The Fourier Series and Fourier Transform Demystified

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Creator - Jade Tan-Holmes
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Music - epidemic sound

[@12tone]

That explanation of the Fourier Transform is probably the most intuitive I've ever heard!

[@duckymomo7935]

12tone, happy to see you here

[@Elenesski]

@@chrisw4578 Oooo a new channel to subscribe to

[@bonolio]

@@Elenesski If you love Up and Atom, 3Blue1Brown will rock your world. I understood Fourier transforms, but until I watched 3Blue1Browns video on it, I didn't truly intuitively understand it. He has an amazing way of not just showing you how it works, but visualising the why in ways that really expand how you think about math.

[@timandersen8030]

@@bonolio I still prefer this video over Grant's because he makes it more complicated to understand.

[@ahadamin7361]

True

[@ohnonomorenames]

When ever I watch one of Jade's videos i feel like I am watching and adult version of Playschool (an Australian kids educational TV show) there is a level of enthusiasm and 'you can do this' that comes through that is so wonderful. I often get lulled into a false sense of security and zone out and then have to go back and re-watch remembering that I'm not quite as smart as she makes me feel. Jade I love everything about the way that you do what you do it must take a mountain of work so thank you so much.

[@earthling_parth]

Finally!!! This was my Eureka moment. I've studied Fourier Series and Transformation multiple times during my bachelor's and masters in computer science and each time I only learned the technique and not _why_ and _how_ it's used. This is the best explanation and intuitive explanation of Fourier Series and Transformation I've ever encountered. Thank you so much Jade! You must've researched really hard to come up with the examples and simpler words to explain this. Thank you once again ♥️

[@starfishandroid]

Same. Music producer here. This was my eureka moment.

[@lvmbk3755]

Being a telecommunication engineer I perfectly know how Fourier transforms are ubiquitous, as they are necessary for signal processing an electronic communications. But it is fundamental also for buildings and mechanics because the analysis in the frequency domain allows to understand how materials and systems behave under given inputs. I think nowadays it is as essential as basic math operators like +, -, /, *, etc....

[@markawbolton]

It is also very beautiful.

[@daviskipchirchir1357]

Just got introduced into Fourier Series and transforms. My mind is still blown up tbh

[@Jadstudio7]

I concur

[@mastershooter64]

Integral transforms in general are absolutely ubiquitous. Functional analysis is beautiful

[@festusajayi8917]

Engineering mathematics is core course for all engineers irrespective of the discipline of engineering. Fourier Series is covered in details in engineering mathematics

[@balasubramanianvaidyalinga487]

Thanks! Possibly the clearest intro to the topic. Sharing this with my kid. Subscribed as well.❤

[@Dixavd]

You've become such an amazing educational video creator, Jade! The cinematography amazing: lighting, camera quality, colour-correction, framing, pacing,, etc... You've even mastered how to use these skills to effectively get your point across without it becoming a distraction. I supported you on patreon previously but had to stop for financial reasons and I then didn't keep up with your uploads (mostly because my physics studies became so exhausting, I rarely had the energy to watch physics videos for fun). I'm so glad I looked you up again, though. I'm very proud of how far you've come. Keep it up.

[@SKULDROPR]

This concept blew my mind the first time I learned about it at uni. Until then, I had never realised, or even considered you could transform from one domain to another. I'm now an audio engineer, it's astonishing how ubiquitous, useful and practical the Fourier transform is in the field. I liked the tie in to real world algorithms at the end. I would like to see a video about different sorting algorithms if possible! My personal favourite is the radix sort.

[@hackerguitar]

Isomorphism for the win! it shows up everywhere….I’ve been seeing it more and more in speech recognition algorithms.

[@averagejoebitcoin]

linear algebra. that change of basis vectors and yet still able to Span the entire space "=" sinx and cosx can span the entire "function space"

[@kpriya4739]

Hi I also aim to become an audio engineer. Can you please share your contact details if you are interested in guiding me? Please I have a few queries.

[@kingbeauregard]

Calculus in general does that, when you think about it. Like with simple kinematics: you can describe an object's motion in terms of position, or take the derivative to describe that same motion in terms of velocity, or take the derivative one more time to describe that same motion in terms of acceleration.

[@trewaldo]

This is my most favorite topic in introductory signal processing where signals in the time domain exhibit a certain characteristic in the frequency domain through respective spectral properties. Thanks, Jade, for the animated and colorful video! Cheers! 😍🤓🥰

[@vsalt69]

I really appreciate the way you focused on the real number amplitude components as a way of simplifying your lesson. Not worrying about phase allows you to clearly show the connection between the integral calculations and its amplitude spectrum. This was the clearest of dozens of explanations I've read and watched over the last 20 years. Thank you so much.

[@shaovoon]

I wish I had a teacher like Jade when I learned the Fourier Transform 20+ years ago. Thanks for the brilliant explanation and superb animations that helped me understand!

[@bitsandbytes1s0s]

This is in my math curriculum and i was soo obsessed by them, thanks for this video

[@denkenunddanken5961]

Me too was so much obssed with this in my 2nd year college.

[@daviskipchirchir1357]

It's my second year of college right now I'm so obsessed with this😂😂😂😂 The Fourier of being obsessed at 2nd year correlates with these three souls💀😂

[@denkenunddanken5961]

@@daviskipchirchir1357 🤣🤣🤣 cool

[@divitrajgogia4909]

The best video on KZbin for Fourier transform and analysis! Please make more videos on this part of physics/ engineering. This feeling of understanding and visualization of Fourier transforms is extremely satisfying! Thanks for making a great video.

[@rohank9292]

so may years spent trying to understand fourier series and transform and then this one 14 minute long video comes along and makes things all so clear. Thank you

[@AaronJarecki]

I've come across these concepts before. What I love about this video, and many of your other videos, is that you encouraged the viewer to go beyond understanding that this works and explained how it worked. Super impressed with this explanation. Thanks Jade!

[@BleuSquid]

My favourite usage, and indeed my introduction to, the Fourier Transform is in Mersenne primality testing. The most computationally expensive part of some primality tests is a squaring of a very large integer. By representing the digits of the number as time-series array, taking the fourier transform, squaring the individual elements (this step can be done massively parallel, hello GPU computing!), and then transforming it back, we have effectively squared the original number in a fraction of the time.

[@ehrichweiss]

I've always love the Fourier transform. I first learned of it back in the early 1990s when I was using a "granular synthesizer" that would let you draw a picture and then it'd convert that into sound. It took over 20 years for some software to duplicate that synthesizer. BTW, my wife and I bought your t-shirts and we love them. Keep up the good work.

[@henryukagwu5183]

That's wonderful

[@deang5622]

Fairlight music synthesizer had this capability in 1979.

[@yasscat5484]

1:04 you mean a higher frequency* great explanation exactly when I needed it

[@johnshioli1499]

I’m always excited to see a new UaA video come in, and this one didn’t disappoint! Fourier transforms always seemed like magic to me, but your explanation made it all make sense. Also, beautiful locations! That mountain and lake (river?) scene was gorgeous! 👏

[@loberd09]

Thanks for the video. I’ve been a chemist in industry for 15 years. I learned it back in college but wasn’t great with it. I’ve had to “black box” it (use without a firm understanding) in explanations for instrumentation I use (FT-IR) and some instrument designs I’ve worked on. This is a great explanation. Not sure it’s a refresher for me as I wasn’t solid on it when I learned it.

[@markgoodall1388]

I was thinking the same thing actually, but watching this I do wonder if the technique is under utilised in the chemistry domain.

[@JohnSmith-qp4bt]

But do you really need to understand the mathematical basis? And not rather focus on identification? Are you still working in the lab after 15 years in the chemical industry??? Not a department manager or director by now?

[@markgoodall1388]

@@JohnSmith-qp4bt so many assumptions! Having some level of understanding would seem essential actually. I suggested that the technique was underutilised, meaning I ponder the possibility of using FT outside of the domain of FTIR. Maybe it already is? I mean, I did stop worli in laboratories over 20 years ago. So, yeah, I was just musing. Feel free to now take a dig about not being 'current'. lol

[@user-ee7sc1nu6n]

​@@markgoodall1388😊

[@vector8310]

Your explanations are models of clarity. Just the right amounts of illustration and conceptual elaboration.

[@DeepakGautamX]

Fourier transform, this is interesting. I have studied it my graduation. This could use in various cool projects.

[@adsbhushan5123]

Thanks for cracking open a black box, I've been carrying since college physics. Brilliant exposition and the accompanying video makes it easier to understand.

[@no_justno]

Your editing is PHENOMENAL. Also this is the best explanation.

[@adamharris6557]

Great explanation and graphics. Of all the videos on this topic, your explanation is the most intuitive. You break down everything and explain each piece of the puzzle with great graphics. I'm recommending all my students to this video from now on.

[@adolfos1991]

Thanks Jade for another awesome video!! I wish our lecturers were as good as you when it comes to explaining complex subjects with such simplicity.

[@johnydyroy1576]

I'm so impressed, easy and understandable explications and great animations! Keep up the good work!

[@animalbliss3713]

You are amazing at explaining hard topics. Keep up the great work!❤️

[@shunpinhsu]

Fourier series works mainly on `periodic' functions. Aperiodic functions are treated as periodic functions with their periods tending to infinity. In this case, the Fourier series (in the form of summation) takes the form of integration, which is known as the Foruier transform.

[@triberium_]

Very interesting, thank you! I'm working on a video game and waves are great for generating terrain and this has given me more tools to use with the world generation part of it all

[@brunotrotti6942]

Very good the way it mixed up the intuitive and simple explanation about the matter with the maths jargoons and formalism. Connected different subjects and captured the hole picture in awesome way. Really congrats

[@vctor611]

Amazing video Jade! I learned so much! Definitely needed something like this!

[@facundomazzola7115]

loved the video. the editing and visual effects were amazing!!

[@sparky7915]

Great video! I never heard of Fourier series and transforms. Quite interesting! You make the complicated things easier to understand. I love watching videos on this channel because I am always learning something.

[@briansauk6837]

Great video! One neat trick to solve for series is to consider the derivative or integral of a series that is easier to find. For example, once you have the square wave series, you can trivially solve for a triangle wave, by doing the simple integral of each sin term. That’s because a triangle wave is the integral of a square wave.

[@legendrams548]

This is a superb explanation of Fourier Series and Fourier Transformation. Loved the way you presented this entire video. Highly informative! Thanks a lot to you!👍👍

[@AmitKumar-xw5gp]

Very very well explained. Love the way you explain the topics.. You have a gift to be able to explain a concept in a simple way.. Keep making videos..

[@harshans7712]

This video has one of the best explanation for Fourier Series along with it's application, these types of videos really intrigue every learner about this topic and make them fall in love with the subject, really hats off to your effort 🙌

[@rkamalat]

Wonderful way of explaining Fourier Series and Fourier transform. Have taken a few of your diagrams for my lectures on DSP. Thank You so much.

[@AMANKUMAR-oh1zt]

Reminds of 2nd Year in College. Had a course in Signal Processing and my overall B.Tech. in Electrical Engineering. Fourier series is indeed freaking stuff.

[@tinhoyhu]

Thanks for the video. This really brings me back to 30 years ago during a nerdy summer program where I had a project to modify sound recordings using FFT.

[@ivanliptak19]

Thank you for taking on this topic! I find it wildly fascinating, as with acoustics generally.

[@GarryMah85]

Fourier transform was a topic I could never understood during my undergraduate studies almost 2 decades ago. I'd always skipped any Math examination question that require us to use Fourier transform. While I still doubt I'll ever be able to comprehend the mathematical part of it, your video actually gave me a great idea of what Fourier series and Fourier transform is all about. Thank you. I wished we had resources like this 20 years ago, lol. It helps make sense of all the abstract mathematical concepts we had to learn.

[@abirsadhu5538]

Also you can check fourier series and fourier transform video in 3blue1brown channel. They are amazing.

[@fiNitEarth]

This video is FANTASTIC! I've been using the Fourier transform in data science a lot and thought I had a pretty good understanding of the matter. This video however gave me a whole new intuition for it. By far the best video on Fourier I've ever seen!!

[@bonolio]

If you haven't watched 3Blue1Browns videos then I would suggest. I won't say they are better or worse, but he comes at the intuitive understanding from a different angle. The more ways you can visualise how something works, the better you can intuitively form solutions

[@daviskipchirchir1357]

Hello how do you use Fourier transform in data science?

[@mitchwyatt9230]

Around the 12 min mark, The orthogonality was glossed over a bit here, but it's an important point - the orthogonality is what keeps the calculations for decomposition into component sin and cos waves (relatively) simple. P.S. Fantastic video overall. I really think this is my favorite yet of all your videos. Please keep up the good work!

[@Flovus]

Exactly. Orthogonality is not necessary to describe any vector, a basis is already sufficient. Has anyone ever tried non-orthogonal bases for Fourier-related transforms?

[@Pingviinimursu]

I could have used a high-quality video like this to explain some of this stuff when I studied them, the visualization is a lot better than the ones I saw. I'm happy this video exists now, so others might find it useful and who knows, I might come back to this stuff some day as well :) Thank you Jade!

[@legosteveb]

OMG 10:31 blew my mind! Thinking about the integral as a correlation calculation is the most concise description of FFT I ever heard! Amazing how similar this is to brute force image correlation. Thanks for demystifying the often labeled “magic“ FFT function.

[@xaviergonzalez5828]

It's one of the best videos about Fourier transformation. Thanks!

[@writingsurgeon]

You made my Day!!! A lot of doubts related to the Fourier Series are eleminated. Now, SIGNALS AND SYSTEMS is a fantastic subject for me. Thank You so much,

[@jannickharambe8550]

I love you so much! The way you explain things is breathtaking! You take complicated topics and explain them so easily with simple words - Richard Feynman would be proud of you, that's for sure! Myself, I want to thank you. You help me understand a lot of things that I will be needing/need for my studies. And it's so much fun to watch your videos!

[@FrederickStadler]

Great job with this video, Up and Atom. I thought the material was very interesting and well explained. Keep up the great work!

[@tedblack2288]

Nicely done, Jade. I was introduced to, and began using Fourier Transforms in the 1970's. One of the things I learned is that one does not need the basis vectors to be orthogonal provided they are non-degenerate. As long as each basis vector cannot be described in terms of any other basis in the set, one can still get an absolute description of the phenomenon! When one is examining Complex space, this trick can sometimes massively increase the number of signals that are actually observed. (fyi, Real life detectors simply cannot see spectral lines that have a non-zero imaginary component.) This technique was used to double the observables in early MRI spectra.

[@baljeetin581]

Love you jade, just found you today. Feels good. I am a computer programmer. Your videos seem very helpful to me. Your presentation seem so natural. I do believe to work with your concepts. 😃

[@raghodsa]

Brilliant yet simple explanation! Thank you for your efforts👏🏻

[@DarylBanttari]

The graphic at 5:50 blew my MIND. SO MANY CONNECTIONS. Gonna have to dive into this harder now that it's not just a magic black box. Your videos are amazing, keep it up!

[@wozzlebaby5313]

Wow. By far the best and most thorough explanation of this topic I have ever seen.

[@kaemmili4590]

that was a masterclass in teaching and clarity . would have loved more details and slower pace , but rewatching and google will work , thank you so much

[@jadermcs]

The video editing is improving a lot, really liked the editing of this video.

[@cw9249]

beautiful visual explanation!! well done

[@borisnges4285]

Thanks, I watched your video just for a few minutes and it cleared a lot of doubts I had concerning Fourier. Thank you

[@alangaha1869]

Great Video Jade, thank you. A clear and concise explanation, well presented. I wish I had been able to see this 30 years ago before university.

[@mr.nobody.01]

You know so much how to explain complicated things to us. Thank you and keep going.

[@jeremylaughery2555]

Awesome video! I am using Fourier analysis to help with modeling a general prime number generator or prime number sieve. Fourier analysis is a game changer!

[@jimdevilbiss9125]

It is great to see this being shown. The best part of my electronic engineering college was Fourier analysis.

[@albertopacheco2244]

Very simple explanation of a very abstract topic. You have a gift.

[@meow75714]

Although I know about Fourier series and transforms and have been using it for a few years, this video still added to the basic foundational understanding of it. Much love 💟

[@anantaacharya3019]

Fantastic presentation, you have made it so interesting, giving a very good concept, really enjoyed.

[@zach4505]

A well made video. Thank you for adding some intuition to the formulas.

[@curtpiazza1688]

Beautiful intro. to Fourier Series! Very well explained and presented! Lots of applications to music!

[@blueckaym]

Fourier series & transform are incredibly powerful instrument that can be used in most of our aspects of life. While it's actually not perfect - as it doesn't provide the best possible solution (unless you're ok with applying more and more sine-waves to infinity) it's surprisingly powerful in practical terms. One (of the many) curious things about it, is that it's in the core of Heisenberg Uncertainty Principle (HUP). That's can be very confusing to most people, as most think that HUP is actually related to something physical in the nature of the quantum particles (I thought so too until not long ago) and one of the most popular explanations is that you can't measure a property of a quantum particle w/o interacting with it and affect its other properties in doing so. But this isn't at the core of the problem - it's a practical measuring problem (that we might not ever be able to solve), but doesn't actually say much about the nature of the quantum particles - ie what they do while we're NOT observing them. ... anyway the solution to this problem is still a mystery, and we might never find it (many scientists have already given up, and prefer to "shut up and calculate" what they can), but the current truth about HUP is that physicists are using Fourier series & transform as a tool for their measurements and the uncertainty is actually embedded in the HUP itself - it's a limitation of our Math Tool (no matter that it's indeed really, really powerful otherwise)! It's not necessary limitation of the universe (at quantum level)! That's pretty much the same question - Did we discover Math or did we invent it? - but with quantum physics seasoning :) While in most cases it's not practical to wonder about the philosophical aspect of a given field of science, it's extremely important imho that it's never ignored completely, as most people start to believe that what Math is telling us is what Universe actually IS ... which might be the case sometimes, but isn't really necessary true. Math is like a keyhole and if we sometimes see things take keyhole-shape (as we're looking thru it) doesn't mean that we're seeing the whole picture and that it's indeed their real shape.

[@anthonydefreitas1694]

You always have the absolute best videos!!!! Because of you I read about physics all the time now. Was obsessed with history and politics and rarely go back now. You make these topics so much more interesting. Wish i was better at math. School made it seem so lame

[@araz20091]

I had watched videos about Fourier transform from top science channels and this one did a much better job. Well done ✅👍🏼

[@Whatdouwantmetosay]

I'm eternally grateful! Thank you for the clear explanation!

[@mathwithmurghoobahmedsyed7112]

Excellent explanation, what a remarkable effort to explain the concept , may you have million views and subscribers

[@IntegralDeLinha]

What a coincidence. I was really needing a video about this right now. Thank you!

[@glengineertv1505]

I have worked with an spectrum analyzer arduino code library before. I now understand why the algorithm is multiplying data to a series of numbers which seems correspond to a sine wave. Very good video! Keep it up! Thank you 😊

[@clavierwintergreen5574]

Fantastic video 👏🏼👏🏼👏🏼 Could you make a video explaining (or demystifying) the Laplace transform as well?

[@andrewv.157]

I did not remember of all of this. It was a pleasure to be taught again quick and gracefully

[@EdwinaTS]

Fantastic way of seeing the transform. Many thanks!

[@flyingbluelion]

In cryptography the corresponding transform is called the Walsh-Hadamard transform and it uses linear Boolean functions instead of sin and cos. The maximum value magnitude (= absolute value) in this transform gives the Nonlinearity, NL=(2^n - WWHmax), which is vital to the security of all ciphers. The Walsh-Hadamard Transform is the Fourier Transform when the underlying object is from the set {1,-1} (instead of {0,1}) rather than the set of all real numbers. Use butterfly method to compute it efficiently.

[@alihuzaifa235]

In all i just want to say thank you so much for making such an amazing concept an understandable one, i love that....

[@denkenunddanken5961]

I was waiting for your vedios and for last few days I was looking at your channel for new vedio. And here you comes with a topic i loved so much during my college studies. 🙏🙏 God bless u

[@manurajbharall]

Thanks,it is one of the best way to explain Fourier

[@kanheicharanswain6754]

Excellent teaching about Radio wave or Electromagnatic wave in Mathematical derivatives in practical fields...

[@equesdeventusoccasus]

Excellent video as always. I haven't worked with this in a very long time, and I'd forgotten how fun fourier series and transforms could be. I think I just saw a tooth waving at me so I'll go sin off for now.

[@elmo2you]

Great video and it certainly will help to get a better or more intuitive understanding of Fourier Series and its Transform function. One thing I would like to add though, is that is isn't just a tool. It also has a very real-word importance in (physical) systems. Whenever a transient signal travels through a system, its ability to propagate or sustain itself will depend on how that system responds to it. In physical systems, electronics being a particular important one, signals with different frequencies will face a different resistance/impedance. Those who ever watched a high-frequency digital square wave on an oscilloscope may have noticed that it wasn't quite square. Instead having oscillations around each vertical rise and fall of the signal, similar to the reconstructed square and saw-tooth waveform in this video's animations. This is because real-life systems (including measuring equipment) have a specific frequency response (and often a different one for each individual frequency). Specifically in electronics, sufficiently high frequencies won't make it through a system (often because the physics of the system can't keep up with the rate of change). It is these high frequency components in a Fourier Series that enable signal to have sharp corners (rapid non-gradual changes). That is what makes an ideal square or saw-tooth wave (or essentially anything with sharp corners on a time-graph) impossible to exist/survive in a real-life (electronics) system. It is not just that the Fourier Series and Transform are useful tools, it is also the relationship between transient signals and their frequency components that determines how they will propagate through real-life systems. While I took electronics as an example, there are plenty of other systems for which the same principle hold.

[@timandersen8030]

How does Fourier transform work if you don't have an input function but only raw signal/sound wave as in real world scenario?

[@Wonders_of_Reality]

1:50 - Your graph is so beautiful! I’m so glad to see something light and aesthetically pleasing! Jade, thank you so much! Everyone else is using pitch black as background-it looks like funeral every time. From now, I’ll rather watch your channel.

[@nadirnoorzai7753]

presentation and presenter are both just wonderful. Very well done, love it.

[@cliftoncrooks6389]

This video gives me Flashback to my telecom classes in college. DSP (digital signal processing) with Matlab. Good stuff!

[@parasuraman1155]

I used Fast Fourier Transform in my undergraduate Engineering thesis, which was well received. In the 1990s. My college professor, trained in IISC Bangalore, called Fourier Transform the @hreatest thing ever”. Thanks for this video. Your videos are so well done and inspiring.

[@numericalcode]

Superior explanation and visuals. Well done!

[@UDonotKnowMeBecauseUKnowMe]

One of THE BEST video I saw in related to education 🙌🙌

[@NolanManteufel]

Love the video. Thanks for posting!

[@fredashay]

Thank you, Jade! I've always been baffled how you can take a sound wave and extract each of the individual frequencies...

[@zachariaszut]

You made something which is not that simple look simple. Well done. Cheers.

[@muhsinshah4290]

it's great to see this being shown. The best part of my mechanical engineering was Fourier series and transform

[@yopenzo]

Thank you very much for your great video and its content. One small mathematical pedantic note: more correctly it must be said that all the functions that can be represented with the Fourier series are only the periodic ones; and not even these all of them, but only those where Fourier converges. And as you know very well, the transform is used for all non-periodic functions.

[@victorhugoeh974]

😃🤯wow!!!! There's much more power in the Fourier transform than I ever thought! Excellent video, Jade 😃👍

[@carminesans90]

I wish I had the possibility to see this video back when I was graduating. Great video 👍

[@MmKayUltra1]

My favorite little fact about fourier transforms are that, if instead of a sawtooth wave you have a normalized exponential decay you get the peaks in the same areas but with paired ratios if amplitudes and that the ratio of those pairs can be used to determine the half life or tau of the decay.

[@thegzak]

As usual, great job giving an intuition to complex topics - as Walter Lewin would say, you’re getting people “to see through the equations.” Bravo!