The Discrete Fourier Transform (DFT)
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- Опубліковано 27 тра 2024
- This video introduces the Discrete Fourier Transform (DFT), which is how to numerically compute the Fourier Transform on a computer. The DFT, along with its fast FFT implementation, is one of the most important algorithms of all time.
Book Website: databookuw.com
Book PDF: databookuw.com/databook.pdf
These lectures follow Chapter 2 from:
"Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz
Amazon: www.amazon.com/Data-Driven-Sc...
Brunton Website: eigensteve.com
This video was produced at the University of Washington - Наука та технологія
The amount of free, useful, precise information coming from this channel is remarkable and something to be grateful for. It legitimizes UA-cam education.
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. :)
Can't say this for many videos, but my mind is now blown. 🤯
Finally after years the DFT makes sense.
Awesome!
Steve, you really are the best professor on the planet period ....thank you so much for all these incredible high quality lectures.
I’m glad I have this guy as my uncle
When he said "thank you" in the end I wanted to take a huge mirror and send it right back at him
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}.
Good catch, you are definitely right!
@@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.
@@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
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.
You must also replace indice n by n-1 if you start with f0....f_n-1 etc...
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.
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.
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)
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!
Also, are you writing on a window? ......backwards?!
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!
That's amazing how clear and precise you give the information! Thank you for the video :)
Thanks Steve for contributing on humanity. cheers!
Please never stop uploading useful content like this, nice teaching method!
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!
This is very concise and organized and easy to understand. Thank you for posting it.
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!
Can anybody else appreciate how elegantly he is able to write equations as mirror images 🙄
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 :)
Finally, a real educator...
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
Great suggestion. Let me think about how to do that.
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!
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
Here it's mid night now, but you have opened my eyes !!! Lucky to find this lecture
Hey Steve, your videos are great. I love the format and the clarity of the exposition, keep up the good work.
Thanks!
I agree, it's both clear and enjoyable, you sir are a life savior. Merci !
Thank you. This video really helped me. Thank you for keeping this open and free for everyone.
Always leaning a lot from your lecture! Appreciate it, sir.
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!
You made me feel that I can understand something too!! I’m so glad to understand this. Love and prayers!
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!
This is one of thewonderful lessons I've got, thank you so much for your enthusiastic!
Amazing explanation, absolutely loved the see through board. So cool.
Dear Professor
Thank You so much for your nice explanation!!! 💓
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.
he does not write backwards... it's a lightboard
Absolutely fantastic video, sir! Thank you very much!
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).
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.
Big appreciate Prof. Steven Brunton.
Great description thanks, FFT was a nice bonus.
Very lucent explaination. I love to watch his lecture, His book helped me a lot . Thank you Professor.
Amazing video. Very clear and well presented
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.
I got some insights. Thank you.
The FFT next.
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.
At 0:30 you already solved the question that brought me here. 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!
That is so nice to hear! Really glad it has been useful since then... that must have been my first class too!
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....
wow, that was incredibly well explained, thank you so much!
Damn STEVE...YOU SAVED MY DAY...THANK YOU SO MUCH FOR SUCH A COOL PRESENTATION.
great way to explain. huge respect
Thanks, simple and easy to apply.
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)
Great video!!! It has been so helpful :)
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?
You are a good human.
These videos are amazing! Thanks much!
Thank you very much!! This video is amazing!!
made a lot of things clear, thank you
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.
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?
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...
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?
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
Great professor! Thank you!
Thank you so much for this!
These videos are d**n good. Excellent presentation, great production quality, and very pleasant to watch. Thank you!
Awesome, thanks!
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 ?
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 !
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).
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.
What would be the 2-d version of the DFT system? will the vectors be matrices and the DFT matrix be a 3d tensor?
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.
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)
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?
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!
hey, what are prerequisites for your book 'Data-Driven Science and Engineering'?
Sir could you tell me, How to make you video ?? where i find this smart blackboard ??
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?
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.
great lecture
Thank you so much for this...
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.
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.
Or there should have been n-1 measurements
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 ?
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?
Thanks for the amazing video... however kudos for being able to write mirrored!!
Thanks a lot sir.
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?
you my man are a goddamn national treasure
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.
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!
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
@@HighlyShifty Thank you for your reply!
Awesome! I’m curious if it is too much to expand matrix form for a 2D function, i.e. 3D matrix.
This is coming up soon when we look at the DFT/FFT for 2D images.
Wait wait wait, are you writing all that backwards on a glass pane, so that we see it correctly written?
no, the video is just mirror reversed. (See his hair. It's mirror reversed)
@@samarendra109 I had to pause the video to look in the comments to see if he was writing backwards, It was driving me crazy, small obsessive compulsive attack XD
I've been thinking about how he did that for an hour but still can't get it.
He is writing on a piece of glass and he flips the video after. He is a lefty, which you can see in his early unflipped videos. His part is also the other way.
Its a tech invented by a prof from northwestern university, heard about it while doing a course from northwestern
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
Excellent!
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.
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.
COME ON DUDE LETSGO LETS MAKE ME SMART!!!! i have an exam in the morning it's currently 2 AM and I'm cramminggggggggggg
on a side note... how did you write backwards? or was the video flipped?
or did you actually write backwards.....?
How you are writing in reverse direction???