The Fourier Transform
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- Опубликовано: 16 мар 2020
- This video will discuss the Fourier Transform, which is one of the most important coordinate transformations in all of science and engineering.
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 
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I'm honestly very impressed by the concise nature of your videos and I'm glad that you make these videos accessible on RUclips to a large audience. With a bit of will power and a desire to learn anybody with a sufficient mathematical background can have a better grasp on the extremely powerful tools of harmonic analysis. Great work. I honestly wish my instructors at university had been as understandable.
Thank you!!
@@Eigensteve Seriously. I too wish my instructors had put in half as much effort in being concise and coherent as you have. I've been watching your videos in order for about a week now.
I never thought of actually deriving the Fourier transform this way. This is amazing.
Glad you liked it!!
I don't usually comment on videos but I just want to let you know how amazing this series has been. Thank you!
Finally the chance to commentate under your video: Thanks for the awesome content!
I really liked your lectures! Very clear and easy to understand. Thanks!
this is extremely important also in quantum mechanics! the difference between “free particle” (such as photons) with continuous spectrum and quantized particle with discrete spectrum within potential well is connected to Fourier Transform and Fourier Series.. super important
really good video! streight to the point, quick and easy to understand!
I've finished my university degree 31 years ago. I wish I had this quality of explanation available during my education! Amazing! Thank you!!! His book is excellent as well! Consider it a good investment in your education.
I have watched your videos, they are really good explained,I haven't seen such a good explanation of this topic so far.
This is extremely helpful, thank you!
It's very nice to see Steve makes very technical videos that do really well. Gives me some hope for my vids :)
This an amazing series. Thank you so much ❤️❤️
Glad you enjoy it!
Thank you for this great explanation! Greetings from germany
Hallo! Which city?
Amazing video, thank you!
Very precise and nice explanation ...
I really enjoyed learning from you....thanks sir
This playlist is really amazing.
Thanks!
This is one of the best series of lectures! Question professor:
1) what's the meaning of delta omega? Pi/infinity = 0 here. especially at 7:46 based on what it came with dw integral range from negative infinity to infinity? from omega definition I don't see this integral domain.
2) Shouldn't it be delta K when you converge your first summation equation to integral?
Hope you can help illustrate! Thank you very much in advance!
Amazing, thank you for this lecture!
You're welcome!
Wow, I've heard a lot of explanations and derivations for FT but this by far takes the cake, and I'm not someone who's into pointless compliments but this is really worth it, thank you and please keep up the great work👍
Great video. Thanks
Awesome explanation
Really amazing explanation wow.
I'm amazed that you pronounced ξ as it's supposed to be pronounced. I'm Greek and have studied and lived in both the UK and Australia for long enough to have heard the Greek alphabet being massacred in all sorts of ways hahahah. Props to ya and your amazing lectures :)
"Welcome back" gets me everytime :D
In the Complex Fourier Series video ψₖ was defined as eᶦᵏˣ, I understand that π/L was introduced in for frequency but why did the exponent become negative. It would become more clear to me if someone could explain the general formula for the inner product with period L rather than 2π ( I don't believe he produced this formula in the Complex Fourier Series video).
beautiful ! beautiful!
thanks for ur explaination
Thank you, this was a great video!
Also, just realized you've been writing backwards... That's impressive.
The summation that you turned into an integral was k=-inf to inf ....but you wrote the integral wrt d(dummy variable) and not wrt dk. Could you clarify?
thank you!!
What is the technology behind this transparent mirrored board? What is his using exactly? Thx, excellent video btw
Very good
Do magnitude and phase for particular frequency in continuous frequency spectrum represent exact magnitude and phase of sinusoid with this frequency?
Hi Steve, I've worked with the Fourier transform for years, but I just realized that I don't understand something. Please, consider your triangular function f(x). Suppose I multiply your f(x) by a factor B. Suppose I want a transform that in independent of B. In other words, I want a transform that is independent of a scaling factor. Does that make sense? I have a physics research situation where I believe it should make sense. How would you define a transform that is independent of a scaling factor? Thanks.
Thanks from India
Hello, thank you for the video. Can anyone tell where can i find a more specific explanation of the step using the Riemann sum? I do not understand it. Thanks.
Steve, wish you and your family happy holiday! If you have time, want to bother you one question, but no rush. 7:19 - 8:37 from summation to integral. I am struggling to understand 1) why delta_omega = Pi/L, 2) why the summation is over K (the frequency), but integral is over omega? I thought the idea is to transfer to continuous frequency basis K, shouldn't it be something like delta_omega = delta_K * Pi / L ? How did you come up with omega which kind of wrap up K inside omega. Thank you so much!
Thanks so much
I must agree with some other comments... Gibbs phenomen does not disappear when you go to infinite number of terms but if I remember correctly tends to a constant value.
Thank you for the video but i have this question is pi/0 considered infinity or unidentified?
Sir, what is the difference between π/L and κ in e^ikπx/L?
Aren't they both supposed to represent angular frequency?
This video is very insightful, however, I don't understand how the Fourier transform can represent continuous Fourier coefficients if the term Delta Omega / 2π is omitted or used by the inverse transform? Why is this possible?
Is a Fourier transform of f(x) = x worth evaluating? It shouldn't give you any Frequency, should it?
thank you
Omg ! We have the same name ! I am so happy
What if I take the fourier transform of a periodic signal? Will it be the same as taking the fourier series of this periodic signal?
no, the Fourier transform of a period function will diverge. It can, however, be defined if you know about the delta 'function'.
I am very confused in your derivation as to why you replace all of the k∆w in the summation with w in the integral. If anyone could help me understand this I would very much appreciate it.
He at the beginning took w_k = k(pi)/L, then he wrote it as kΔw. Where Δw= PI/L. In the limit of Δw -> 0, inside the integral, he easily replaced w_k from the first equation. You can also write it as w only in the continuous limit.
From a physical standpoint since omega and time are conjugates, perhaps it would have been better to use “t” rather than “x”.
at timestep 3:03 shuouldn't Ck formula have e^i in positive sign?
now I got it ha... thank you !
Shouldn't the second equation on the board be c_k = 1/2L , rather than 1/2pi as we're on the domain [-L,L] rather than [-pi,pi] ?
yes it had to be so!
Indeed!
Ok, it helps a little bit, which is a lot, but every time you say ok I have so very many questions. So many.
I miss from most lectures like this a more precise definition of what you mean in this context by taking the limit as the period tends to infinity. If you let 'N' be the absolute value of the upper and lower bound of sum index 'n' then what you get to see is both N and the period tend to infinity. It is very hard to think about how this all converges because of the 2 variables. If I wanted, I could always choose a larger period and a larger 'N' so that the frequency domain passed on to the Fourier Transform Function will not increase. You need to imagine the tendency of the period and 'N' so that you always get a larger AND denser frequency domain so that your sum can be seen as a Riemann sum and it indeed converges to the limit defined by the final inverse integral from minus infinity to infinity. If you take in this very sense "take limit as period tends to infinity" then yes you must obtain the Fourier Transform but it is not straightforward.
When can you interchange summation and integral in Fourier series?
The Riemann integral is simply defined by the limit of a Riemann Sum as the delta variable approaches zero. The instructor just recognized the form and replaced it with the integral notation. Not sure if that answers your question.
@@kylegreen5600 I don't understand. When can you integrate a Fourier series term by term?
@@NoNTr1v1aL Could you respond with a rough time in the video where you're not following the steps and I'll try to help.
@@NoNTr1v1aL Your question is different from what Dr. Brunton did: like Kyle said, he's just recognizing a Riemann sum that already exists, he's not integrating the Fourier series. It's just a definition: as L -> inf, the series turns into an integral.
Your question is different and gets into deep waters. The short answer is: if f is "nice enough" (e.g. C^inf smooth.) The long answer: this is the entire subject of Fourier and harmonic analysis. In general, integrating and differentiating series term by term depends on the analytic properties of the f being expanded, the types of convergence (uniform, pointwise, etc.) and it only gets more subtle with Fourier. The Gibbs phenomenon is already an example of this: a jump discontinuity in f screws up uniform convergence of the fourier series, and if you remember, Dr. Brunton mentioned that the "top hat" function was related to the derivative of the "triangle hat" function (which was continuous, but not continuously differentiable), so already you can see some of these subtleties creeping in. Check out Stein and Sharkachi's books, Tao's books, etc.
when the integrable function is uniformly continuous, then we can do like this interchanging.
How does he write k times delta omega as omega?
Same question here. The part of taking the Riemann integral makes sense on the expression as a whole, but I'm confused as to what justifies that substitution (hiding the fact that the delta omega has shrunk to zero). Probably has to do with the fact that the k is infinetly summed (and then transformed to Riemann integral).
... is that backwards in your perspective
9:10
:( I did not get it. Any background I need to understand this topic? this guy seems to explain clearly but not clear for me
Sorry to hear that, but don’t despair! A little more background in linear algebra and vectors would likely help. You could check out my courses to see where this fits in: faculty.washington.edu/sbrunton/me564/ and faculty.washington.edu/sbrunton/me565/
@@Eigensteve Thanks man!
This is legit.
Deviation
Круто
my brain is melting
I want to focus on the video, but all I'm thinking about is how he is writing on the board backwards with perfection...
bro wtf its just mirrored camera, of course he's writing in the right direction
@@alegian7934 jokes
@@Thespookygoat oof I wooshed rather hard there... sry this is math after all
The first five seconds of this video made me realize that CBD actually makes u high
arifin manchestera attığı golü arıyodum buraya nasıl geldim amk
23k people view this but only 500 likes. This is why we have a pandemic.
If you need brain surgery, let's hope your guy is as good as Dr. Brunton. It could get a little hairy, but he will be able to pull you through the complex stuff. Hmmm, I think I may have constructed an unintended pun here.......
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This is some first-rate BS.
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