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An Interactive Introduction to Fourier Transforms
- SlowRobotAhead 8y agoWow. This is by far, hands down, the best way I have ever seen this explained. Just excellently done.
- peterhil 8y agoExactly my thoughts also. Thank you Jez!
- unao 8y agoVery impressive end enlightening indeed. SmarterEveryDay published a video on this topic few weeks back: https://www.youtube.com/watch?v=ds0cmAV-Yek https://www.youtube.com/watch?v=ds0cmAV-Yek
- jacobolus 8y agoIn the animation at the top, should use a different color for the circles vs. the hand/pen.
- FrankDixon 8y agoBeautiful! How long did it take to make this blog post?
- jezzamon 8y agoOh man, it took me a while. I started in August and have been chipping away at it on the side, until now.
- fdsak 8y agoEpic ...thanks
- sbergjohansen 8y agoAt the time of writing the Gibbs phenomenon -- the overshoot occurring at discontinuities of a square wave, which does not go away no matter how many harmonics are included (see https://en.wikipedia.org/wiki/Gibbs_phenomenon https://en.wikipedia.org/wiki/Gibbs_phenomenon) -- is not correctly represented in these visualisations, implying that important aspects of the underlying maths are obscured. In my opinion this limits the didactic value of an otherwise impressive presentation.
- jezzamon 8y agoYes, apologies I made some approximates in the article! Interestingly if you're talking about discrete time fourier transforms (which is what we usually deal with in computers with jpegs, mp3s, etc), then you can perfectly represent a signal with say, 1024 samples using exactly 1024 sine waves without worrying about that effect. It's only in the continuous time variant that you have to worry about things like the Gibbs phenomenon (which you do run into once you start translating to real world output)
- sbergjohansen 8y agoRight! I did assume a continuous space because that's what it looked like to me. Hence my pedantic comment which was sort of comically at odds with the light-hearted spirit of your article :) Very nice work. Thanks for sharing, and for the clarifying remarks!
- pjc50 8y agoWhat's not correct in the animations? (arguably the whole point of a tutorial like this is to do it without the maths)
- jamesbrownjr 8y agoSlightly unrelated, here's a beautiful mathematical approach to the discrete fourier transform with evaluating a polynomial at the nth root of unity in the complex numbers: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-046j-design-and-analysis-of-algorithms-spring-2015/lecture-videos/lecture-3-divide-conquer-fft/ https://ocw.mit.edu/courses/electrical-engineering-and-compu...
- Waterluvian 8y agoThis is incredible. If I wanted to collect things like this to teach my kids, where do I look? How do we gather amazing resources like this in one spot? I particularly love that it focuses on one topic and nails it.
- Sean1708 8y agoSomething that's become quite popular in recent times are "awesome lists"[0, 1]. This isn't the common use-case for them, but it might be worth looking into. [0]: https://github.com/sindresorhus/awesome https://github.com/sindresorhus/awesome [1]: https://awesomelists.top/ https://awesomelists.top/
- amitp 8y agoFor interactive explanations check out: https://explorabl.es/ https://explorabl.es/
- agurk 8y agoThis is great as it doesn't just explain the basics, it also shows how they are used in MP3 and JPEG files. I always had a very vague "throws away data that's not perceived" idea of how MP3 works. With the background from this I was able to really understand the wikipedia page and found this great Ars[0] article from 2007 to cement my understanding. [0] https://arstechnica.com/features/2007/10/the-audiofile-understanding-mp3-compression/ https://arstechnica.com/features/2007/10/the-audiofile-under...
- ryandvm 8y agoReally fantastic work. Best interactive write-up I've seen of a technical topic in a long time.
- whytaka 8y agoThat was the most impressive opening demonstration. My eyes popped out of my skull.
- travisgriggs 8y agoThis is really cool. I especially like the rotating circles visualization. Another video that helped me finally "get" Fouriers is the 3Blue1Brown video: https://www.youtube.com/watch?v=spUNpyF58BY https://www.youtube.com/watch?v=spUNpyF58BY
- zero_kool 8y agoAmazing work! Thank you :)
- jeffwass 8y agoVery cool. One idea that would be cool to see is a repeat of the video where the number of ‘harmonics’ is clipped. Eg, show the video with only the first 5 circle components. Then the first 10, the first 15, and so on. Will see the drawing approach the final image, and get the idea the low-freq harmonics do the bulk of the work but high-freq ones give the small details to make the hand a hand.
- ccvannorman 8y agoThis is included in the article. Slide the slider below the animations to control how many circles will be used.