The Discrete Fourier Transform (DFT) 

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

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

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

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

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.

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.

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!

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)

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

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.

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

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!

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}.

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

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

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

Thanks Steve for contributing on humanity. cheers!

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!

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

Finally, a real educator...

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

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

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

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

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 :)

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

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

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

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

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!

Absolutely fantastic video, sir! Thank you very much!

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

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

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.

Amazing video. Very clear and well presented

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!

Big appreciate Prof. Steven Brunton.

Great description thanks, FFT was a nice bonus.

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

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!

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

great way to explain. huge respect

Thanks, simple and easy to apply.

Great video!!! It has been so helpful :)

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

Thank you very much!! This video is amazing!!

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

These videos are amazing! Thanks much!

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....

made a lot of things clear, thank you

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).

You are a good human.

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

Great professor! Thank you!

Thank you so much for this!

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!

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.

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

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!

great lecture

Thank you so much for this...

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.

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?

Thanks a lot sir.

Excellent!

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)

so brilliant

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?

Thank you so much

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.

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?

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

Thanks for the amazing video... however kudos for being able to write mirrored!!

you my man are a goddamn national treasure

I'm just here because I wanted to make an audio visualizer as an add-on for my gui exercise in c++. Guess I underestimated it.

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.

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

More impressive than the math is that Steve is writing mirror-imaged. Leonardo DaVinci would be proud.

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?

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.

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

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?

thank you😇

Thanks sensei

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

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 !

the matrix for the Fourier coefficients and the f function samples should also go up to n-1 and . If someone was confused about it.

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.

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 ?

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?

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 ?

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)

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

New subscriber

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...

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

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.

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!

great!

I love you Steve