The Discrete Fourier Transform (DFT)

[@Mutual_Information]

The amount of free, useful, precise information coming from this channel is remarkable and something to be grateful for. It legitimizes KZbin education.

[@gabrielnicolosi8706]

It is not "free". Most likely, Professor Brunton has these lectures as one of the deliverables of many of his NSF grants. Thus, this is paid by the US taxpayer. :)

[@greensasque]

Can't say this for many videos, but my mind is now blown. 🤯 Finally after years the DFT makes sense.

[@Eigensteve]

Awesome!

[@ahmedgaafar5369]

Steve, you really are the best professor on the planet period ....thank you so much for all these incredible high quality lectures.

[@gmoney6829]

I’m glad I have this guy as my uncle

[@OrdnanceTV]

I have absolutely no clue what you're talking about but I love listening. Even without understanding it's very evident you're a talented and efficient teacher.

[@WahranRai]

You must also replace indice n by n-1 if you start with f0....f_n-1 etc...

[@funkflip]

The video is very nice. Thank you! Just a small remark: The indexing of f and f hat in the matrix vector multiplication is wrong. Should count up to f_{n-1} not f_{n}.

[@Eigensteve]

Good catch, you are definitely right!

[@VarunAgrawal11]

@@Eigensteve Or conversely, shouldn't you simply make the summation from 0 to n? Since for f_0 to f_n you now have n+1 sample points, and x is an n+1 size vector. By making your summation to j=0:n, it is summing over n+1 points which is the standard notation used in approximation theory.

[@eric_welch]

@@iiillililililillil8759 you can change summation range if you pull out the j = 0 term and add it in front of your sum :) similar to how it is done in series solutions for certain differential equations

[@wtfftwfml98]

I have to give you credit for giving the absolute best educational videos I have ever seen. The screen is awesome, the audio is great, you explain thoroughly and clearly, you write clearly, your voice is not annoying and everything makes sense. Thank you mr sir Steve.

[@srikasip]

Oh my goodness! Stumbled onto video 1 in this playlist this evening. and I can't stop. Steve, you're amazing. I actually finally feel like I understand what a fourier series is and why it works. can't wait to get to the end. This is easily the best set of lecture on this topic i've ever experienced. HUGE thanks!

[@srikasip]

Also, are you writing on a window? ......backwards?!

[@LydellAaron]

I like your insight that this should actually be called the Discrete Fourier SERIES. Thank you for your way of relating the matrix to the computation. Your perspective help me see how the matrix is related to the tensor and quantum mechanics.

[@masoudsakha9331]

Thanks for great lecture. However, I think the last element of vectors must be F_n-1 instead of F_n.

[@erikgottlieb9362]

Mr. Brunton. Thank you for clear, concise, organized presentation of DFT. Appreciative of how much time and effort such a presentation / explanation takes to create and deliver. Appreciative of the format you use and precision in getting explanation correct. Explanation of terms and where terms originate has always been helpful in your presentations. Going through the whole DFT, FFT series again to refresh my thinking on the topics. Thanks again. (Erik Gottlieb)

[@javadvahedi6278]

Dear Steve I really enjoy your teaching format and also your wonderful explanation. Just one suggestion, It would be great if you could have at least one practical lecture at the end of each series of lectures, e.g for Fourier series transformation lecture designing one lecture which shows a real problem is great and enhance the level of understanding. Stay motivated and Many thanks for your consideration

[@Eigensteve]

Great suggestion. Let me think about how to do that.

[@gloiremumbere9262]

I always struggle in order to understand deeply what Fourier transform really is, but now after watching your video I'm very confident in what's really is .Thanks a lot

[@Eigensteve]

Glad to hear it :)

[@user-iw1dv3rw4t]

Thanks Steve for contributing on humanity. cheers!

[@anantchopra1663]

Excellent video! The video was conceptually very clear and to the point. You are an amazing teacher, Prof Brunton! I loved your control systems videos too!

[@pranav2pta]

Here it's mid night now, but you have opened my eyes !!! Lucky to find this lecture

[@joakiti]

This is by far the best explanation I’ve ever seen. Thank you Steve, I hope to find reason to buy your book soon.

[@zaramomadi5569]

When he said "thank you" in the end I wanted to take a huge mirror and send it right back at him

[@nitinshukla6751]

Your ability to explain something this abstract in such a simple manner is simply astounding. However i was more impressed by your mirror writing skills. hats off sir..very very good video.. Subscribing to you.

[@vitormateusmartini3946]

he does not write backwards... it's a lightboard

[@sashacurcic1719]

This is very concise and organized and easy to understand. Thank you for posting it.

[@MboeraKisaroKimambo]

It took me 5min and 55sec to discover that you're writing correctly, I was wondering why are you writing the inverse way! Thank you for the great presentation!

[@BurakAlanyaloglu]

Finally, a real educator...

[@nrdesign1991]

I *finally* understand it. Memorizing it for exams is not good enough for me, i want to *get* it. Now I do, and see all the great applications for it. Filtering out specific frequencies, isolating specific frequencies, or the same with a broad spectrum of frequencies will be extremely easy with it. Either just calculate a few values individually, or just take/throw away a chunk of the resulting vector. Great videos!

[@mariogutierrezdiaz3366]

Hi Professor Brunton, Just wanted to let you know I took your AMATH 301 course at UW in 2012. It really kicked my butt but learned so much. I still use the RK4 for work once in a while. You and Prof. Kutz were both outstanding. Wish you both well!

[@Eigensteve]

That is so nice to hear! Really glad it has been useful since then... that must have been my first class too!

[@soorkie]

Thank you. This video really helped me. Thank you for keeping this open and free for everyone.

[@duameer6832]

You made me feel that I can understand something too!! I’m so glad to understand this. Love and prayers!

[@miguelaugustovergara4185]

Please never stop uploading useful content like this, nice teaching method!

[@ZetaCarinae]

The last time I tried to give a similar lecture I messed up the indexing much more than this, it was a little comforting to see you do it too. It made me wonder if it was worth it to count from 0 always when teaching linear algebra (probably not).

[@Eigensteve]

Thanks for the feedback... yeah, I know that when I make mistakes in class, it actually resonates with some of the students. I hope some of that comes through here.

[@AKASHSOVIS]

Omg, when I first learned DFT in class I was so confused, but I watched your video and now everything makes sense. Thank you so much. Please continue to make videos!

[@joeylitalien1355]

Hey Steve, your videos are great. I love the format and the clarity of the exposition, keep up the good work.

[@Eigensteve]

Thanks!

[@julesclarke6140]

I agree, it's both clear and enjoyable, you sir are a life savior. Merci !

[@abhishekbhansali1377]

Can anybody else appreciate how elegantly he is able to write equations as mirror images 🙄

[@subratadutta7710]

Very lucent explaination. I love to watch his lecture, His book helped me a lot . Thank you Professor.

[@LL-ue3ek]

Thank you for the presentation with clarity and intuition. I have a question, @ 9:14 you mentioned something about the fundamental frequency wn. If we are given a piece of signal like you drew, how do we decide what frequencies to look for in that signal? and hence how do we decide what fundamental frequency we can set wn to be? In other words, how do we know if we should look for frequency content from 10 - 20 hz instead of 100-110hz?

[@JoelRosenfeld]

Heya! I really enjoy the pacing of your lectures. It's also nice for me to get a quick recap of some signal processing before assembling my own lectures. It is also helping me fill in the gaps of knowledge I have around data science, where my training is in Functional Analysis and Operator Theory. This past fall I dug through the literature for my Tomography class looking for a direct connection between the Fourier transform and the DFT. Mostly this is because in Tomography you talk so much about the Fourier transform proper, that abandoning it for what you called a Discrete Fourier series seemed unnatural. There is indeed a route from the Fourier transform to DFT, where you start by considering Fourier transforms over the Schwartz space, then Fourier transforms over Tempered Distributions. Once you have the Poisson summation formula you can take the Fourier transform of a periodic function, which you view as a regular tempered distribution, and split it up over intervals using its period. The Fourier integral would never converge in the truest sense against a periodic function, but it does converge as a series of tempered distributions in the topology of the dual of the Schwartz space. Hunter and Nachtergaele's textbook Applied Analysis (not to be confused with Lanczos' text of the same name) has much of the required details. They give their book away for free online: www.math.ucdavis.edu/~hunter/book/pdfbook.html

[@doneel.5338]

Thank you for the explanation focused on the implementation of DFT. Fourier series makes much more sense to me in general as well! Now I will attempt to code it :)

[@jsm640]

Thank you,sir. I really got some new knowledge from your videos,which I never know when I studied this theory in my class. Maybe that's because my terchers just want us to understand the theory without applications,but in yout videos I just found a new world of how to use the mothods of math to solve problems in the real world. Thank you again!

[@olayomateoreynaud9956]

At 0:30 you already solved the question that brought me here. Thank you!

[@johnnyhsieh0208]

Big appreciate Prof. Steven Brunton.

[@ozzyfromspace]

One of my friends posed me an interpolation problem and I instinctively decided to try a DFT. I used some for loops and got the job done, but I never thought that you could build a matrix using fundamental frequencies. That's clean. Then when it came time to using the algorithm, I realized that it was super slow! Granted, it was an interpolation on some 2D data, but still. My laptop couldn't handle an interpolation over fairly small grids (at 35x35, I was waiting seconds for an answer), which blew my mind. But on further inspection, a for loop (or matrix multiplication) is like O(n^2) but likely all the way to O(n^3) after naive implementation details, so it makes sense. What I'm trying to say is, I can see why you think so highly of the FFT, and I'm super excited to learn how it works, and maybe even implement it myself 🙌🏽. You rock, prof!

[@Kay-ip9fy]

This is one of thewonderful lessons I've got, thank you so much for your enthusiastic!

[@ziggly0018]

Some videos ago I was concerned at the implications of this being called the DFT, as it not repeating would be problematic for me, and from my understanding of others' implementations, it is supposed to repeat, so I was happy to hear you clear up the easy to make mistake that this was an actual transform and not a series. Things make sense again now. It's still weird that its mislabeled though.

[@effulgent_imr]

9:15 why is the fundamental freq an exponential function and also why it has a negative sign

[@michaelpadilla141]

A nice way to think about the mathematical sums, which Prof. Brunton doesn't explicitly mention, is that each of the n+1 rows in the matrix as a vector that functions as a basis function, together which span the space of all n+1 element vectors. Hence all you're doing is taking the inner (dot) product of the original signal with each of those n+1 basis functions (the vectors), i.e. projecting the orignal signal against each of those basic functions to see how much of it is along each of those (vector space) directions.

[@christiaanleroux4016]

As far as I understand, when we take the inverse discrete fourier transform, we end up with the function values at x_0, x_1, x_2, ..., x_n, but how would you determine what the values of x_0, x_1, ... ,x_n are? I need to know this for my masters thesis please help me if you can.

[@manuelaayo4199]

Thank you so much for this series of videos. Just a small suggestion; to be consistent, it seems that the vector should have points from f_0 to f_(n-1)

[@Martin-lv1xw]

Damn STEVE...YOU SAVED MY DAY...THANK YOU SO MUCH FOR SUCH A COOL PRESENTATION.

[@kele1969]

at min 11:56 when you corrected the F0 instead of F1, shouldn't you have corrected also Fn-1 instead of keeping Fn as last value?

[@euyin77]

I think the summation should go from 0 to n because you have n + 1 rows in the pink column vector and n columns in the yellow matrix.

[@recomoto]

Or there should have been n-1 measurements

[@muhammadsohaib681]

Dear Professor Thank You so much for your nice explanation!!! 💓

[@Foxie-1]

3:44 - It's a really interesting idea to perform the car diagnosis like this! But what stage goes after the FFT one, is it a neural network or something else?

[@maksymkloka7819]

Great video. One of the better ones. I wish you explained the exact meaning of the coefficient in the exponent though ... e.g. I never really understood the relationship between sample frequency and number of data points (N). Seems like they will always be the same.

[@AG-cx1ug]

At 14:55 shouldn't the last value be wn ^ (n(n-1)) instead of wn ^ ((n-1)^2) Since the value is at the fnth value row wise and jnth value coloumn wise?

[@AG-cx1ug]

At 5:56 if its only going till fn (the coefficients) and thus the number of weighted signals, how is it an infinite sum of sinusoids? I'm a bit confused

[@sealedwings6788]

Does Mr. Brunton have a more conceptual video on why that fundamental frequency is defined, why we sample it with harmonics proportional to it etc.? Thanks

[@alireza98325]

You are a good human.

[@AG-cx1ug]

13:06 the number of 1s for the first row of the matrix will be j ones right? the same number as the number of data points in the signal (or n for that matter)

[@alexeyl22]

Awesome! I’m curious if it is too much to expand matrix form for a 2D function, i.e. 3D matrix.

[@Eigensteve]

This is coming up soon when we look at the DFT/FFT for 2D images.

[@KurohiNeko]

Amazing explanation, absolutely loved the see through board. So cool.

[@huangwei9664]

Very useful lecture. Thank you so much, Steve! One question by the way, why the number of f hat equals the number of f ? I can't really understand the point here. In my opinion, the number of calculated Fourier coefficients can be different from the one of sampling points.

[@garekbushnell3454]

Sounds like a good question to me. Maybe some of the values are so small that they can be neglected? I'd be interested for him, or someone else who knows this math, to talk about it here in the comments.

[@ehabnasr6925]

What would be the 2-d version of the DFT system? will the vectors be matrices and the DFT matrix be a 3d tensor?

[@thatoyaonebogopa9483]

Thanks, simple and easy to apply.

[@mz1rek]

At 10:49 corrected the matrix size to be n but then the vector size became n+1; needs another correction but I'm still watching! Edit: I saw the same catch in the comments below, but I think the solutions given weren't the best: My solution is as follows: n should be kept the same as it is the number of samples, also the summation should go until n-1 to give n points and nxn matrix size, but the summation formula should contain f_{j+1} keeping everything else the same. This way you don't even need the x_{0} data point. Still liked the video a lot...

[@FFLounge]

one thing i don't really understand is why there is a "j" in the exponential e^{2\pi1k/n}. Aren't e^{2\pi1k/n} sort of like the basis vectors we are projecting onto? Why do we need to raise each of those to the j's?

[@YYchen713]

I think I'm just going to watch all your videos for my machine learning course this semester instead of my professor's lecture which was so painful and frustrating....

[@oroscogold]

Hey great video and super clear explanation! I have a question regarding the indexing. Since we are indexing from 0 shouldn't the data and Fourier coefficient vectors index to "n-1" instead of "n"? Otherwise we would have "n+1" entries to the data vector. Understanding that it's just indexing, however, the dimension of the matrix and vector wouldn't match for the matrix multiplication. I think as it stands it's a "n X n" matrix and a "n+1 X 1" vector.

[@SreenikethanI]

Absolutely fantastic video, sir! Thank you very much!

[@kn58657]

These videos are d**n good. Excellent presentation, great production quality, and very pleasant to watch. Thank you!

[@Eigensteve]

Awesome, thanks!

[@masoudsakha9331]

If I am not wrong we collect the sample of data from x(t) in time domain so the elements of the second vector (red one) are not the signal frequencies and just the amplitude of our signal in time t?

[@JamesB-yh2xx]

Amazing video. Very clear and well presented

[@Tyokok]

Hi Steve, at 13:07, if your increase your sample data to 2n, then your DFT matrix first row will be 2n of 1s, and f0_hat will be doubled, is that right? Thank you!

[@miklosbence3852]

Hi, great video. Question: you say you multiply the vector with the matrix, but to make dimesions match, shouldn't you multiply the matrix with the vector ?

[@UmutKaradabann]

Hello, I did not understand the sizes of the matrices. I think the bottom element should've been fn-1 on the first and last vector. Can you please explain why it goes to fn?

[@LydellAaron]

How would an efficient DFT look, if I have a series of n-coefficients λ0, λ1, λ2, λ3, ..., λn which are prime numbers (2, 3, 5, 7, ..., P(n)) times a factor (f0, f1, f2, f3, ..., fn). And each factor is a positive integer, including zero?

[@garekbushnell3454]

This is excellent, thank you very much. A question - does it matter if the spacing between your independent variable samples isn't even/periodic? If it does, how do you approach that scenario?

[@sir_charlie]

you my man are a goddamn national treasure

[@Tyokok]

Hi Steve, do you have a lecture to the connection between fourier series and DFT? their form seem so alike. do they actually connect each other? interpretation wise. Many Thanks!

[@HighlyShifty]

They do! The important thing to notice is the continuous FT is described as an integral (an infinite sum) whereas the DFT is defined as a finite sum. Otherwise they're almost identical Would recommend 3blue1brown's video on this

[@Tyokok]

@@HighlyShifty Thank you for your reply!

[@nwsteg2610]

Note that the samples f0,f1,f2,...,fn are equally spaced in x.

[@mikefredd3390]

I got some insights. Thank you. The FFT next.

[@mehdiheshmati1258]

Are the vector dimensions correct, shouldn't the coefficients be indexed from 0 to n-1?

[@pcooper-chi]

Thank you Steve! I am still not 100% on how we get from the Fourier series coefficients to the DFT coefficients (f-hat_k). If someone could explain that or share a relevant resource, I would greatly appreciate it.

[@ryannoe86]

Insightful… also, how in the world did you write backwards on that glass and make it look so good??

[@CigdemO279]

i thought maybe its mirrored

[@MinhVu-fo6hd]

Professor, I have a question. Since I often notice that a lot of fhat are zeros, can we use a different number of basis (preferably less) than n?

[@bhargav7476]

hey, what are prerequisites for your book 'Data-Driven Science and Engineering'?

[@rhysparker6998]

Great description thanks, FFT was a nice bonus.

[@resu2381]

Great video! I have one question. Why do we have multiple images of our signal in time domain after performing DFT?

[@Eigensteve]

I'm not quite sure I understand your question. If you are asking why the DFT/FFT has multiple "mirror" copies, this is because the DFT/FFT is complex-valued, and so there is redundancy in going from "n" real valued data points to "n" complex valued Fourier coefficients.

[@resu2381]

@@Eigensteve So that is why after DFT our signal is periodic? Or it is because we have discret spectrum.

[@Eigensteve]

@@resu2381 Yeah, the DFT is assuming we have periodic data, so you can't build a DFT model that isn't periodic.

[@resu2381]

@@Eigensteve Thank you!

[@Saens406]

I dont understand how you can have information about the presence of a certain frequence. How come there are discrete frequence?

[@alt-f4666]

In DFT, you can tell there's a linear system of equations (whose dimensions are n*inf) that's being solved through inner products, by eliminating all terms except 1 on each equation, since the complex basis vectors are orthogonal to each other. Thats pretty straightforward and intuitive. However, when f is continuous, Fourier treats it the exact same way, which seems wrong, since the e^(iωx) and e^(i(ω+dω)x) vectors arent orthogonal to each other anymore, so even if we use inner product, there will still exist some non-zero 'remainders' on each equation which we cant get rid of. Also, any F.T. of a function f in the [-inf,+inf] domain is problematic, since the inner product of any pair of 2 basis vectors diverges. Do we assume then, that we extend our domain to [-inf,+inf] in such a way that the I.P. remains 0? Unfortunately, noone explains those.

[@oliviajulia7913]

Hello ! Thanks for your video. I had a question : So if you start with datas from a periodic analogous signal x(t) of period T, frequency w and you want to discretize it with sampling frequency f_s. I know you use DFT but how to you link the frequencies of your discrete and analogue signals ? Is the frequency w_n you're showing here the frequency of the continuous signal ? Thank you !

[@Eigensteve]

Good question! There are deep connections between the discrete and continuous Fourier transform, but you can derive the discrete from continuous and vice versa (taking the limit of infinitesimal data spacing).

[@eju1316]

Always leaning a lot from your lecture! Appreciate it, sir.

[@nami1540]

When i try to discretize f_hat from the continuous Fourier transform, I can't figure out how dx disappears. Shouldn't some delta x be part of the f_hat function?

[@svenjaherb6001]

wow, that was incredibly well explained, thank you so much!

[@p.z.8355]

so how do I do a complex matrix multiplication on the computer f.e using c++ ? just store sin & cos for every entry or is there a better way ?

[@mbisavunma662]

Dear Prof. Steve. I think there are n+1 data points (starting from "0" to "n"), but you have calculated the frequencies for (f1,f2, f3, .., fn) total "n" points. I think that one point is missing? Is something wrong?

[@ephimp3189]

How is something like this recorded? is he writing on transparent glass or mirror? how is the background removed?

[@ishtiakhasan8397]

great way to explain. huge respect

[@harsh_hybrid_thenx]

One thing i want to point out i suspect the DFT matrix is a symmetric one ..... Is it ?

[@Eigensteve]

Yes

[@BloodHuntress99]

COME ON DUDE LETSGO LETS MAKE ME SMART!!!! i have an exam in the morning it's currently 2 AM and I'm cramminggggggggggg

[@BloodHuntress99]

on a side note... how did you write backwards? or was the video flipped?

[@BloodHuntress99]

or did you actually write backwards.....?

[@MohamedMostafa-gf7rc]

Why does we limit the frequencies that the signal consists of to only from zero to k/n ,shouldn't we measure all frequencies to infinity

[@shlimon7667]

are you drawing everything mirrored? That's impressive if so

[@harsendevsisodia22]

How did you write it??? I mean it seems you are standing behind a clean glass, that means you must have to write everything from right to left,sort of a mirror image of a normal writing............that's so cool, I really wanna know if that's how you did it??? (Also yeah I'm supposed to concentrating on DFT instead of the mirror image writing, but that's me,I can't help it...)

[@JohnVKaravitis]

It's called a "lightboard." They are writing normally on glass, and recording the person writing. You have a choice: Capture the work in a mirror, and video the mirror, so everything looks normal writing, OR, record as they write through the glass, and then put the video into Microsoft Movie Maker and "FLIP HORIZONTAL." The glass is low-iron glass, so no reflections, there are LEDs at the top and the bottom of the glass. The light gets trapped in the glass, and, as they write on the glass, the marker ink makes a path whereby the light can escape. Also, black backdrops behind the writer and the camera. Easy once you know the trick behind the magic.

[@harsendevsisodia22]

@@JohnVKaravitis OOOHHHH Thanks brother, I thought he must have trained his brain to write in reverse, which would have been pretty impressive, but this was cool too , thanks