What is a Discrete Fourier Transform? | Week 14 | MIT 18.S191 Fall 2020 | Grant Sanderson

[@siddhant_yadav]

He is the best mathematics teacher I have ever seen.

[@octaviaraizel9240]

*Lokpati sir cries in corner*

[@siddhant_yadav]

@@octaviaraizel9240 I myself Suggest him to watch 3 blue 1 brown

[@BlackHoleForge]

It was nice to put a face to the voice I've heard so many times.

[@anch95]

Was pretty unconvincing for me when I first saw his face. The brown π is what I'll associate this voice with.

[@mariasoubra7080]

My mind was like: where is the animated math we used to see out there?😂

[@lidiyapriyadarsinik2815]

He kinda reminds me of that MS Word paper clip. In a sweetly inspiring way

[@Titurel]

That’s just his avatar. He’s really a Pi figure.

[@5ty717]

He is a modest lil thing yeah? Lol

[@homomorphic]

This is literally the best tutorial on fourier transform I've ever seen.

[@venkataramayya4266]

I wish that this talk was available when I was doing research into the analysis of Cutting Force Signals in 1973-1976!!!

[@hoaxuan7074]

Weren't you all using the fast Walsh Hadamard tranform back in those days?

[@jakebruner2719]

was not expecting that major7 chord to sound so pretty

[@adityakhanna113]

They are niiice

[@skeletonrowdie1768]

the ninth always sounds so cute to me

[@jeane0253]

This man is an educational god! He easily beats my professors in signal and system modules in Uni in terms of explaining what the whole thing is about. Magnificent

[@homomorphic]

He definitely is amazingly gifted as an educator.

[@mahpat]

Magnificent indeed!

[@truthphilic7938]

don't dare to associate partners with the god. Our God is one and greater than everything you can imagine.

[@mustafamosavy4458]

he is a great online teacher you have to use his videos in every lecture.

[@rvmishra9881]

Man love these videos. School made me HATE physics but you made me fall in love with physics again.

[@johubify]

Indian schools can do that to you 😂

[@rvmishra9881]

@@johubify true enough

[@forloop7713]

@@johubify or any school

[@xzhao3888]

How can a real human being sounds so much like a π.

[@dqrksun]

XD

[@redox5269]

cause he's real, not necessarily rational

[@CosmoShinobi-h5w]

😂😂😂

[@basavasagarkpatil8973]

Being an electronics and communication student, I have to say your videos gave me an insight of communication systems and signal processing and helped me immensely

[@mahpat]

He actually explained fourier series in a way I understood it. Years of studying this in college and I had no idea what it is. Thank you so much.

[@norbertn752]

Sadly I have similar experience with my university. Teachers had tendency to explain things in the most complicated way so nobody cant easily understand, which force us to learn it from different sources. For example I was interested in Introduction into SQL, expecting I will learn how to operate some specific database engine, based on SQL. But it was all about relational algebra theory, where most of the time I had no idea what they are talking about. So it was just waste of the time.

[@DarthZackTheFirstI]

guess your professor also had no clue like his students. only memorized it from books XD

[@jcbritobr]

same here.

[@jay_sensz]

11:20 It might be important to note that you can't generally overlay multiple FFTs like that and have the frequencies align nicely. This only works if your input signals are all the same length (and sampling rate) because the frequency domain spacing is inversely proportional to the length of the time series.

[@a-mann]

Two things: 1. This is one of the best videos to give you a natural intuition of the DFT that I have ever come across. I wish I saw this back in university. Thank you so much. 2. The fact that you are using Figma to illustrate some equations and concepts is amazing.

[@giridharsd1272]

Just love the way you explain mathematics. ❤

[@AshishSingh-753]

His voice gives me relief

[@scorpio19771111]

Redundancy is good for slower learners like me. Thank you for making the extra effort.

[@brockobama257]

Grant, no one creates quality videos of complex mathematical concepts better than you. I can’t thank you enough.

[@chenmarkson7413]

19:55 this point is where I finally understand everything. Perfect explanation!!!!!!

[@alfredo1valenzuela]

Oh, my God. You can see in his face what exceptional parents this person had. It's beautiful.

[@gansogames4927]

Blown away by the quality of your videos. Also blown away that you pronounce them as wahv files🤯

[@lilapela]

until 1:46 I thought he was reading from a script lol. Grant is so well spoken and has an impressive way with words

[@alan2here]

3b1b grant? :)

[@plekkchand]

Why?? it's perfectly normal/average...

[@zackglenn2847]

Very nice video. One of the interesting things about Fourier analysis is that there are so many different ways of approaching it, so even after getting intimately familiar with this field I still enjoy new takes on these basic concepts.

[@peetiegonzalez1845]

I had questions on Reducible's channel about the link between FFT and frequency domain because I was confused about the relationship between his polynomial example and the frequency values. Having now watched your video I got my answer. So great to see new channels explaining difficult concepts clearly, and especially using manim.

[@miro.s]

26:19 It would be safer to count maximum difference than mean value, because we work with large datasets and mean value is hiding peak differences. Maxima norm is more handful for more precise comparing while working with big data. Anyway, to understand that from different perspective, maxima norm works like boundary operator ∂ or derivative revealing globally peak points, vertices or so called extremes and on the other hand, avereging is its inverse dual operation that is integrating = hiding irregularities.

[@ravitheja012345]

Man your "3Blue1Brown" channel is amazing

[@Yasharvl]

Grant Sanderson on Julia Programming channel, there's still hope for 21th century!

[@abdelmajidfathi1481]

This can be demonstrated by taking the FFT of an arbitrary signal, and then running the frequency spectrum through an Inverse FFT. This reconstructs the original time domain signal, except for the addition of round-off noise from the calculations. A single number characterizing this noise can be obtained by calculating the standard deviation of the difference between the two signals. For comparison, this same procedure can be repeated using a DFT calculated by correlation, and a corresponding inverse DFT.

[@myboigetflick372]

This will help a lot for my Fourier Series and PDE's final exam I have this week! Thank you!

[@gumpiball5911]

i will never not imagine grant humming at my sample when taking an NMR spectrum from now on

[@APaleDot]

Yooooo, I just watched that FFT video by Reducible not too long ago. It's so good.

[@akmzahidulislam2764]

What an educator you are! Long live Grant

[@ifnspifn]

Great video! I had a question though. I was watching the Reducible video on polynomial multiplication, and it struck me as somehow very odd that the DFT/FFT had anything to do with polynomials at all. After all, the FFT of a series of values gives us some description of the frequency domain of a function, but for polynomial multiplication, we’re running it on *coefficients* of something decidedly non-frequency related, and it’s returning the *values* of that function. Do you have a feel for the intuition of why it applies to polynomials at all? Or put another way, is there an intuitive relationship between polynomial coefficients and the coefficients of a fourier series?

[@Pietro-qz5tm]

Mathematically polynomials are just the sequences with value 0 from one index onwards; "finite sequences of coefficients" could also be a way to understand them. But the notation as polynomials (with "powers" of a symbol called "variable") is very useful since (pointwise) sum and (convolution) product are easily defined in term of summing coefficients of variables with the same exponent and distributive property. The evaluation morphisms are also pretty immediate: substitute the variable with the value and simplify.

[@Pietro-qz5tm]

So polynomials are just a very useful formalism to work with finite sequences, like signal samples.

[@javrancheng9917]

just want to raise a point that FFT is not just more efficient, but more accurate - a naive implementation performs more floating point operations, which means inaccuracy builds up more quickly.

[@zuhairmehdee]

So wait, fourier transforms are important for making convolutions fast and convolutions are important for making fourier transforms fast? Huh.

[@hetsmiecht1029]

That sounds... Convoluted

[@zuhairmehdee]

@@hetsmiecht1029 Ayy I laughed at that

[@lobais]

@Grant Sanderson, when you do a lecture like this, do you script it, or is some of it improvised?

[@poohshmoo9892]

Reducible ... yes, quite well explained , credit is due there

[@ZinuBatul]

Please add LaPlace transformation too. Thank you

[@ronakparikh]

This is so much better than my linear systems and signals teacher

[@hoaxuan7074]

You can view the FFT as a bunch of dot products and sometimes that's all you want out of it. If so you might want to stop it taking a spectrum by applying a fixed random pattern of sign flips to the input data. That is quite quick however for hardware implementation it stiil needs a lot of transistor hungry multiply operations. An option is to use the fast Walsh Hadamard transform where the cost per dot product is log2(n). For a dot product of just over a million terms the cost is 20 add subtracts🤗

[@moulin3818]

Wow this is the crossover I haven't seen it coming!

[@thetommantom]

I once saw a picture coded into a frequency described with an algebra equation and thought if data was coded like that information density and data transfer would be off the charts

[@MinecraftingMahem]

Correct me if I'm wrong but this seems to be the same thing you did for the counting of subsets of [1,2,...,2000] that add up to 5. I think you used the roots of unity and generating functions there to select the correct subset amounts. Seems roots of unity are used here to pull out info on the frequencies. Seems kind of similar technique.

[@bipeenj]

Thanks for doing these videos appreciated ! I would like to see video on Quantum fourier transform as well because you have a gift of explaining things nicely.

[@teinili]

Seeing him make those little mistakes while coding makes me very happy :D

[@alan2here]

Some of the content is not like the others. :) It's a secret that makes being subscribed soo worth it. :) Branch off to your own channel!

[@skeletonrowdie1768]

wow that autodocumentation is awesome

[@kenbeta9376]

Great man with nice soft heart

[@nafrost2787]

I wanted to ask. Regarding the examples you gave at around 17:00 when you explained how we can use the DFT to decompose the signal into waves. In those examples, the signal may look like a wave when you put all the values next to each other. But it isn't really a wave isn't it? If the signal was a wave, then shouldn't its magnitude be constant in all its points, and its phase change between different points, which is exactly the opposite of what happened in the examples?

[@beauxq]

When you play the sound for which the transform showed the fundamental and most prominent frequency around 200, I thought: Why do I hear something closer to 100 Hz? Then I thought about how you explained the DFT algorithm. It looks at divisions of the original length, kind of assuming that the original length is 1 unit of time. But the original sound files are not only 1 second long. They are closer to 2 seconds. That's why the DFT reports a frequency of 200 when the sound wave frequency is around 100 Hz. So if you want to translate the output of DFT to Hz, you have to consider the sample rate and length of the data.

[@beauxq]

That sound file is 1.8923541666666666666666666... seconds.

[@jcbritobr]

Thats incredible stuff. Amazing.

[@AiryDiscus]

You can write a DFT that is as fast as mkl FFT for N

[@saketsingh6804]

3Blue1Brown is one of the reason that made me interested in math and physics.

[@GodzillaGoesGaga]

Beautifully explained. Thanks.

[@gdclemo]

This is a very good explanation of Fourier transforms! Thank you. I have one complaint though, which is that the audio of the samples is much lower than your speech and I can hardly hear them. They really need to be louder than they are.

[@TheCloud553]

Amazing as always Mr. Grant :D

[@venkataramayya4266]

Can you come up with an overview of multi-spectral signal/data analysis similar to that can be done by the BMD Package!!!

[@ozgunozerk]

I still do not get why: decomposing waves (turning the time-domain base into frequency-domain base) is the SAME as computing the evaluation form from the coefficient form? Ok, the math and equations hold (of course), but what is the intuition behind those two are the same thing from the aspect of the Fourier Transform...

[@journeytotheinfinity440]

Eagerly waiting for your video..

[@Andrew90046zero]

A mere length of 90 thousand! Also I love this, helps me so I can code my own transform function

[@sanjaux]

28:39 is this a form of memoization? I'm super new to all of this but that's what the pattern seemed like to me at first glance!

[@pauljohnson6377]

Would it be possible to teach or learn Fourier series, convolution, and characteristic functions (of probability distributions), not in terms of signal/time domain analysis, but of just a pure probability and the corresponding density function? I bring this up, as early in my student career I shrugged off Fourier series and transforms as I wasn't interested in wave analysis, not realizing it's applicability far beyond signal processing but now seeing it comes up in heat transfer, particle diffusion, AND PROBABILITY THEORY ITSELF. I have a hard time not thinking of the Fourier transforms, NOT in terms of frequency analysis, but there is a clear base importance in the way moment generating functions are written because of the Fourier transform.

[@ACLindustrial]

Are you the guy of 3Blue1Brown? I love that videos. Excellent explanation!. Thanks so much.

[@ignaciocordova1325]

Thanks for the video! What is the mic you use?

[@Jaan2103]

very cool and nicely explained! What is the notebook or program you are using? I already thought about switching to the Julia language

[@Jaan2103]

Found it, it is Pluto.jl

[@Chrls5]

Broooo brooooo brooooo brooooooo, this sounds like the solution I was looking for to compress my FFT algorithm for my little microcontroller, Broooooo!!!!! Thank youu!!!!!

[@nathanjaroszynski6210]

Thank you it's an amazing lecture.

[@area51xi]

But why does the first data point not walk around the circle at all. This keeps bothering me.

[@nraynaud]

1) somebody discreetly snuck a windowing function in their samples before taking the DFT 2) I see native English speakers have the same issue as me (I'm French)to spell "lenght" or "lenhgt" or "len..." size! :)

[@LydellAaron]

Good example of superposition

[@ayuminor]

I tried the same thing in python but packing the values for zeta into an array only halved my computation times and it seemed to scale proportionally with more samples.

[@OtherTheDave]

3:05 “Wave”, according to its Wikipedia article. en.wikipedia.org/wiki/WAV

[@venkataramayya4266]

The only book that had any information in 1973-1976 was the book on "Random Data" by Bendat & Piersol!!!

[@gheffz]

Thank you. _By the way, nice easy listening voice!_

[@petruericstavarache9464]

But this doesn't answer the most important question: Why do we only use the frequencies 0, 1, ..., N - 1? How can we be sure that the original signal doesn't somehow contain the frequency 2.714? And we are wrongfully ommiting it?

[@GodzillaGoesGaga]

This is one of the issues with discretising frequencies. The way you get better resolution is to double up the signal (ie copy every element and insert it next to it's original) and then run the DFT again. You now have N*2 samples but you've added no new information but you have doubled the frequency resolution. At least this is how I recall it is done. You have to use integer multiples of the original number of elements. Obviously the DFT calculation time takes longer. In your example multiplying by 2714 would give you the resolution to pick out that particular signal. Maybe 27 would be adequate for the application ?? The discrete buckets are known as "bins".

[@jay_sensz]

@@GodzillaGoesGaga That's not correct. Instead you pad the signal with zeros at the end to "increase" the frequency resolution (really just interpolating the spectrum). But that obviously doesn't add any new information to the spectrum either.

[@GodzillaGoesGaga]

@@jay_sensz Thanks for clarifying. My memory is rusty on this stuff.

[@Dhanush-zj7mf]

can we get access to the source code....

[@BottleOfCoke]

You also play guitar? Leave something for the rest of us!

[@tehpostman7]

Did he choose the letter "s" for array of numbers on purpose?

[@solutionstate]

Great lesson! can you share the source code/notebook please?

[@tiago.engenheiro]

Is Julia worth learning? honest question

[@victorcalderon8496]

I didn't know Tom Misch was making math classes

[@Dhanush-zj7mf]

will the fourier transform work when the function is non periodic

[@webgpu]

difference between a discrete and non discrete is that the former doesn't scream.

[@namlehai2737]

waiting for FFT!

[@alidanish6303]

The explanation with phasor/vector on a unit circle somewhat convoluted and in my opinion not to the point. I understand that with complex plane and a unit circle you are trying to explain to superposition or more like a convolution/superposition with sines and cosines. It would have been much better if you could have shown original signal x(n) on complex plane with convolving phasor that when hits the correct frequency bin gives the max magnitude. I mean this is what you ll learn from any standard course book on DSP in summary. Your view seems more like generalizing a mathematical aspect may not get the point through to the some audience.

[@chhayakankariya2748]

How to get such auto- documentation on side??anyone know?? Btw Thank you 3b1b for great explanation you are awesome😎😎

[@vishalchovatiya1361]

Excellent, Thanks

[@zskater1234]

Where is he running this Julia code? Looks like jupyter, but I believe it isn't. Could someone tell me?

[@IulianVasileCioarca]

Check out Pluto.jl. It's a reactive notebook.

[@VincentLatzko]

Thank you Grant for your lectures and homework. It was inspiring to watch you and your calm reaction to hiccups teaches me every day ;D

[@RD2564]

Dog, this video really didn't get going until you started using the "Grant engine" at 12:57 to explain how all the terms in the sample combine to tranform into the frequency domain representation. That walk around the complex plain was very helpful for me and I handn't see DFT described like that before, very nice. This all means the first 13 minutes where a lot of introductory overhead and in some sense it makes sense to me to put that "example" stuff at the back but that is obviously just one man's opinion.

[@guyfromkerala3577]

What is the name of that type of frame he wears?

[@karanabraham7906]

It's so weird seeing his him talk after only listening to his voice for in other videos😂

[@jackmcgaughey4388]

What if there are changes in frequency over time? Like in human speech

[@IulianVasileCioarca]

You run short time fft on windows of 20ms. It's considered that over such a small interval the signal is stationary, meaningg spectral components are time invariant.

[@Sky-pg6xy]

Its Grant!!!

[@ioannisgkan8930]

This lecture was quite amazing understanding fft through visualisation Great job

[@reilbenedictobinque9650]

Your hair is gorgeous

[@aayushbajaj2260]

damn, this guy can code

[@mechwarreir2]

MAKE A VID ON UR GUITAR PLAYING PLS

[@michael.forkert]

What’s that good for?

[@damnguen1726]

love you man

[@missionprodigy]

Waits 46 seconds for a discrete fourier transform to finish... Ok I don't have time to go into the fast version now 😂 just kidding though you are a great teacher

[@AbhikalpUnakal]

What DAW is he using to record ?