Fourier Series: Part 1

[@byynee]

Fourier series as inner products and projections onto orthogonal basis. I'm in awe! Can't thank you enough Prof. Steve Brunton!

[@andrewgibson7797]

Not sure yet if he talks about this in other videos, but this lets you understand things like Bessel's inequality and Parseval's identity as just the Pythagorean theorem. Stein and Shakarchi's "Fourier Analysis" gives this same presentation on page 78 with really nice diagrams. It really is the right way to think about all this.

[@liboyan7010]

Prof. Steve Brunton is one of greatest teachers in this world!

[@joshelguapo5563]

I love when you can put anything in linear algebra terms, it makes everything simpler

[@minghanzhu6082]

I learned Fourier Transforms 5 years ago and today is the first day I fully understand it from the vector space point of view. Thank you so much.

[@kamalabouzhar8469]

It makes a lot of sense to think about the coefficient A_k and B_k as projections on the orthogonal basis formed by cos and sine. Never thought about it this way ;) it always good to learn and relearn the same concept from different perspectives.

[@BoZhaoengineering]

Thinking of sine and cosine as basis, function space is span ( sine , cosine). This really make sense when I was trying to gain a solid understanding of Fourier ‘ s method. Thank you very much for your instruction video.

[@iwakeupsad]

Hi Steve! I've been a sound designer working and building with Fourier EQ's and Compressors for years and years BUT never has someone explained it to me like this. I can't thank you enough. I feel like I get my job and my art now.... and it feels great.

[@jeremystookey7270]

The best Fourier Analysis videos taught by a floating torso on KZbin. Thank you Steve. You've made this topic much easier to understand.

[@rhke6789]

I have read dozens of books and viewed dozens of videos on this topic. This series is the only one that marries intuition with process and procedures and explain why. Congrats. I highly recommend to everyone.

[@amribrahim7850]

That's literally the best explanation I've ever heard about Fourier series. Thank you!

[@sandlertossone1813]

After all the calculus, physics, signals and systems courses I have taken, the only thing I understood is how to do them and not what they actually mean. I was worried about not getting answers from the books, online, or my professors because I want to teach in the future. This is a game changer and was like inventing the light bulb to me. I can't thank you enough for not only explaining it in detail, but doing it in a simple and understandable way.

[@erockromulan9329]

I just got the book and I'm starting Chapter 2 with a major head scratch. Then I watch this video and it's like taking a blindfold off. Thank you, Steve!!

[@gauravbhokare]

This is the most simple, elegant and beautiful explanation (IF YOU KNOW LINEAR ALGEBRA) I've ever seen. Thank you Prof. Brunton.

[@majdabouakil8878]

Suddenly these formulas appear to be very simple and I'll probably never forget them. Thank you very much.

[@djmips]

I'm not trying to impress anyone, but after a lot of thinking over the years (I'm a game programmer) I came to this idea on my own but I was never sure if I was correct. This is the first time I've seen a presentation that clearly confirms what I had conjectured. I'm learning a lot of details now. I will continue on with this series as I am quite curious to know more and confident that I can understand your presentation format.

[@farhanmadilfaraz]

You made me fall in love with mathematics again. Great work, Thank You so much.

[@shagunagrawal2003]

Same

[@mohamededbey]

I've just found out about this awesome channel. You sir are wonderful. Please don't stop the good work 😉

[@Eigensteve]

Thank you very much!

[@pairadeau]

If there is such a thing as pedagogical wizardry you're witnessing it. This channel is superbly good.

[@chopnchoopn13]

This guy is actually the best on KZbin!

[@whenmathsmeetcoding1836]

What a great interpretation of Fourier series with vector spaces, I am proud to say I do follow quite a few channels and your's work is best among them.

[@dragoncurveenthusiast]

I'll join into the choir and just want to say that this is an amazing explanation! Thank you so much!

[@mugiwara-no-luffy]

The analogue for projecting f onto orthogonal vector basis to explain projecting f onto orthogonal functions was really helpful and made everything click for me. Thanks!

[@quantabot1165]

I have never seen anybody so excited to teach something. Amazing

[@isaacharris7534]

The intuition you provide regarding the inner products is incredible. So helpful in understanding this difficult topic. Wish you were my Prof. when I learned Fourier Series in class!

[@jairjuliocc]

I really like maths and machine learning and after long time trying, i finally understand fourier series with this video .Thank you so much!!. Greetings from Peru

[@Eigensteve]

All the best

[@oelocram2364]

What a great and clear explanation. I graduated this past spring with my B.S. and am currently working now. I wanted to just brush up on this topic just for fun and I didn’t realize how unclear my professors were during university. This is definitely helping me regain the understanding of the fundamentals of FFT.

[@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)

[@mipmap256]

After watch Steve's online videos. I don't know any reason to pay the tuition for Stanford. Actually your videos is the only reason I upgrade my KZbin account to premium account. Really appreciate for what you are doing. So far you and Gilbert Strang are the source I got my math foundation in English. I got my education in China, Which didn't explain everything very well and all the math vocabulary is in Chinese, which make it extremely hard to read papers. Thank you again.

[@dk_3562]

Professor Steve Brunton, you are a GEM! I can't thank you enough.

[@Monroe18999]

Very helpful. I wish my other professors would have explained it this way... Thank you Steve.

[@Eigensteve]

Glad you liked it!

[@Angel00Exia]

Beautifully explained and illustrated. Thank you very much for this video!

[@johnwt7333]

YES we get it Prof. Brunton! Fourier Series is writing f(x) as an orthogonal basis of sines and cosines of increasing frequencies. I will use that phrase instead of "The mitochondria is the powerhouse of the cell".

[@yangao4321]

at 8:15, should it be the norm's square or should it be the 2-norm(Euclidean norm)?

[@jossec1344]

Steven L. Brunton i love you, thank you for everything you saved some lives and you helped people being hired

[@TotanDas1991]

Man you make my mind blow ... please make a video on reference frame theory ....

[@AntonioLima-uo2un]

Magníficas aulas do Prof. Brunton e seu excelente livro merece ser lido. Magnificent videos of Prof. Brunton and his excellent book deserves detailed reading.

[@herooftimesergio]

This guy is a genius. The way he explains this complex concept by comparing it to linear algebra and projections is incredible and unique. Thank you very much!

[@pedrorebelo9445]

Great explanation! Never thought about it that way before!

[@paulisaac3489]

I love math so much, I might be changing it a lot with an idea I have... A new coordinate system based on koopman linear complex plain transformation, where our number line disappears completely into polar but gives a different view to enable the elimination of apparent randomness in several fields. I think I came up with something that if correct may be the most significant mathematical discovery in the century..... If I'm right, i could win a fields medal and more as well as crack all elliptical curve cartography and create a system that should be able to provide solutions to at least four of the millennium problems... Not bad for a guy who went to alternative school and hasn't finished traditional postsecondary. Thanks for the knowledge man!

[@paulisaac3489]

for obvious reasonsI need to be sparing with the details, but some problems with our system of complex analysis exists with the origin of the complex plane along the real numbers plane, I don't think it's zero... And specific transformations like halfing the radius of projection for modeling, and specific particularities with the conceptual complex planes, digging into an new way of looking at numbers without an infinite number line real number line in the system.

[@lalitkaushik1991]

That's great explanation.. I think i will never forget that...

[@tejesdas3896]

Thanks Professor, for such lucid presentations!

[@bentley2495]

Thanks, papi. Completely forgot most of the EE stuff I learned from my CompE degree, this is a nice refresh without reopening my Analog signals & systems (Kudeki & Munson) book. Lol

[@alexcartwright9120]

Absolutely mindblowing, thank you Steve.

[@johnwt7333]

Prof. *

[@mahdighane3162]

Thank you for the amazing video! I have a question about normalizing inner product of with ||cos(kx)||^2, if resembling to vector space inner product, I would expect ||cos(kx)|| instead of ||cos(kx)||^2; could you give some hint about the source of power 2? it is possible to see that we are looking for a transformation that if a cos(kx) enters then the results should be 1 for the cofficient, then it makes sense. but to make an analogy with vector inner products, if it is not the unite vector, it is divided by L2 norm. Thanks in advance.

[@phamhongvinh550]

Awesome intuition explained about Fourier transform! Very good video

[@MiguelLeonRomero]

Great channel and great explications. i love the way to explain the concepts of him.

[@Eigensteve]

I appreciate that!

[@mortezakhoshbin]

you always have things to learn from you.thanks

[@putyavka]

It's marvelous! I guess I finally undersand the core meaning of fourier series. It just an abstraction of coordinate system of orthogonal functions

[@nidashabbir754]

Just found your channel, I must say you are Amazing MAN :)

[@curtpiazza1688]

Never heard FS taught like this! 😊 Great!

[@gadeyashwanth]

Thank you so much for sharing your knowledge in a more clear way and I loved the explanation.

[@rylanschaeffer3248]

I think there's an error in this video. Around 8:15, Steve is talking about normalizing vectors to make them have length 1, and he says to divide by the norm squared. I think to convert a vector to a unit vector, one should divide by the norm, not the norm squared. Could someone clarify? www.khanacademy.org/computing/computer-programming/programming-natural-simulations/programming-vectors/a/vector-magnitude-normalization

[@yangao4321]

i think he meant the 2-norm(Euclidean norm)

[@kariemhagag973]

Wow, I am working With FMCW radar and FT is the heart of its working theory and this video helps me a lot why Chirps is increasing linearly over the time with respect to band width woow thanks a lot !

[@DavidKirwanirl]

Thanks for explaining this, looking forward to the rest of the playlist!

[@blzKrg]

You are a very good teacher, you know that!?

[@Eigensteve]

That is really nice of you!

[@WuhaoJiao]

Thanks for interpreting! I have a question which also appears several time in the comments: Why is / ||cos(kx)||^2 not just divided by ||cos(kx)|| (why divided by norm square but not just norm) ?

[@samyakmarathe3434]

It's an interesting question. Here's the answer. In Fourier series we're representing any function as a sum of sins and cosines. Consider sin and cos terms as a different coordinate system. So all the cos terms are like the vectors made from the basis vectors of our hypothetical cos coordinate system where the basis vectors are cosx, cos2x, cos3x.... Similarly all sin terms come from the sin coordinate system when f was projected onto it. While projecting f in each of the coordinate system we divide by the norm of the basis vector (cos and sin). We have to normalise not just the cos(basis vector) but also the values of f thay are projected. So f(x) is also normalised by the same amount because it's in the cos coordinate system (or sin) not the usual Cartesian coordinate system. It's like transformating a vector into other coordinate system. U don't just change the basis vectors, but also change the vector itself. That's why not just the basis vector are normalised but also the resulting vector (f) as we're in the coordinate system of sins and cosines. The unit vectors, although the word unit means 1, doesn't have the length 1 in cos and sin system. It has to be divided by some magnitude to make it the unit vector that we use. And in doing so, we also divide f as we have to represent it in terms of cos and sines. f vector on left is our Cartesian f that we will use. f vector on right is the f we plotted in sin and cos coordinate system. So it must be divided by the norm of that coordinate system (both the basis vector and the vector itself) to get the f in our coordinate system (Cartesian one)

[@leopardus4712]

Great finally understood fourier series with linear algebra I'm lost at ||1/cos(kx) ||^2 = 1/pi or the sin

[@contourintegral4992]

1/||cos(kx)|| normalizes the inner product.

[@ioannisgkan8930]

Wearing a black top behind a black screen, he is like a ghost with a head and two hands He is brilliant by the way, great explanation of Fourier series

[@zrmsraggot]

5:00 Can someone confirm that this 'cos direction' as a vector is the equivalent of the basis vector i[0, 1] in cartesian plan ?

[@samyakmarathe3434]

No. It's similar to [0, and √π] we divided it by the norm to make it like [0, 1] as we understand unit vectors. That's why we divided by pi and not √π as we also have to normalise f since we're representing it as a linear combination of cos vectors (and sin ofc).

[@dhanushka5]

I understood for the first time, wonderful

[@Eigensteve]

Happy to hear :)

[@cosmic_kid2868]

I might have missed it but when you scaled the A_0 term by 2 you mentioned we'd see why later, was this in reference to the constant outside of the integral for the Fourier coefficients? Just wanted to make sure I asked this to clarify why you decided to scale the A_0 naught term. Thanks for the awesome recap on Fourier Series!

[@greenpumpkin172]

Really thank you for these videos !

[@agh1750]

So why is A naught scaled by 1/2 ?

[@legendary_egg]

World class teaching!

[@91KKiran]

Excellent video, thanks Steve ! Makes me wonder if the fourier series can be constructed with arbitrary functions as basis (instead of sine/cosine)?

[@Eigensteve]

Absolutely, there are lots of choices for orthogonal function bases. Fourier makes the choice of sine/cosine functions. Orthogonal polynomials are big for some applications. Hat functions or delta functions are also commonly used in scientific computing.

[@91KKiran]

thanks ​@@Eigensteve I suppose the approximation quality in practice would depend on the choice of basis? this is very interesting, what subject would you suggest studying to dig deeper into this (functional analysis?)

[@Eigensteve]

@@91KKiran Yeah, I think there is a lot of interesting work on this due to modern Wavelet theory. The books by Mallat and Debauchies are great. For the more mathematical perspective, then functional analysis would be good (building on real and complex analysis). Also a lot of good stuff in scientific computing around "Galerkin projection". Lots of numerical methods rely on choosing a good basis.

[@user-iiii234a5gc]

I could not understand the concept of orthogonal in a non vecter funtion all the last term. It is crazy explanation!

[@MrIrresplaceable]

I am still trying to understand if you are writing backwards or the glass is converting it into backwards so can we see it normally

[@fanny3865]

This video was very helpful! Thank you so much!

[@koushiknaskar4751]

Recently found this channel. Really liking your teaching. Can you make a video about lanczos algorithm?

[@MandeepSingh-ny9ok]

how do you write on a glass board where text appears visible to you as well as in the video?

[@hashirroshinvaliyaparambil70]

Prof Steve,I clearly understood how Ak and Bk come but I wonder why Ak and Bk are again multiplying with sine and cosine to get f(x)?

[@fnegnilr]

Very well explained!!

[@woulg]

God I wish I didn't go to art school sometimes, this is so much more interesting than what I did in school :(

[@Eigensteve]

I think the grass is always greener. But never too late to learn something new! Some of the best mathematicians I know went to art school!

[@woulg]

@@Eigensteve :) thank you for saying that.

[@dp0813]

Go back to school

[@hubstrangers3450]

Thank you for your thoughts and time

[@JoelRTLCosta]

A little correction: the unit vector consists of the vector divided by its norm, not its norm squared.

[@leonardobrasil7032]

Amazing explanation, thank you !!

[@Eigensteve]

Glad you enjoyed it!

[@junaidqureshi8283]

Thanks for the video but for unit vector,, shouldn't it be norm of x or y rather than norm of x or y squared?

[@haticehocam2020]

Mr. Brunton What material and program did you use while shooting this video?

[@varunkumarsingh758]

sir ji ek number!

[@chemicalsandemotions]

Thanks for the video. But are you writing from right to left? How does that set up works? Couldn't focus on the subject thingking about that. Please don't tell me you are writing in reverse to make it look normal from the other side of the glass. :D

[@ruongu]

This video is great! I love it! But one week earlier would be great, because I wrote an exam about it and I didn't went well.

[@Eigensteve]

Thanks for the kind words... hopefully next time we will get the timing right...

[@mybean1096]

So are the different swigly lines ocean waves?

[@Eigensteve]

Anything periodic, like ocean waves, are well approximated by a Fourier series.

[@cabdolla]

Great video Steve!

[@tcratius1748]

Remind me I came up with an equation to solve when image is greater than one in unsupervised SVD for recreating an image with guassian noise. Though I will say, from memory, the mean and standard deviation of some images do not equal one, and therefore was confusing. So I placed guassian noise on your dog picture and recreate the image from that. Hence the whole variable being greater than one.

[@exibian757]

you are so cool,I love your course

[@janferkl7649]

Dear Profesor, thank you very much for your work. I would like to ask you, why is orthogonality of basis vector so important? In ME565 lectures you mentioned, that cosines are orthogonal to other cosines, sines to other sines but that they are not orthogonal to each toher (with coresponding frequencies). That is possible to be shown with inner product. But what is the final effect on expressing general functions as a sum of these basis fnctions? Why is orthogonality so important, since it is possible to express general vector as a sum of non orthogonal basis vectors? Once again, thank you very much. I do enjoy your lectures so much!

[@jacobvandijk6525]

Even I understand this now! :-) Very fine explanation. Thanks.

[@rafvissersraf]

Can he write that good in morror or is it a computer program helping?

[@nesslange1833]

What about some Videos on Fourier Optics?

[@therockshack]

Nice presentation!

[@MrMuyu0117]

Quick question. If one set of Ak and Bk can explain f, why do we need to sum an infinite number of them?

[@MrMuyu0117]

What I am confused about here is that the orthogonal basis is not sine and cosine, the orthogonal basis is (cos, i*sine) as you proved in the next video. Or am I missing something here?

[@grantalbertus2757]

Brilliant explanation

[@MathFinUSA]

Nice explanation!

[@yomaru_1999]

How can you write on the other side?

[@Eigensteve]

Years of practice :)

[@altbeb]

the fourier weighs the contributing trigonometric functions by the value of f(x) at that particular value of frequency... sound like a weighted mean.. which makes sense.. as it the amount/quantity of the frequency's contribution to the overall response... could you do a series on the acutal derivation of the series, or if that has been done.. could someone point me in that direction. Always worthwhile for me, to get the development history..like to know how it came into existence.

[@nhathuynhhuu3698]

Hello, this is a great video. I want to know which references you use for this section, can you recommend me some books about Fourier series and Fourier transform? Thank you so much 😀

[@vgycreations6525]

Master piece ❤❤❤❤

[@P3.14i]

How can you do the mirror writing...

[@JohnVKaravitis]

Why is the 40th video in this playlist PRIVATE???

[@superbros1690]

Can you do example problems of fourier series and transform. Like examples we would face in class.

[@Eigensteve]

Good suggestion. Let me think about this.

[@engr.israrkhan]

Great content

[@jaikumar848]

Is there any application of fourier transform in orbital mechanics ??

[@Eigensteve]

Lots of good applications here. In fact, there are some strong historical connections too. Gauss essentially invented the FFT to speed up his mental math calculations related to astronomical observations. Ptolemy's doctrine of the perfect circle, which was the leading theory of planetary motion for 1500 years until Copernicus, was essentially a Fourier transform of the planetary data.

[@jaikumar848]

@@Eigensteve thanks a lot sir ! I would like to explore this. Could you provide some resources / links related ...or suggest some Google key words ??

[@Eigensteve]

@@jaikumar848 I don't have a lot of great resources on hand, but if you google any of the phrases above, you will probably find something interesting.