| harmful_?
| bitwize wrote:
| Coming 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 wrote:
| _" The unreasonable effectiveness" is all you need_
| seba_dos1 wrote:
| The unreasonable effectiveness of considering something
| harmful.
| pizza wrote:
| _Lies, Damned lies, and Unreasonable Effectiveness_
| rbonvall wrote:
| Lies, Damned lies, and Unreasonable Effectiveness _For
| Fun and Profit_
| flufluflufluffy wrote:
| Lies, Damned Lies, and Unreasonable Effectiveness: How
| Lies in Titles are Damn Near Unreasonably Effective
| threethirtytwo wrote:
| The Unreasonable Effectiveness of LLMs.
|
| Ironically a very relevant and accurate title.
| tverbeure wrote:
| Not at all relevant in those instance because the blog post
| was not LLM generated slop.
| observationist wrote:
| Unreasonable effectiveness is all you need.
| temp123789246 wrote:
| I lol'd
| nextaccountic wrote:
| "Unreasonable effectiveness is all you need" considered
| harmful
| Nevermark wrote:
| Don't let unreasonable effectiveness become the enemy of
| reasonable effectiveness.
| AlienRobot wrote:
| A closer look at unreasonable effectivenessness.
| cbracketdash wrote:
| A rigorous study of the "unreasonable effectiveness"
| method.
| Qem wrote:
| Why Johnny can't be unreasonably effective?
| alberto_ol wrote:
| What we talk about when we talk about the unreasonable
| effectiveness
| ggm wrote:
| News at 11
| greenwallnorway wrote:
| 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
| tgv wrote:
| Funny. You might want to do that modulo capitalization, and
| perhaps some other common substitutions (LLM/LLMs/Large
| Language Model/Large Language Models, it's/ it is,
| what's/what is, I am/I'm), but they change the number of
| words, so better opt for the shortest alternative.
| chaboud wrote:
| Someone will end up writing "Scholastic Parrots"...
| Yodan2025 wrote:
| The Antipode of unreasonable effectiveness ness
| SAI_Peregrinus wrote:
| 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.
| rcarmo wrote:
| I would heartily recommend Sebastian Lague's latest video, which
| covers this in a very approachable way:
| https://www.youtube.com/watch?v=08mmKNLQVHU
| derektank wrote:
| 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
| pastrami_panda wrote:
| 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
| cogman10 wrote:
| > 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 wrote:
| But 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 wrote:
| It'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?)
| ggm wrote:
| I posit Gauss knew he was a GOAT and had ego. But I also
| posit he loved his children.
|
| So, a nepotistic delivery was beneficial for his family,
| and advising his son to seek excellence outside the
| shadow cast by Gauss himself wasn't stamping on dreams
| (in my view) it was seeking the happiest outcome.
|
| Without overdoing it, the suicide rate for rich kids with
| famous parents isn't nothing. There are positive
| examples, Stella McCartney comes to mind. She isn't
| wings.
| ambicapter wrote:
| What does "She isn't wings" mean?
| refulgentis wrote:
| Paul McCartney started a band called Wings and she was
| also in it. I think the idea is "she received nepotistic
| spoils, lived in the shadow of dad even in his backup
| projects that 0.01% of people who know the Beatles even
| recognize." (This elides a very successful career as a
| fashion designer, as well as the awkward question of what
| _would_ have guaranteed her more "success", as well as a
| lack of understanding of how you feel after grand success
| you were chasing for it's own sake (empty))
| ggm wrote:
| I meant, she isn't defined by wings and being daddy's
| kid. She struck out into a field neither Paul nor Linda
| had occupied. She isn't (defined by) wings.
| derektank wrote:
| For what it's worth, his children were quite successful
| by all accounts. Two of the boys became successful
| businessmen after emigrating to the US and one of the
| boys became a director of the railway network in
| Hannover. Seems as though they weren't harmed by their
| upbringing.
| zipy124 wrote:
| I mean just like most scientists at the time Gauss was
| rather wealthy, so it is unsurprising they were fine.
| ambicapter wrote:
| > But what he deemed being posited as true
|
| Implicit in the "correctness" of this motive is the idea
| that unless you're #1 in your field, you are nothing
| (depression implies strong feelings of worthlessness).
|
| I don't know if you think that's a great lesson to teach
| your kids as a parent, but I don't.
| PunchyHamster wrote:
| Not wanting your child to be permanently compared to you
| for their entire career is entirely understandable
| aquafox wrote:
| 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.
| libraryofbabel wrote:
| 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.
| rcxdude wrote:
| Well, in that time it was more or less how mathematics
| worked. It was a way of showing off, and often it would be
| a case of "Hey I've solved this problem, bet no-one else
| can". It was only later it became a lot more collaborative
| (and a bit more focused on publishing proofs).
| libraryofbabel wrote:
| You're correct that the culture of mathematics has
| changed a lot, and has become much more collaborative.
| The rise of the modern doctoral training system in
| Germany later in the 19th century is also relevant. So
| really Gauss's example points primarily to how much
| mathematics has changed. But at the same time, I think
| you could reasonably take Gauss to task _even applying
| the standards of his own era_ - compare him with Euler,
| for example, who was much more open with publication and
| generous with his time and insights, frequently
| responding to letters from random people asking him
| mathematical questions, rather like Tao responding to
| random comments on his blog (which he does). I admire
| Euler more, and he was born 70 years before Gauss.
|
| Of course, irascible brilliance and eccentricity has an
| honorable place in mathematics too - I don't want to
| exclude anyone. (Think Grigori Perelman and any number of
| other examples!)
| srean wrote:
| There's also this notion of holding themselves to their
| own standards.
|
| They, Newton included, would often feel that their work
| was not good enough, that it was not completed and
| perfected yet and therefore would be ammunition for
| conflict and ridicule.
|
| Gauss did not publicize his work on complex numbers
| because he thought he would attacked for it. To us that
| may seem weird, but there is no dearth of examples of
| people who were attacked for their mostly correct ideas.
|
| Deadly or life changing attacks notwithstanding, I can
| certainly sympathize. There's not in figuring things out,
| but the process of communicating that can be full of
| tediousness and drama that one maybe tempted to do
| without.
| srean wrote:
| Weird typo in what I wrote. It's past the edit window.
| This is what I had meant to type:
|
| There's _joy_ in figuring things out, but the process of
| communicating what has been so figured can be tedious and
| full of drama -- the kind of drama that one maybe tempted
| to do without.
| PunchyHamster wrote:
| not sharing something you had no time to properly check is
| entirely understandable
| hinkley wrote:
| Someone blew my mind by convincing me to read Bush's "As we
| may think" which was published in 1945. Then I started
| digging into him and discovered he was also the second
| president of the ACM, was instrumental in shaping the
| formation of the National Science foundation (mainly by
| critiquing their initial plans as unworkable) and also
| Claude Shannon's doctoral advisor. Because of course he
| was.
| libraryofbabel wrote:
| Not to mention instrumental in getting the Manhattan
| Project going, along with many other research projects
| during WWII. He basically knew _everyone_. I didn 't know
| he was Shannon's advisor though!
| 3abiton wrote:
| > 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.
| bell-cot wrote:
| Imposter syndrome? If _I_ was a PhD-level student (back
| then) and had an idea - and it turned out that Gauss had
| also thought of the idea, then written it out, and he kept
| the notes right in his desk - _yeah_. I 'd take that as
| proof that I was one of the world's top mathematicians.
| brcmthrowaway wrote:
| How was Gauss so productive with 6 children?
| deepfriedchokes wrote:
| Probably sexual division of labor.
| obezyian wrote:
| Behind every successful man, ...
| kridsdale1 wrote:
| Chutzpah.
| wolfi1 wrote:
| no TV
| pantalaimon wrote:
| Nor Internet
| defrost wrote:
| 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.
| kowbell wrote:
| That's 30 more fingers he could count with.
| tzs wrote:
| 30? Did they only count on one hand back then?
| skinkestek wrote:
| 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.
| 0xdeadbeefbabe wrote:
| He wasn't as productive as he could have been. This seems
| like Chuck Norris and Jeff Dean territory after all.
| hahahahhaah wrote:
| Gauss is gonna Gauss.
| njarboe wrote:
| 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.
| petermcneeley wrote:
| I dont like the Fourier Transform. It is infinite which makes it
| coarse and rough and it it gets everywhere.
| ted_dunning wrote:
| 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.
| petermcneeley wrote:
| Yes but they commonly dont end up as FT/FFT. For example
| wavelets and DCT.
| RossBencina wrote:
| 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.
| globalnode wrote:
| not like here.
| monuszero wrote:
| Not sure if an oblaque tomb raider reference or math metaphor.
| andrewflnr wrote:
| It's both a math metaphor and a Star Wars prequel reference.
| hedgehog wrote:
| 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.
| cogman10 wrote:
| 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.
| astrange wrote:
| 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.
| Chrupiter wrote:
| I don't think pulsing skin (due to blood flow) is visible from
| a webcam though.
| moralestapia wrote:
| You can do it with infrared and webcams see some of it, but
| I'm not sure if they're sensitive enough for that.
| javawizard wrote:
| 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.
| zamadatix wrote:
| My dumb ass sat there for a good bit looking at the example
| in the first link thinking "How does a 30-60 Hz webcam have
| enough samples per cycle to know it's 77 BPM?". Then it
| finally clicked in my head beats per minute are indeed not
| to be conflated with beats per second... :).
|
| Non-paywalled version of the second link
| https://archive.is/NeBzJ
| jpablo wrote:
| You will be surprised of The Unreasonable Effectiveness of
| opencv.calcOpticalFlowPyrLK
| lanstin wrote:
| Which is a special case of mathematics.
| hedgehog wrote:
| 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.
| rcxdude wrote:
| 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.
| slow_typist wrote:
| The frequency resolution must be pretty bad though. You
| need 1 minute of samples for a resolution of 1/60 Hz.
| Hopefully the heartrate is staying constant during that
| minute.
| hahahahhaah wrote:
| I have seen apps that use the principle for HRV. Finger
| pushed on phone cam.
| 3eb7988a1663 wrote:
| 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...
| hinkley wrote:
| I befriended the guy in high school who built a Tesla coil.
| For his next trick he was building a laser to read sound
| off of plate glass. The decoder was basically an AM radio.
| Which high school me found slightly disappointing.
| zipy124 wrote:
| It totally is. Look for motion-magnification in the
| literature for the start of the field, and then remote PPG
| for more recent work.
| neckro23 wrote:
| Sure it is. Smart watches even do it using the simplest
| possible "camera" (an LED).
| Rzor wrote:
| >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.
| isolli wrote:
| There is also a loose analogy with finance: act (trade) when
| prices cross a certain threshold, not after a specific time.
| lutusp wrote:
| 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.
| RossBencina wrote:
| 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.
| wtetzner wrote:
| I would think you could only patent a particular usage of it.
| lutusp wrote:
| > 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.
| RossBencina wrote:
| Moreover, The Unreasonable Effectiveness of Linear, Orthogonal
| Change of Basis.
| srean wrote:
| You are spot on.
|
| More here https://news.ycombinator.com/item?id=46553398
| Jun8 wrote:
| 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 :-)
| hahahahhaah wrote:
| The self loading dishwasher would be the greatest marriage
| saving invention since car navigation systems.
| PunchyHamster wrote:
| 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
| bschwindHN wrote:
| > 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
| Jun8 wrote:
| 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.
| bschwindHN wrote:
| Ahhh that makes sense!
| Minor49er wrote:
| Do you two ever play friendly games of Tetris against one
| another?
| hammock wrote:
| It's literally the Heisenberg uncertainty principle, applied to
| signal processing.
| btilly wrote:
| 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.
| hahahahhaah wrote:
| Thats. I always assumed it was more a quirk of the universe
| than something driven by pure mathematics. Amazing.
| hammock wrote:
| Yes that's fair to say. The tradeoff is mathematically
| inevitable. Physics just dictates the constants.
|
| It's also the kind of thinking that can throw a wet
| blanket on the "beauty" of e.g. Eulers identity (not
| being critical, I genuinely appreciate the replies I got)
| Gannon6790 wrote:
| thank you for that reminder/clarification. I forget
| sometimes how much we think we have clear pictures of how
| things like that work when really we're just listening to
| someone trying to explain what the math is doing and we're
| adding in detail.
| ComplexSystems wrote:
| 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.
| OscarCunningham wrote:
| 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.
| PunchyHamster wrote:
| 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.
| shihab wrote:
| 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).
| fracus wrote:
| 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.
| bobbylarrybobby wrote:
| And even non cycling functions (e.g. a gaussian)
| alden5 wrote:
| 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
| rocqua wrote:
| 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)
| srean wrote:
| 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_.
| alexlesuper wrote:
| 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.
| patentatt wrote:
| 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.
| madhadron wrote:
| 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.
| fc417fc802 wrote:
| 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 )
| nihzm wrote:
| From the abstract and skimming a few sections of the first
| paper, imho it is not really the same. The paper is moving
| the loss gradient to the tangent dual space where weights
| reside for better performance in gradient descent, but as
| far as I understand neither the loss function nor the
| neural net are analyzed in a new way.
|
| The Fourier and Wavelet transforms are different as they
| are self-adjoint operators (=> form an orthogonal basis) on
| the space of functions (and not on a finite dimensional
| vector space of weights that parametrize a net) that
| simplify some usually hard operators such as derivatives
| and integrals, by reducing them to multiplications and
| divisions or to a sparse algebra.
|
| So in a certain sense these methods are looking at
| projections, which are unhelpful when thinking about NN
| weights since they are all mixed with each other in a very
| non-linear way.
| srean wrote:
| Thanks a bunch for the references. Reading the abstract
| these used a different idea compared to what Fourier
| analysis is about, but nonetheless should be a very
| interesting read.
| srean wrote:
| 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.
| nihzm wrote:
| > Sometimes models that seem nonlinear turns linear if
| those nonlinearities are pushed into the basis functions,
| so one can still hope.
|
| That idea was pushed to its limit by the Koopman operator
| theory. The argument sounds quite good at first, but
| unfortunately it can't really work for all cases in its
| current formulation [1].
|
| [1]: https://arxiv.org/abs/2407.08177
| srean wrote:
| Quite so. Quite so indeed.
|
| We know that under benign conditions and infinite
| dimensional basis must exist but finding it from finite
| samples is very non-trivial, we don't know how to do it
| in the general case.
| madhadron wrote:
| I'm not sure what you mean by a change of basis making a
| nonlinear system linear. A linear system is one where
| solutions add as elements of a vector space. That's true no
| matter what basis you express it in.
| Xcelerate wrote:
| 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.)
| srean wrote:
| Indeed.
|
| It's disconcerting at times, the scope of finite and infinite
| dimensional linear algebra, especially when done on a
| convenient basis.
| tsoukase wrote:
| 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.
| hinkley wrote:
| 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.
| physicsguy wrote:
| That 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.
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