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Thanks.

Why you changed the title and thumbnail, it was pretty cool

thank you

Oh!!!!!!! This is the most understandable Fourier video I've ever seen.

I can't believe how intelligent Gauss was, it's just incredible

hence the phrase "he's good but he's no Gauss"

It is related with his works on magnetism.

Don't read profile photo ok

here's another example of Gauss being a pure genius: "The Prime Number Theorem was conjectured by Legendre in 1798 and proved a century later by de la Vallee Poussin and Hadamard in 1896. However, after his death, a notebook of Gauss was found to contain the same conjecture, which he apparently made in 1791 at age 15. (You sort of have to feel sorry for all the otherwise “great” mathematicians who had the misfortune of being contemporaries of Gauss.)"

I’m smarter

imagine discovering the FFT and not bothering to publish it. legend

Gauss couldn't imagine it either. He was a perfectionist. The formula for FFT was one of the many incomplete papers he didn't publish, there were many more.

👽we do it all the time. Some things humans should never understand. Like how we shut down their nuclear launch facilities.

@A Awe are eternally indebted to you

Fuckin' Gauss.

Absolute sigma

I wish I had this when i was in college studying DSP. Had no idea back then where and why this was used and we were only cramming the theory for the exams. This is probably one of the best visualisations of FFTs that I've come across. So simple to understand such a complex topic. This will help students of the subject understand the theory so much better when its real world application is known and its impact.

We remember during our engineering, our maths professor just came in, just wrote and wrote nonstop on the board. The writing was so fast that we named him "machine gun" (the sound of the chalk on the board made rat-a-tat-a-tat-tat sound like that of a Thompson Machine gun). At that time we wondered why we were being taught this esoteric technique.

That's Indian education for you

That's the difference between good professors and mediocre ones. Thank goodness I had an excellent professor for communications theory.

Ah the dreaded DSP !

Thank God ! DSP and this video came at same time for us.

I am an electronics and communication student studying digital signal processing. it was the best FFT explanation I ever came across . This is probably one of the best visualisations of FFTs that I've come across

Facts man. I’m bioengineer planning to study my master in digital signal and image processing, and back to my bachelor days, the FFT was one of the biggest references we had to analyse the signals such as ECG, EMG and image from Fourier spectrum. So impressive the amount of applications we find with these algorithms. Edit: Also shout out to Veritasium for this amazing explanation of the FFT.

@Santiago Oliveros exactly..... and more complex or sensitive the data becomes with the help of FFT we could do far better analysis of data And like this kind of visualisation helps us explore a particular theorem or idea could be implemented at various places where we might generally not implement them

Learn about Spring and Spring Dampener Algorithms

same

Thank you this was a excellent description of FFTs. I’m 73 now and my PhD is in theoretical atomic physics. After school I worked for awhile with seismic data processing and we used FFTs extensively in acoustic scattering. This is the best explanation I’ve seen so far.

Do you regret anything in your life

The outside is always regretable in any age. Even Christ regrets humans.

Oh thats intresting work. I've been working with acoustic sensors and FFTs for a while now; I'm with you in saying this is really a great explanation of FFTs.

Another phenomenal video. As a teacher, your ability to convey this super-high-level information is impressive and encouraging.

Can you teach me this please

Love the video as always! 2:16 I'm just happy the nuclear fallouts stopped at the Canadian border, otherwise we could have had it bad here too! 😁

Before this comment exploded into thousands

Wow! Electroboom

Animator stopped it.

That's simply an artifact of the border. Upon landing on Canadian soil, the fallout particles said "sorry" and stopped emitting harmful radiation.

Stop knit picking Derek's video. Any appreciative comments this awesome video on FFT? I guess you still haven't recovered from the fallout with Derek after the 1/c problem video. LOL.. :)

I study neuroimaging with MEG, and for signal processing, FFTs are very critical. I came across your video by chance and it was a supreme way of visualizing it! Kudos to you sir! The best explanation of the FFT for sure!

3blue1brown

Same with me for EEG!

3blue1brown

As a resident of southern Alberta, Canada, I am so thankful that the radiation fallout from all that American nuclear testing decided not to cross the border into Canada, as shown at 2:15 and formed a neat line along thr 49th parallel. 😅 Hats off to the CBSA for holding up the border like that.

it wouldn't dare cross the border😏

I find it incredulous how many people these days take these geniuses for granted and give more attention, credit and admiration to some singer or sportsperson. We are so fortunate to stand on the shoulders of these people

Agreed However... most people don't think very deeply or put much effort into understanding the world we live in. Many don't have the capacity to understand anything in depth or are simply focused on survival or obsessed with trends & fads, consumption, simple novelty or real & imaginary drama. Hence popular culture flourishes, sports attract sponsorship & fans, commercial media, marketing & PR (propoganda) proliferate - religious, cultural & historical belonging remain huge parts of identity rather than spirituality, philosophy & community focused forward looking projects aimed at inclusiveness & utility. Humans are complex creatures & perhaps all we can do is be thankful for brilliant minds in varied fields & hope that in time such people rise to prominence based on merit, effort & output

You have no idea how much love I have for this man and his team. This is arguably the best channel for people like us who want to know so much about this cut don't have the time or capacity to understand this on our own. This really is such a gift to the world.. So many interested minds given the knowledge they need thanks to these people. I thank you team veritasium I thank you.

Let's not forget the big leap that has been done in the didactic field (this wonderful channel is a major example of it). In the mid 80s, when I was first confronted with Fourier and the related analysis, it was explained by the teachers and in the books in ways extremely difficult to comprehend. For me and many other students Fourier, Bode & Nyquist were a trinity of nightmares looming on every test or class exercise. But, thanks to Derek, today's students can finally enjoy just sweet dreams. Or are modern mathematical nightmares now caused by other concepts that still require better explanations?

I really can't overstate how appreciative I am of these science history videos. It's easy in the STEM fields to forget the history soaked into the ideas we take for granted every day. I would like if Math classes gave a little glimpse into this - especially in primary schools. Maybe more kids would appreciate the importance of math and "when we would ever need this in real life".

Same. I'm a huge history nerd, and it wasn't until college, when I took history of math courses that I really began to appreciate how awesome Math is and became somewhat close to actually being good at it (though I always knew it was important). If math had been taught to me that way at a much younger age, then I might've developed more math-friendly habits early enough that I could actually be a "math person" today.

The current generation of teachers (@ any given point...) are the product of their own teaching/grasp of context. Oddly, it wasn't my history teacher, but my English teacher in HS in the 80's who was intuitively combining Core principles (cross-disipline teaching).. Fun class, cool trivia that Really pulled one in... Thanks, Mr Murphy. 🙏

@Patrick Jordan You were fortunate, then. My teachers in the '80s made the subject as dry as the chalk in their hands. Math taught with the same enthusiasm as Ben Stein in *Ferris Bueller's Day Off* shut me off completely.

Yess, Math history actually sounds interesting too

I think if taught in schools, kids would just fall asleep. The important part, as Veritasium has discovered, is being able to be a good storyteller. There's a reason Derek has a PhD in Physics Education, he's basically spent his life on being able to teach STEM this well. I wish it was possible to have a Derek in every class, but unfortunately it's not. A good alternative though, might be for school boards to commission videos like this that teach the curriculum in more interesting ways. And there's already evidence this works, via Bill Nye the Science Guy. Who can say they actually disliked him or got bored of his videos as a kid? Not many, if any. If all of STEM was taught so interestingly, with stories that pique curiosity and experiments that amaze kids, less people would find STEM boring or difficult and more would find it a place to explore. IMO at least.

I still remember 10 years ago in college. Signal and Systems was a course which was very difficult to understand at first and little did i know that all the courses going forward (Network theory, DSP, Analog circuits .....) required at least a decent understanding of this one concept called Fourier transform and by extension Laplace Transform. The professor at that time made sure that at the end of every class I was left even more confused (though this is on me, I suppose). And as expected I barely passed this course, but in the back of my mind I some how knew that I would miss out on something huge if don't understand FT. So, as if I was possessed by a ghost, I spent the next summer reading half of Oppenheim and Willsky and trying to understand how everything worked. After multiple problem solving sessions and Matlab simulations later, I got a decent understanding of the concept and it was also during this time that I felt that a new world had opened for me. It felt as if I was using mathematics to look at reality while wearing a completely different hat. It was beautiful. It remains one of the best decisions I have made in college as it made future courses much more enjoyable as a bonus. Though I have forgotten most of the stuff now, the intuitions about the world that I developed at that time still remain. Thanks Derek for taking me down the memory lane!

These newer half an hour documentaries you are doing are just amazing!!!! What a high level of production for the rest of this platform to strive for!

I think my lowest grade in college was in my linear systems and signal processing course. I felt such a lack of confidence in my understanding from that course that I took another elective on similar subject matter so I could better grasp the material. I wish I'd had this video back then lol. I really like how your video interleaves historical context alongside the theory - great video!

I love how you include all the historical context in your videos. It makes the already awesome explanation of an interesting topic that much better. Kudos.

I'm a physics student and this is the first time I've properly understood FFT's. Thanks so much, I finally get it! (wish I had just understood it early, would've drastically improved my marks lol)

I am a sound engineer, I work a lot with DSP. Knowing the background story of the FFT, and having it explained so smoothly leaves me speechless, you did another masterpiece. Thank you!

haven't heard such profession, can you explain a bit, tell things about it and job opportunities?

@Ed Weibe No, he is probably an ordinary student, possibly college level and definitely brighter than average that he both watched this and read the comments. . Be thankful for your professors and the environment in which you live.

@Ed Weibe When you finish laughing, give Boran a couple well chosen sites you believe best exemplify your profession.

@Ed Weibe Why did you even make such a immature comment, what is your motivation? What are you, 10?

Ughhhh this makes me really regret not taking math more seriously in school. Little did I know I'd want to get into DSP later in life and would need it. Its not nearly as easy going back to learn, but its worth it! They need to show more real world examples like this in school!

We use global atmospheric models for our work and thus we use the Fourier Transfrom to apply grid based data in a spectral space global model, or the other way around. This video is really a nice help to understand where it comes form and what it does!

New theory: Supporting antisocial trolls and assorted pirates is so expensive for YouTube that they are desperate to ramp up the ads. Are you seeing more and especially offensive ads from YouTube? Maybe it's just me? And is it related to the surge in hate-filled comments from the trolls? But in any case, I want to know if it's because my surfing with good privacy practices has starved the beast, so their ad picker is befuddled, or if they know it's me and it's just targeted retaliation and harassment because I keep commenting in public about how EVIL the google has become. The gun ads and racist t-shirts ads are especially effective at being annoying, but not at selling. If I ever notice myself shopping from a google advertiser, then I plan to stop it. And if you see this comment after many YouTube videos, that's because I saw another offensive and irrelevant ad.

@Shannon Jacobs Ad-block plug-ins are available for most of the major browsers. They naturally cannot stop ads incorporated into the videos but they can block the rest of them. And don't feel guilty about using one; you have the right to determine what you watch and what you don't and if companies make their advertising obnoxious they are pushing users in the direction of ad-blocking. I would prefer that YouTube get advertising revenue so they can continue to fund this free service and in that spirit I used to tolerate the ads. But as they became more objectionable I inevitably went to an ad-blocker and I have not seen an ad on YouTube in just over two years. YouTube's management are bringing this on themselves.

I couldn't stand ads, that is why I use revanced and microg. Very effective and it even skips self promotion and other various things that you can customize. I know this sounds like an ad to get rid of ads, but I promise I am not affiliated in any way with those companies.

​@Shannon Jacobs racist t-shirts?

I may not be a college professor or a mathematics expert, but this video was still immensely comprehensive and inspiring. The sponsor also helped inspire me to do good. I love Veritasium and love the direction this channel is heading. Keep it up!

I had studied electrical engineering for several years, more precisely communications engineering, where digital signal processing and discrete signal analysis like DFT were the main focus. If I had had such fantastic visual explanations available to me as in this video, my studies would have been successfully completed after perhaps two years. Having to understand higher mathematics without visualization using only formulas is very tedious. Thank you so much for this method of bringing knowledge to the world. Keep it up.

Fourier Transform is one of the most powerful mathematical tools which many modern technologies rely upon. Even the famous Heisenberg uncertainty principle was derived from it.

I studied Electrical & Computer Engineering. Needless to say Fourier Transform was our bread and butter. It is such a pity that usually the historical context is missing in the educational process. It helps so much put things into perspective and get a sense of the purpose and the significance of what you are being taught. I know so many students that are getting discouraged and disappointed just because the educational process only involves sterile mathematics brutally being thrown into your face without any context whatsoever.

Why did you reply at least 6 times to this video?

@Lappro internet likes

Which is a crime from my perspective.

i totally agree, if you first explain it simply and show the big picture first you at least have a sense of what you're supposed to learn. Most schools just start from the beginning and build blindly from there so you only really know what you're doing by the end of the course

ABSOLUTELY correct. If only this video existed back when I studiedm it would have motivated me much more.

I'm going to have to stop here and say this is the single most catchy, fun to watch, visually illustrative and informative FFT video I have seen, I wish I had a video like this 3 years ago it would've really helped. You're making a real difference here with videos like this and I hope you keep up the great work.

IIRC an FFT is also used to analyze the FID (free induction decay) signal in an MRI machine. It has been 30 years since my MRI classes, and the FFT aspect was dumbed-down for us "non math majors". This explanation is light-years ahead of what we received. Well Done Veritasium!

That’s an incredible story, I feel only one thing is missing the most, I would’ve loved to hear more of that scientist and his thoughts on the matter, I was really surprised when you said you asked him himself but honestly, a little disappointed when we didn’t get to hear much of what else he has to say about it, but still a really amazing video as always

Wow - that's truly impressive. I'm in awe by all the geniuses coming up with these algorithms that we take for granted. Thanks for the interesting explanation!

I have a degree in Music Tech, and I've been working with digital and analogue sound manipulation for many years now. I've got to the point where I can tell what kind of frequency structure is likely when viewing a complex soundwave. But I use FFT in my work on a daily basis... and I STILL don't fully understand how it does what it does so quickly! But anyway, your video made things much clearer, so thanks!

I currently study signal processing at university and without this video I could've never imagined that the fourier transform was such a big deal. It's really sad how colleges don't take time to explain the importance of the taught topics before starting to lecture students on them. Thank you very much for making me see the big picture and realize how important are my current classes. This is the best kind of motivation to keep going.

I don't think a lot of people would appreciate having part of their precious expensive class time with a teacher taken up by a history lesson instead of the subject they're supposed to be teaching.

@Cracked Emerald Properly motivating a topic is a pretty crucial part of good teaching though. It piques curiosity and engagement, and helps to anchor what you are actually teaching. I definitely think a lot of times university lecturers could just default into immediately jumping into a complex topic without properly motivating to the student why they should even care about it to begin with. Doesn't have to be a 20 minute history lecture, but just spending a little bit of time could go a long way. But honestly though, a lot of the technical how's and what's can be obtained through books anyway. The job of a lecturer is to teach, not just to regurgitate what the books says.

Right? Most lectures are just like reading a list of definitions and formulas to memorize and that's it, completely voiding the subject of any context, relevance and usage. Good teaching is extremely rare.

@Cracked Emerald Wrong. That is a massively important part of the lecture, and typically the least boring part.

Welch me to the University of YouTube where the best minds teach and express their curiosity .

Part of my senior research project for my physics degree dealt with infrasound. Fourier Transform was difficult for me but after watching your incredible explanation of it, I may give it another shot.

Darn... I went in wanting to know more about how an audio equalizer and spectrum visualizer worked... I wasn't expecting to learn about nuclear weapons and cold war politics. This was an impressively good video, as usual! Yours has to be one of the best science dissemination resources out there for sure. Thank you!

This is the most intuitive explanation of Fourier transforms I've ever seen. You're so right that Fourier transforms are everywhere: I use them daily in my work as an electrical engineer of course. Interesting backstory about Tukey I had no idea about!

you do have a way for making very complex material accessible to laypersons. Content like this will go a long way towards making specialized knowledge accessible to everyone, so it will never be lost again. It's just silly that something this complicated yet revolutionary had to be figured out twice in human history.

Numerous FFT algorithms have been discovered dozens of times in the last hundred years. In the late 80’s Henrick Sorensen, a grad student of Sid Burrus, at Rice University wrote a short paper on the FFT chronicling it’s history and they decided to stop when they reached Gauss. I suspect that Cooley and Turkey’s discovery was better timed with the development of programmable computers

Gauss discovering FFT even before Fourier published transforms is the most chad moment in history

Dude literally doodled on his notebook and said trash

But he was kinda built different

kronos Jean-Baptiste Joseph Fourier, author of the analysis in his name, had the greatest impact on Science (& Technology now) like none else among the Scientists. There seems to be no alternative to Fourier's track.

Your production quality and presentation organization are insane. I learn things from you, thanks for your hard work!

I love the FFT. After college I got into writing music visualization apps like winamp. I wrote my first FFT algorithm on windows 95 with a pentium using floating point math so I could create dancing images which reacted to drums and cymbals separately. I had a blast. I ended up writing a fixed point version in x86 assembly before compilers could do a better job. The only problem with the FFT is it's linear and the human ear hears logarithmically. So no matter how complex the FFT, you'll always have far too much high frequency data, and far too little low frequency data. If only someone could devise a logarithmic FFT which processed frequencies into the notes we actually hear.

The visual at 18:09 is absolutely gorgeous. Also, I can't describe how informative this video was and in fact helpful in understanding how data is retrieved for my research! Thank you!

It would make a really nice poster! His current video thumbnail is also beautiful, I want to know where he got that art.

I wish I had these kind of content while I was studying. I still wonder many other things I had learned but I see no use in my career but now I still find fascinating , how this is (or was) applied in real life. Thank you for the video

We had finished the FFT in just few days without knowing the impact and use of FFT. Story behind the FFT is really a treasure for a communication engeneering student like me Thank You for this amazing video.

This is a SUPERB video. I’ve worked in Fourier space for over 40 years and this is one of the best explanations that I’ve seen. I was also unaware of a lot of the history as well. VERY well done!!

Small Note: For digital images, compression algorithms (like JPEG) actually use Discrete Cosine Transform which I think is basically Discrete Fourier Transform but for only the cosine (real, even) part. I tried to skim through its wiki to see why we don't need the sine (complex, odd) part, but I couldn't figure it out. My bachelor degree is in Digital Media Engineering and Technology, and I feel kind of ashamed for not knowing everything about this 😅 I remember learning DFT, but I don't remember learning the optimization (FFT), only that it exists and that Matlab has a function for it. So either we didn't study what optimization is actually done or we did but it was a morning lecture so my memory formation was questionable 😂

Great video, very well explained and the fact that Gauss discovered the FFT it in 1805 is mind blowing. The FFT makes most of the technology we use today possible.

As a sound engineer, killer explanation of FFT! Best visualization I've ever seen. Vera never misses 👌

It wasn’t ever really explained that well in the computer vision course I was thrown into in fourth year, nor did I have the math prerequisites required to properly understand the concepts competently before entering the course. But after mucking through the assigned problems, the results and performance we got out of the convolutions were almost magical. Albeit that these were prepared by the professor ahead of time. I can’t imagine the creativity, ingenuity and tribulations the predecessors had to go through to come up with these elegant solutions.

As a Electrical Engineering student who has taken digital signal processing, this is a beautiful high level understanding of fft. Love your videos man!

As another electrical engineering student, I couldn't agree more. It's beautiful in ways I can't describe

@Kevin as another electrical engineer, I am akin to Nikola Tesla so if you have any questions just ask me

Damn it. That was word for word the same comment I was about to write

Good time memories staying up until 2AM doing FFT and Laplace.

Ah signals and systems, memories.

I remember the testing in the atmosphere. I was a child then and there were announcements when a radioactive cloud would be coming ie strontium 90 and others. My mom and my hair fell out at that time. I have never heard this story about FFT before. It would be interesting aside for DSP class but time consuming so its good to hear right here.

was that in the 1950s? this is terrifying, Kim. i keep hearing these things about the US nuclear experiments back in the day and i wonder how the hell people survived that or got through it all okay sickless and how come all those affected states (those downwind of the fallout clouds) are still inhabitable. Nice to hear from someone who actually experienced that.

in addition, heck, how come Nevada is still habitable after all those nuclear experiments - Las Vegas lies in it and people go there from all around the world. Also, underground testing - i always wonder whether the soil and all what grows in it is still safe, whether the lakes and underground water and rivers and whatnot are still fine..

@JabrHawr if it was nuclear weapons exactly but I avoid anything from the locker in general. oceans and lakes are way too dirty to eat from pretty much everywhere just from seepage

I wish I had these video 5 years ago. Several math courses, radiophysics and unimaginable amount of Fourier transforms and only now I really understood FFT. Thank you, Veritasium team! Not even saying about obvious historically valuable info in this video

Gauss lived in an age when, after solving the problem, the automatic second thought was - how do I reduce the computation. Expecting everyone else to do the same he probably regarded the Fast in FFT as trivial. Great Video!!

I was lucky to get introduced to FFT algorithm early in my career around 2k3. It was a revelation on reducing complexity of our world into singularity. Afterall east thinks of everything both animate inanimate objects in this world as just waves that started from singularity of the Dark Matter. It is a fantastic visual treat explaning on FT, DFT and FFT. Thanks a ton Veritasium.

Really great explanation on a difficult/abstract concept. I think it goes deeper, too. Differential equations that fit Sturm-Liouville theory (like the separated wave equation) form a series of orthogonal solutions (orthogonal basis), which looks a lot like a Fourier transform, and I don't think that's a coincidence. Just similar math owing to being a similar system. I'm not a PhD mathematician so I can't comment further, though. But it's wild, fundamental stuff.

As a 66yo electrical engineer... it took me the last 20 years to finally understand the FFT as well as you've described it here. I recently realized that unfortunately the colleges do a horrible job teaching this subject... and I agree that it's one of the most important topics of math for anyone going into physics or engineering. I put together a radar 2D imager (ISAR) but using acoustics to keep the cost down. For the last 2 years I've been trying to get my local high schools interested in starting an after school geek club to teach kids basic signal processing using this acoustic imaging application. So far... no one is willing to even talk to me. It's too bad. Maybe your video will help. I really think that with the right application (acoustic imaging) this subject can be taught to 11th and 12th graders (multiplying sinusoids is really all you need to understand). This would prepare high schoolers beautifully for college science/engineering.

If you want reach, creating a short well made video and post it on all social media will help you a lot! its not hard.

@bluetorch13 Yup, I've thought about that. I'm not sure how to approach it... I have 100+ slides of discussion/description... probably 10 different applications that show the LFM pulse in slow motion so you can hear it, simulation showing the xmit to rcv correlation, simulation of forward and inverse FFT, sine cosine for phase angle, noise reduction through FFT size, resolution vs bandwidth, etc.. Then I have the imager itself. I've thought about showing the imager but that will only hook you if you're already interested in understanding the math. Animations like Veritasium puts together are REALLY difficult to do. His video is great... but let's face it... only us engineer/math geeks really followed his description.. it was great... but the non-geek probably did not really grasp the "area under the curve" or "sine vs cosine gives the phase angle", or the Euler expression. It's a tough subject to convey, or get someone interested enough to commit time. The beauty is that in an hour I think I can totally hook high school kids by showing the imager, then a brief discussion of why the xmit pulse looks like it does, then a simulation showing what happens when you correlate the xmit with the rcv from one, then more than one scatterer, and finally the magic of the FT that is basically a correlation with multiple frequencies.... and imaging can be described using range and velocity for a rotating target (ISAR).

I learned the basics in college then researched on my own. What helped was writing code and running tests for both audio and images

Great idea from a mechanical engineer.

@Engy phonermp You could do a van Biezen or Brian Douglas type video series.

My uni lecturer spent three terms in the 1980s describing the FFT. You students today do not realise these incredible graphic videos!

I'm shocked that colleges don't tell us about the necessities that made humanity invent the concepts that we learn. I am an ECE student and this by far the most complete explanations of DFT and FFT that I could ever come across.

I learned Fourier Transform in "Signal and Systems" course in my 4th semester of computer engineering, it completely changed the way i see the world.

Great content as always. I'd like to point out a mistake though : When taking 8 samples (over a second) then the size of the DFT will be 8/2 + 1 = 5. Not 8. You have the DC component plus the correlations with 1, 2, 3 and 4 Hz sine/cosine. Indeed, when sampling at 8Hz one cannot detect frequencies exceeding 8/2 = 4Hz, the Nyquist frequency. The graphics for 8Hz frequency is a bit off but if you correct those you'll see that the cosine should be 1 at every point thus multiplying at those points would yield the same result as the DC component. The same way that the correlation with 5Hz will yield the same result as the correlation with 3Hz cosine (that is quite related as why we sometimes see wheels spinning in the other direction they are actually spinning, but let's stop here).

Great video ! I was wondering about what is said at 12:50. If I understood well what is said, a discret signal of n sample has a frequency spectrum of size n (at most). I encountered during my studies what I think is known as "Shanon theorem" (I'm not exactly sure how it is called in english, as I studied in France) which state that in order to sample correctly a signal with a spectrum of size n, you need to sample your signal with more than 2n sample (the sample rate has to be higher than 2 times the highest frequency of the signal mesured). So, I expected to find a spectrum of size n/2 (at most) for a signal encoded with n sample. Is there a simple explanation of why it is not the case ? (I remember not having seen the demonstration of Shanon Theorem, I think it was because it uses distribution theory which I didn't know at the time)

Taking a complex problem, and decomposing it into the history, science and specific use cases is refreshing to see and should be explored more often in STEM education. Thanks Derek, and the team!

Taking something complex and decomposing it into a bunch of simple things...sounds like the FFT!

Don't put history on the test, plz.

Crystallography is a fun example, because we measure the intensities but cant measure the phase, so some really clever experiments are needed to guess and then refine the phases for FFT and inverse FFTs and solve the crystal structure. It's called solving the Phase Problem or Phasing a structure

The timing of this video is so freaky. My professor had us watch some video on discrete Fourier transform and it was really hard to follow and understand, and this thing comes along to describe it to me in such an entertaining way. And I'm about to graduate very soon too, so the sponsor of this vid is definitely going to come in handy. This video just gets better and better!

Another masterpiece from Veritasium. Please never stop making videos :)

I am nowhere near the mathematical literacy to fully comprehend this, I know I would need to sit down for a year (or years!) to get there. However, I can appreciate how insanely amazing this discovery was, and even more impressed with the discovery in 1805, fascinating video and extremely well made.

You really have a great impact;fan from India!You have to be one among the great explainers .With no background of waves or physics or even proper programming I was able to get the gist and essence of the FFT! Hats off;really continue making impact.

What an amazing video, I'm blown by the combination of storytelling, breakdown of complex math, connect to real life applications and of course the drama. This is top tier content.

(digital) signal processing (via FFT/DFT) was one of my favorite classes in the early 90's electrical engineering. It was cool in the sense of understanding the "new" field of computer audio, and later in the 90's helping out a DJ friend with digital sound theory for his dance remixes (which was cutting edge in the late 90's and helped get him on the map. He's still DJing/remixing today.)

Really appreciate the time put into this video. Thank you for fueling my curiosity, makes me feel alive.

great and simple, congratulations! Multiplication for sine/cosine functions, and analyzing areas according to sign laws is a more clear approach that using exponentials and complex numbers! FFT is the discrete-math version of FT, using finite intervals, windows and samples...windowing is key! Please do a video about windows in FT vs. wavelets!

Heh, I discovered the correlation part on my own after knowing what the FFT was. I just couldn't wrap my head around it but this video ratified my understanding of it. Pretty damn cool what the FFT can do

This video is just incredible. Although I already learned a topic a bit, including a brilliant work of 3b1b, it still explained a lot for me. Thank you!

This was, quite impressively, a much more clear explanation of how Fourier series & transforms work than I ever got in school.

The Fourier discussion was deep and well explained but the discussion on the antagonisms between the USSR and USA here is really shallow and short. Both sides did have people attending these conferences who legitimately wanted to contain nuclear weapons. After all, they were mostly academics and scientists. But back home both sides had to contend with the political sphere: the senators and representatives that make up the government. These political actors on each side were not as thoughtful and were driven by fear and distrust. So even had the multiple talks about nuclear containment and non-armament ended well I doubt either side would have honored them. In the USA we had senators were absolutely certain the Soviets were planning world domination and in the USSR members of the Supreme Soviet (their senate) were absolutely certain of the same. Also it was a commonly held belief among many Soviet politicians that the Americans were literally insane and ready to use nukes at any moment.

If only professors did that before just throwing equations on board.

@dezmodium yeah. Neither the USSR nor the USA were even willing to adopt the "No first use" policy. So I find it very hard to believe that the USSR, USA, etc would have stopped the nuke arms race if the FFT was discovered earlier etc etc. And I agree the Americans were and are insane in terms of nuke usage management - compare the USSR's Perimeter/Dead Hand with the USA's "The US President can launch nukes even if he goes crazy and nobody is supposed stop him" system. Search for "Harold Hering and the forbidden question that cost him his career."

What kind of school are you talking about? I graduated university with mathematic degree and we didn't study Fourier transform.

@Max Jacoby Any kind of school where you learn about signal processing. Digital electronics is where I learned about it.

I had about 1 year at university about Fourier Transformations. Now, 26 minutes made me understood how and why. Thanks for this video!

Hi Veritasium, I saw the explanation of the fast Fourier transform and I liked it very much, however, it is a wish of mine that you can make a video about control system algorithms like kalman filter or PID control. Thank you very much for spreading science, greetings from Peru.

Thanks for sharing this very interesting story! This is a must see for anybody interested in math, algorithms, scientific history and much more!

I did my Master's degree in trying to find Black Holes and Neutron stars in neighboring galaxies. I used Discrete Fourier Transforms to look for orbital periods. What is really interesting is that signals can be smeared out by the window function. In most cases the window function is a reflection around the y-axis of the first few bins or periods in an FFT. My data was poor quality and very noisy so all the results got smeared around and were difficult to interpret.

A few years back, when Fast Fourier Transform was added to Engineering Services Examination Syllabus(India), I thought that it was just another gimmick to expand the syllabus for students. Now, I realize how dumb I was and how vital these transforms are be it Fourier, Laplace, or Z-transform.

Scrolling through the comments I realize that my feelings toward this video as someone who is currently a graduate student in Electrical Engineering is not unique at all. It's amazing what great animation and very well articulated and easy to understand words can do when trying to understand a complicated subject. I've taken probably 6 or so courses at least that use the Fourier Transform, along with classes that utilize the FFT, and never once fully understood what the hell I was really doing and why it worked. Thank you very much Veritasium, this has genuinely helped my understanding of FTs 5 years deep into my college education lol.

There's a lot of "shut up and compute" in some courses. Taking an advanced math class where the prof was covering Bessel functions (in gory detail for like 3 weeks) someone finally asked "ok, Dr Smith, we've spent 3 weeks studying these, what are they good for?" and the prof replied "I dunno--engineers use them for something, I think". Didn't know, didn't care. It was kinda comical taking an applied math class from a person who really had no idea how to apply the math--guess that was left as an exercise for the reader. One EE grad class the prof (guy who co-invented the Discrete Cosine Transform) decided we really all needed to learn to derive the FFT algorithm and do it on a test. And then a couple weeks later, decided we needed to be able to derive on the next test how to use DFT's to perform a DCT. Did that help us understand this stuff? Heavens no. It just forced us to rote memorize some magical math. But through reading enough stuff outside the textbooks, I finally (sorta) came to have an innate understanding. And yeah, well-done videos can go a long way toward providing that mapping from pure math to logical understanding of the concepts.

@FroToe Yes, there's really too much of that, that's frustrating.

Painful stuff... Differential equations was enough for me during my time as a computer engineering undergrad. Will I use diffEQ again? Probably not. Did it make me have a deeper sense of understanding and appreciation of mathematics? Yeah sure I guess.

Yeah this vid would defo have helped me understand Fourier Transforms quicker when I was trying to grok what the hell they were about in my undergrad electrical engineering degree. I did finally get it though, and then it was a glorious light of wow, how epic is this stuff!

I highly recommend the following book for an excellent pictorial, intuitive and 'gentle' mathematical introduction to the basics of the FFT. The copy I have is: Title : The Fast Fourier Transform Auth: Brigham, E. Oran ISBN: 0-13-307496-X Pub: 1974 by Prentice-Hall Inc.

I love how you make sense subjects easier to learn but MAN this one would be ridiculously complicated seeming if I didn’t understand some of the maths that I had classes for. It’s definitely dense

It's such a mind boggling to know that the same algorithms used in JPEG compression could've been used to stop the nuclear race. Thanks for the great video!

I really enjoyed Mr. Garwins reply to your question, thanks for that! I have a question for you - Isn't a Fourier transform a way to digitally do the same thing as an analog spectrum analyzer? An SA would show the frequencies and relative amplitudes of the seismic wave form. I think analog SA technology was well known at the time.

@Veritasium: Thanks for the video, you gave me a better insight into FT, DFT and FFT in less than 24 minutes than my professor in over 45 minutes. I missed the part how the scientists use the frequency component to estimate depth or power of the nuclear explosion. Do you have a simplified explanation what was done to know if there is a violation of the test ban? Thanks!

I never could appreciate DFT and FFT, even when I was studying and one of the most feared (and knowledgeable professors) talked about it like it was the golden child of mathematics. Only the sheer amount of applications and more time spent dealing with it helped me appreciate it.

FFT is the reason we are able to diagnose the problems with industrial machinery (pumps, compressors, turbines). As a mechanical engineer, I absolutely loved this explanation, but have to watch it again to understand it fully. Thanks, Derek for this work! 🙏

CSI 2140

Me too. I do vibration analysis on dyno systems and auxiliary equipment. I also worked in the cable industry and very lightly used it in signal analysis for troubleshooting.

FFT is used a lot in signal processing... 🥰

That makes sense! I wouldn't have guessed but now that you mention it I see how they could apply in say motion detection software to that tests the durability of machinery (forgot the name but there's a video on that)

As a young engineer in at UT Austin 1976, I was required to write program to study changes in heart rate for astronauts. The FFT algorithm was still new back then, but I felt it had an intuitive logic to it that made it much more accessible than the ponderous derivation by Fourier that I learned in math class. And so I learned that the magic of pure understanding withers under machine iteration.

When I was learning FFT, I just kind of bashed through the algebra and accepted it with roots of unity. But seeing that points on a sine wave have similar evaluation makes so much more sense than the roots of unity tbh

You have absolutely outdone yourself with the animations involving sine waves and the Fourier transform itself. I feel like these things often go unappreciated, I can only imagine how much work went into this.

I learn or understand things better when it's part of a narrative. Thanks for the video, I think I know how the Fast Fourier Transform works now.

Thank you for not shying away from such a dense topic. I do wish you covered a more relevant use case a but more. I think your style is perfect for teaching people about Wi-Fi and how we have found so many ways to increase the bandwidth

As a senior electrical engineering student in college, i can say this is absolutely the most accessible and well explained video of the Fourier series/transform i've ever seen.

I wish I'd had it during my EE college days too

Long before Excel and MathCad we EE students did homework assignments using only a pencil and hand-held calculator to learn about DFT and FFTs. This video woke up a few thousand neurons which still had a whisper of that info after 50 years.

@Darrell Hambley I have no idea how you guys did it. Whenever I’m doing homework I’m glued to my computer looking up how to do it and how it works. I seriously commend your generation of engineers

@Darrell Hambley I graduated in 2019 and never used Excel or even know what MathCad is lol we had to do everything by hand with hand-held calculators. The only time we were allowed to use anything else was for EM lab

Spot on

I used the FFT in graduate work in 1974. We collected 1000data points from an average evoked potential from the spinal cord of a cat. The work was aimed at detecting injuries at different levels of the spinal cord using statistical (stepwise discriminant) analysis. We didn't have enough cats. If I remember correctly, the transform I used took less than a minute for 1000 data points from a 300 msec signal.

In '93 I was part of a ultra-highend synthesizer project which was using fourier transforms and inverse fourier transforms as core of its sound generation. The latter was so important that we designed custom chips to do the hard lifting because even a highend RISC processor back then was not fst enough to do the job. In the end the project was to expensive for our small company so it was cancelled but the signal processing know-how continues to live on as core of today's successor of that company. What I didn't know this all started with Gauss. Wow.

Best explanation of FFT, I have ever heard. I have used FFT's through out my 45 year career and I finally get it, thanks.

The FFT is just an efficient DFT. Cooley and Tukey wrote the seminal paper observing the redundant calculations in the basic DFT algorithm and came up with an algorithm to remove the redundant calculations. Nice explanation BTW. Next you should discuss the effects of data windows like the 4-term Blackman-Harris window.

The FFT made convolution (the foundation of every computer vision model) much faster and easier. Thanks to it we do a convolution by transforming the signals ( images) on a Fourier base, performing a matrix product, and use the reverse FFT (IFFT) to get a convoluted signal.

Man.. the education system was so bad for me when I did my Electrical & Electronics degree at uni 25 years ago. I struggled to grasp the purpose and concept of Fourier Transform and subsequent lectures on DFT. Now looking at your visuals and explanation with historical relevance, it looks simply amazing and makes so much sense. Awesome work Derek.

I had the same feeling when I studied electrical engineering. The lecturers managed to suck all enthusiasm out of the subject.

Same. However it wasn’t that well understood 25 years ago and was only kept in the hands of the top level researchers. More people have now digested it to make it easier to understand. Also we have the internet to get access to the people who can explain things properly and unambiguously.

@Bob our lecturer gave this topic of FFT for self-study when there were no online videos and free course ware.

Engineering sometimes focuses a bit too much on application. I know it doesn’t make sense to teach every math proof but a bit of conceptual understanding of underlying algorithms would be hugely beneficial

Bahahahaha that's why I switched to CS where we just talk about probability and number theory, but don't do actual math

Fantastic! I’ve dabbled in signal processing via digital audio and synthesis since the first Sound Tools (now ProTools) application in about 1990-1991 but I’ve never looked into the details of an FFT. They just work, you ask the computer to do one, and you get the frequency domain, phase and amplitude information that can be manipulated and resynthesised. So I know about all this from a practical POV, not the algorithmic and theoretical. A lot of stuff just dropped into place watching this. I’m off to watch Mr BlueBrown after this :D.

Oh, for some reason I’d always assumed an FFT was a quick and dirty version of a full discrete FT, with some approximations and extrapolations but it seems not. Not in the digital world anyway.

Great video, I believe speech recognition is all made possible with FFTs too. What software did you use to do your sin and cos animation?

For some years I've been struggling, trying to catch a glimpse of how to learn the fourier transform. Whenever I find information, it's a black box peek into how it's used and why it's useful and general glance, I never get to the point where I could confidently say I know what to do and could apply it on anything. I just know it's practical because it changes the functions to trigonometric wave functions. Or not necessarily to even those, but it's like magic. Some people have secret knowledge how to actually use it to calculate something.