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The Fourier Transform, explained in one sentence (2014)
- Hnrobert42 4y agoCool, but unreadable as a colorblind person with deuteranopia.
- MontyCarloHall 4y ago3Blue1Brown video that actually animates this: https://youtu.be/spUNpyF58BY https://youtu.be/spUNpyF58BY
- highdeserthackr 4y agoThis is very cool.
- Bost 4y agoThat color coding is nice. Not just as a reference to the parts of the math formula, but also within the sentence itself.
- stephendause 4y agoAgreed. It's a nice, succinct way of tying the natural language to the components of the formula. I don't know how well it would work in longer form content, but as a one-off or perhaps occasional technique, I think it works really well.
- onos 4y agoI see this as a confusing explanation of the algorithm for calculating coefficients, not as a line that explains what the Fourier transform is. As for a single line on intuition, can you beat: “method of decomposing a function into a sun of sines and cosines?”
- hgsgm 4y agoThen you have to get into why you are talking about sines and cosines, adding unnecessary complexity to the explanation. The OP is better. It's the frequency that matters for intuiook, the shape of the periodic function is a technical detail.
- cousin_it 4y agoSines and cosines are the best choice here because they're the "elementary periodic function" for a given frequency: the sum of two (possibly scaled and shifted) sinusoids at some frequency is also a (scaled and shifted) sinusoid at the same frequency. For other shapes of periodic function it's not true: for example, if you sum two triangle waves of the same frequency but offset from each other, you'll get a shape that's more complex than a triangle wave. Same for two square waves etc. So the most natural way to decompose a signal is into a sum of sinusoids rather than triangles or squares etc.
- PaulDavisThe1st 4y agoMinor note: "sinusoids" (as you say yourself in your final sentence) is enough. No need to mention sines & cosines as distinct types.
- symmetricsaurus 4y agoInstead of energy, it should say amplitude(and phase since it’s an inherently complex thing).
- Etheryte 4y agoA good example of an explanation that only makes sense once you already know the thing being explained.
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- benreesman 4y agoIMHO an under-explored bit of intuition is crops in the frequency domain. If you want to try this for yourself grab a megabyte of photos of “conventionally attractive” selfies and a megabyte of selfies chosen at random and gzip them. I’m personally not all that symmetrical so I don’t compress all that well. But I suspect you’ll find that “pretty” people compress better. Same experiment with pop music vs serious jazz, same result. I have a pet theory that everything from caffeine to alcohol is cropping in the Fourier domain. It’s a fun experiment, try it out!
- amuresan 4y agoWhat do you mean by crop in frequency domain? Low pass filtering i.e. removing higher frequency components?
- benreesman 4y agoI hope I adequately advertised this as personal metaphysics, certainly I don’t have figures that I can share. Asymmetry, blemish, teeth a different color than the corpus all consume entropy in the Shannon sense. In a very real sense, movie stars are the most boring people to take still photos of.
- hgsgm 4y agoI don't believe that, especially because jpeg is already compressed using Fourier analysis before you gzip it using non-Fourier compress.
- benreesman 4y agoI was being a bit flip when I said gzip, a lossless mechanism is unlikely to give much insight! I should have said JPEG or 264 or something: a DCT is what I meant. Thank you for the correction.
- KineticLensman 4y ago> "To find the energy at a particular frequency, spin your signal around a circle at that frequency, and average a bunch of points along that path" You're going to need a longer sentence... [Edit] If this sentence genuinely did what it said on the tin, the rest of the article - which give a lot more detail - wouldn't be necessary.
- dghughes 4y agoMany here probably have seen The Engineer Guy's series of videos showing a Fourier Analysis machine. If not here it is it's a great watch. https://m.youtube.com/watch?v=NAsM30MAHLg https://m.youtube.com/watch?v=NAsM30MAHLg
- sebzim4500 4y agoThis reminds me of an old joke in the Haskell community, where people who struggled to understand Monads would finally get it after a while, and would assume that whatever the last sentence they heard was the only necessary one for the explanation.
- dlandau 4y agoThat was my experience studying statistical physics in Uni. For "some reason" the fourth book I read was the "only one" written in a way that made sense. No relation to the order I read them in, just written better.
- sebzim4500 4y agoYeah statistical physics is another example where suddenly things just kind of 'click'.
- jn5 4y agoIf you lose something, you always find it in the last place you search... because why would you keep searching after you found it
- dec0dedab0de 4y agoIt took me an embarrassingly long time to realize that it was a joke when people said "It's always the last place you look." Like well into my teens. But ever since I figured it out, I always look at least one more place after finding something.
- TedDoesntTalk 4y agoDo you find it again?
- quercusa 4y agoI think that there are many people who don't recognize it as a joke and pass it on as great wisdom. I was also pretty old when I realized the tautology.
- Moodles 4y agoThe best explanation I’ve ever seen of the Fourier transform is from 3Blue1Brown: https://m.youtube.com/watch?v=spUNpyF58BY&vl=en https://m.youtube.com/watch?v=spUNpyF58BY&vl=en
- rectang 4y agoThe key insight for me on this topic also came from 3Blue1Brown, but in a different video: it’s that e^x is NOT best thought of as repeated multiplication, but instead as the function exp(x) = 1 + x + x^2/2 + X^3/6 + x^4/24 + … After being relieved of the burden of that misconception, I was finally able to understand the role of complex numbers in the Fourier Transform. https://www.youtube.com/watch?v=ZxYOEwM6Wbk&t=439s https://www.youtube.com/watch?v=ZxYOEwM6Wbk&t=439s
- fortran77 4y agoOr even more simply, if you know that e^x on the complex plane rotates you around the origin.
- rectang 4y agoPeople told me that over and over but it didn’t help — because it didn’t make sense why repeated multiplication would cause rotation! Later in that video, we see a visualization of the rotation. I was able to grasp how the exp function could yield rotation where I’d never been able to understand why e*e*e*e… did. https://www.youtube.com/watch?v=ZxYOEwM6Wbk&t=2178s https://www.youtube.com/watch?v=ZxYOEwM6Wbk&t=2178s
- nuancebydefault 4y agoI think you need to study the Euler equation to understand the relationship between goniometric functions and exponential functions when calculating with complex numbers. One easy to remember formula that connects those functions.
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- tel 4y agoHonestly, it's not so bad. It's easy to pick any such attempt apart. This is close to my favorite pithy way of explaining it, too, which is to break it down component-wise using the idea of filter banks. It's not a single sentence, but here's what I tend to say: Any signal—like sounds or electrical signals, or even images—can be thought of as having a certain amount of 'energy' at any choice of frequency. This makes the most sense in music where we might thing of a 3-note chord as having 3 distinct packets of energy at 3 different frequencies. For any given frequency, we can compute the amount of energy a signal contains at that frequency by comparing the signal with a test signal, a "pure tone" at that frequency. Pure tones are signals that have the unique property of putting all of the energy at exactly one frequency. The Fourier Transform is an equation which packages this idea up, showing us how to represent all of these measurements of energy at all frequencies. The natural idea of a "pure tone" might be a sine wave. This is what we think of when we think of a musical pure tone and it certainly exists only at a single frequency. But sine waves make for bad comparison signals due to the problem of "phase": two sine waves played together can perfectly support one another and become twice as loud, or they can perfectly interrupt one another and become silence. This happens because sine waves oscillate between positive values and negative values and positive things can cancel out negative things. When you look at the equation for the Fourier transform you'll see an exponent of a complex number. This is an improved version of a 'pure tone' which avoids the phasing issues of a sine wave. It does this by spinning like a clock hand in two dimensions, remaining always at the same length. This extra dimension lets us preserve enough information so that things never cancel out like with sine waves.
- sam_lowry_ 4y agoI like your explanation in that it gives reason to the rotation.
- glitchc 4y agoThat's better than the blog post actually (ECE background here).
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- munchler 4y agoI’m a layman, but I always think of the Fourier Transform as “an algorithm that converts amplitude over time into frequency intensities”. I guess that’s more of a What than a How, but it still seems good enough for a single sentence.
- thrdbndndn 4y agoIt's close enough. But then, how is it different from Laplace transform? (I have to admit, I actually learned about both during my uni time. But now I totally forgot them all.)
- linhns 4y agoI'm currently in uni, learnt about both but has already forgotten what the Laplace transform is about, but still remember Fourier vividly as I got to play with it a lot. If you read more into it's history, the DFT actually helped a lot during the nuclear arms race as it can detect any nuclear test that happened anywhere except underground.
- Jensson 4y agoLaplace transform is for exponentials, fourier is for the sine function (ie a complex exponential). So they do the same thing just with different weight functions. All of this is just linear algebra, with different linear transforms. A fourier transform is a linear transform that transforms point space basis to harmonic oscillation basis.
- deleted 4y ago[deleted]
- bee_rider 4y agoLaplace transform is for when you are going to do something professory or otherwise tricky, Fourier transform is for when you are going to do something engineery. Fast Fourier transform is for when you are going to do something so engineery it becomes tricky again.
- nuancebydefault 4y ago
- philippejara 4y agoJust like "A monad is just a monoid in the category of endofunctors" it'll only make sense and click once you already understand the concept, at least the haskell one was satirical. Those simple phrases are I admit a fun experiment to see how much you can compact concepts, sometimes it is actually useful to come up with new words to represent common phenomenons in a given field to use with the people that already know of the concepts(hence why you pretty much need to "expand" almost every couple words of the phrase for the laymen to understand). There is value in doing this expansion to teach the laymen, and the author does make the expansion somewhat in his post to be fair.
- shreyshnaccount 4y agoI find this kind of math translated to natural language very useful to visualise things. It adds tremendous value for someone like me. The comments saying that if the sentence was enough you wouldn't need the article are in a very "ha gotchu" way and it's counter to both the people who want to understand it and the spirit of HN
- meindnoch 4y agoOr just say that the (complex) amplitude at a particular frequency is the dot product of the signal and a (complex) sinusoid of that frequency. Then you’ll immediately understand how the Fourier transform can be just a change of basis (a matrix multiplication in the discrete case).
- voidhorse 4y agoI think the best semi-intuitive, non rigorous explanation I've seen of the Fourier transform is still one that first explained signal correlation in the time domain and then described the transform as basically performing correlation on the signal for all the possible sines at different frequencies. Essentially you're just testing for the presence of individual sine waves (of different frequencies) within the signal. Unfortunately I can't remember where I saw this explanation. It might've been in the Scientist's and Engineers Guide to DSP book.
- ajross 4y ago> best semi-intuitive, non rigorous explanation [...] signal correlation in the time domain [...] basically performing correlation on the signal for all the possible sines at different frequencies That's only going to make sense for someone who understands your jargon usage of "correlate", and who groks that integrals of sine curves are orthogonal under addition. That's precisely the hard part the linked explanation is trying to explain. If you've already gotten that far then the DFT is just calculator math.
- voidhorse 4y agoTrue! Though tbh, I did not even know that property of integrals, since I've studied this stuff only in the discrete realm. At the totally basic level, another thing that I think can really trip people up and that a lot of texts are not clear about is that the transform is about moving between representations and that we're not actually transforming anything. Not enough maths texts take the time to explain how the representation connects up with the objects we wish to study and how there are multiple possible ways to represent them, and in some cases, like Fourier, useful ways to "translate" between different representations.
- tonmoy 4y agoTo me the sentence in the OP is the same as the sentence you posted (your sentence is easier to parse since I already know the meaning of correlate)
- teldau 4y ago
- itcrowd 4y agoThis is the discrete Fourier transform (DFT) not the Fourier transform.
- marosgrego 4y agoFor the continuous Fourier transform, you would just take a continuous average, i.e. an integral.
- pid-1 4y agoI'm found of seeing the Fourier transform as a fancy way of changing basis / coordinates, not sure how mathematically correct is that. In high school physics some problems got way easier by redefining coordinates as x' and y', solving for those, them going back to x and y. That's what the Fourier transform does, but for functions. Looking at its formula, we can see it looks like we are projecting a function into a series of complex exponentials, just like we need to project a vector in x' and y' for a change of basis in the euclidian space. Then, the Fourier coefficients can be thought as "how much my function looks like a complex exponential of this given frequency". Compare the formulas for the Pearson Correlation and Convolution, they are almost the same. Since the Fourier transform is just a particular case of changing basis, there are many, many other integral transforms that might be useful in different contexts. We can even create our own. But unis teach about the Fourier transform because it is really good for solving differential equations, which turns out to be a popular way of modeling problems in engineering.
- necroforest 4y ago>I'm found of seeing the Fourier transform as a fancy way of changing basis / coordinates, not sure how mathematically correct is that. It's exactly a change of basis
- walnutclosefarm 4y agoIt's a mathematically correct way of looking at it as long as you're clear that that the bases are functions, and you're considering how a particular mathematical artifact (the signal function) is represented as a sum of basis functions. In the naive signal trace, the basis functions are one-hot functions (zero everywhere except at a single point, where they have value 1), in the Fourier basis, the basis functions are sine waves of amplitude 1. So the signal is the sum of the basis functions multiplied by a unique amplitude for each basis function. Sine waves aren't the only basis with which you can make this transformation. Your basis can be an arbitrary set of periodic functions as long as it meets certain requirements. Decomposition into wavelet functions is commonly used in seismic signal analysis, for example.
- dimatura 4y agoYes, that totally makes sense - you can think of a function as an "infinite-dimensional" vector, which you can express in the (orthogonal) basis of a bunch of sines and cosines of different frequencies.
- hprotagonist 4y ago“most of the time, you can take apart a continuous signal into the sum of a bunch of sine waves. The fourier transform tells you how tall and how shifted each of those sine waves are for a given signal, for as many of those sine wave components as you care to calculate. If you want those summed sine waves to usefully model your input, you’ll need to add up sine waves that are wiggling at least two times as fast as the most rapidly changing feature you want to capture is. “
- rurtrack 4y agoEver saw those EQ lines jumping around as the time goes by and music sounds? That is the direct result of drawing the information the FFT gives. There are many interesting applications, for example, voice recognition https://towardsdatascience.com/understanding-audio-data-fourier-transform-fft-spectrogram-and-speech-recognition-a4072d228520 https://towardsdatascience.com/understanding-audio-data-four... It is really amazing
- nixpulvis 4y agoMaybe it's just because I'm sorta an audio guy, but this is how I first understood it.
- necroforest 4y agoI got it by first thinking about the inverse Fourier transform, which is just saying that I want to write a signal as a weighted sum of sinusoids: `x(t) = sum_k w_k sin(2pik*t)` Then you just ask the question: given x, solve for w. The next question is "why sinusoids", and the answer is that because for any linear, time invariant system acting on x: y(t) = F[x](t) that system diagonalizes over (complex) sinusoids: y(f) = F(f)x(f)
- Misdicorl 4y agoIME explanations of the Fourier transform focus far too much on the specific mechanics of it without motivating the reason. I find people often don't even realize the time space and frequency space functions are the same function! I dive in like so "There are many different ways to write down the number 5. Tally marks, the sigil 5, 4+1, 2.5 times 2, 10/2. Each of them is more or less useful in different circumstances. The same is true of functions! There are many different ways to write the same function. The mechanics of taking a function in one form and writing it in a new form is called a transformation." Above needs to be said a hundred times. Then you talk about why a frequency space function can be useful. Then you talk about how to transform a time space function to the frequency space representation.
- spfzero 4y agoI like this explanation. It would be good to include a little bit about why we even care to do frequency-domain analysis, and what a "transform" is.
- panabee 4y agothis is a wonderful explanation and illustrates the importance of fourier transforms in a simple way anyone can grasp. coupled with voidhorse's comment, it appears like "transform" obscures the concept whereas "translation" would be more appropriate and clearer.
- prof-dr-ir 4y agoDoes "blog post learning" ever really work? I have taught these kind of undergraduate subjects and, in the context of a course, the Fourier transform never struck me as something very complicated. First, it feels completely natural to write down a Fourier decomposition for periodic functions; the inverse transform to determine the coefficient is just (real or complex) calculus; and all that is left is to convince the audience of completeness which is best done through a few examples as well as the time-honoured method of "proof by intimidation". (Note that the blog post does not do a better job here.) By contrast, these posts seem to fulfil a demand where people look for an isolated explanation of concept X, like Fourier transforms, matrix determinants, monoids, etc. But they are necessarily too isolated to really convey the subject. For example, one does not normally teach (complex) Fourier transforms without first dedicating significant time to topics like complex exponentials and integrals of trig functions, and likewise one normally teaches monoids only after introducing functors and applicatives. In other words, without having the necessary background at your fingertips it is hard to really grasp the core concepts. That is why, IMHO, reading these blog posts may feel good but will rarely make things really stick.
- avip 4y agoI think you’re confusing proving with understanding. The two are almost orthogonal.
- posix86 4y agoDisagree. If you truly understand something, you must have prooven it to yourself, everything is just a mental help ("donkey bridge" in german) that allows you to remember the statement better - weather it's right or not, you can only know once you've prooven it; and before you did that, it can often happen that your intuition on what's right is actually wrong.
- adlpz 4y agoThat makes no sense to me. Or maybe your definition of "understand" is completely unrelated to mine. Under your thesis one cannot "understand" anything physical (as in physics, chemistry, etc) as the underlying cannot be "proven", at all, by definition.
- ourmandave 4y agoVeritasium has an explainer of the Fast Fourier Transform on youtube. From how it's used and an interest history. https://www.youtube.com/watch?v=nmgFG7PUHfo https://www.youtube.com/watch?v=nmgFG7PUHfo
- m3kw9 4y agoWe need an explain everything in one sentence website.
- blippage 4y agoHere's my explanation: think of linear regression. Now ask yourself, what if I applied the idea of correlation/regression to a sine wave. An (audio) signal can be broken down into an addition of sine waves having different frequencies and phases. That, conceptually, is what we're trying to do with the Fourier Transform. What frequencies is the sound wave composed of, and of what amplitude. Phase angle is a consideration, but we are less interested in that. OK, that's a paragraph instead of a sentence, but it does motivate the whole raison d'etre of "why would I care anyway?"
- mejutoco 4y agoIf anyone wants to understand the Fourier transform, I can recommend the book "Who is Fourier: a mathematical adventure". It is supposed to be a children's book. It was very clear and intuitive. https://www.goodreads.com/book/show/6969906-who-is-fourier https://www.goodreads.com/book/show/6969906-who-is-fourier
- two_handfuls 4y agoIt's useful to test explanations on people unfamiliar with it. So in this spirit, I want to share: the sentence didn't explain it to me. After watching3Blue1Brown's video, I get it. So I can share with you what confused me: It's this part of the sentence: "average a bunch of points along that path". I know what averaging it, but I don't know what averaging "along a path" means. Also I was (wrongly) expecting the output of this function to be a real number - I just expect "energy" to be a number. Instead, here what we're doing is averaging complex numbers so the output is a complex number. What a complex energy means is left as an exercise to the reader. Here's my suggestion for a sentence that would have worked better for me: "To find the energy at a particular frequency, spin your signal around a circle at that frequency, and find the center of mass of the plot." Presumably then we have to take the distance from the origin to get an amount of energy.
- xedrac 4y ago3Blue1Brown's video is the best explanation I've ever seen.
- julianeon 4y agoThank you, this is the epitome of a helpful comment and it helped me.
- wardedVibe 4y ago> I just expect "energy" to be a number. Instead, here what we're doing is averaging complex numbers so the output is a complex number. What a complex energy means is left as an exercise to the reader. They shouldn't be using the term energy, really. That should be reserved for the magnitude of the complex number (so re(z)^2+im(z)^2). You can write a complex number either as a real and imaginary part, or as a amplitude r (the distance you're referencing) and phase (the rotation from r+0im). > "To find the energy at a particular frequency, spin your signal around a circle at that frequency, and find the center of mass of the plot." As someone who's spent a lot of time with the subject, I like this better than the original.
- based2 4y agosrc: https://betterexplained.com/articles/colorized-math-equations/ https://betterexplained.com/articles/colorized-math-equation...
- qez 4y agoAs someone without a lot of math background, the most confusing part of this is k. Is k a number, like 7.45 or whatever? Or something else? Everything else is just normal mathy stuff.
- torginus 4y agok is the parameter of the function as in the 'x'-axis so instead of saying y=f(x), the formula says y=f(k) You can see in the formula, that it's plugged in as the frequency of the signal the original signal is 'spun-around' with.
- gyorgytamas 4y agoSmart. But please give my PCBs back.
- raydiatian 4y agohttps://youtu.be/spUNpyF58BY https://youtu.be/spUNpyF58BY The last Fourier Transform explanation you will ever need.
- fortituded0002 4y agoI love the color coded approach for this, but to problems for me: 1. I can't tell the difference between the pink and the purple in the formula. The text is fine, but the formula isn't. 2. I have no idea what "signal" means in the "spin your signal" so the single sentence explanation is completely useless to me. This bums me out a bit because I think there's something really great here.
- kristopolous 4y agoI found that animation and visuals are really helpful here for people who struggle with it. There's plenty of tutorial videos on YouTub An epistemological hurdle that really helped me was realizing these things like frequency and time domain aren't "real" as much as they are extremely useful mathematical models for manipulating things.
- queuebert 4y agoTitle should say "The Discrete Fourier Transform, explained in one sentence".
- ur-whale 4y agoFor me, the most intuitive explanation of the Fourier transform is geometric and is as follows: If: . you have a vector space equipped with a dot product - say R³ . you have a specific orthonormal basis of said vector space - say (i, j, k) in R³ . you have a vector V - say V ∈ R³ . you want the coordinates of V in that basis . then all you have to do is take the dot product of V with each member of the basis (i.e. you *project* V onto each basis member). . in other words: V = Sum_over_basis_members[ individual_basis_member * Dot_product[V, individual_basis_member] ] And magically, that's all the various Fourier transforms are: . the discrete Fourier transform (both signal and basis are discrete and finite, dot product is the usual suspect) . the Fourier series (basis is discrete and infinite, signal is a continuous periodic function, dot product is the continuous version of the standard dot product, i.e. an integral) . the Fourier transform (basis is infinite and continuous, signal is any "well-behaved" function, and the reconstruction of the "vector" involves a double integral, one for the dot product and one for the summation over basis terms) For example, for the Fourier series: . The "Vector" is any well-behaved periodic function (don't recall exactly what well-behaved entails, but not the most important part of the story anyways) . The vector space is the set of well-behaved periodic functions . The basis is the family of functions Exp[2*π*i] where the parameter i = {0, 1, 2, 3, ..., infty} . The dot product is the cross-correlation between the function and a basis member: i.e. Integral[x=0, x=2*π, function*basis_member] . You rebuild the function as: function = Sum_over_basis_members[ basis_member Dot_product[function, basis_member] ] And that's it.
- motohagiography 4y agoThere should be a competition for pithy, colour coded layman explanations of theorems. This is everything I needed to know about it, especially after watching the wavelets video posted to HN a few weeks ago: https://www.youtube.com/watch?v=jnxqHcObNK4 https://www.youtube.com/watch?v=jnxqHcObNK4
- ear7h 4y agoKind similar is this demonstration on a record player: https://youtu.be/mRi23ueU7Zk https://youtu.be/mRi23ueU7Zk
- deleted 4y ago[deleted]
- hasmanean 4y agoThe DFT is the correlation of a signal with a bunch of regularly spaced apart single-frequency vectors. The fft is what you get when applying dynamic programming (memoization and recursive subproblem division) to the DFT.
- wardedVibe 4y agoI don't really like this explanation, tbh. "spin your function" doesn't really make sense. If I didn't already know what they were trying to say, I'd have no idea.
- taylorius 4y agoOk, my knowledge of the Fourier transform is limited to the discrete version, but I might as well add my take into the mix. :-) I've always thought of the Fourier transform as a series of projections (as in high dimensional dot products). You dot product your signal with a set of sine and cosine waves, which are selected to form an orthogonal basis over the domain of the signal. So it's just a basis transformation. The phase of the wave is captured by the fact that a sine and a cosine of the same frequency, when added together, will always remain a sine function, but with a different phase, depending on the relative amounts of sine and cosine that are added together.