
An Interactive Introduction to Fourier Transforms - a1k0n
http://www.jezzamon.com/fourier/index.html
======
agurk
This 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-
under...](https://arstechnica.com/features/2007/10/the-audiofile-
understanding-mp3-compression/)

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Waterluvian
This 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
Something 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/)

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SlowRobotAhead
Wow. This is by far, hands down, the best way I have ever seen this explained.
Just excellently done.

~~~
peterhil
Exactly my thoughts also. Thank you Jez!

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unao
Very 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)

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sbergjohansen
At 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
Yes, 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
Right! 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!

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jamesbrownjr
Slightly 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-
compu...](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/)

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jeffwass
Very 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
This is included in the article. Slide the slider below the animations to
control how many circles will be used.

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travisgriggs
This 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)

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ryandvm
Really fantastic work. Best interactive write-up I've seen of a technical
topic in a long time.

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jacobolus
In the animation at the top, should use a different color for the circles vs.
the hand/pen.

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whytaka
That was the most impressive opening demonstration. My eyes popped out of my
skull.

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FrankDixon
Beautiful! How long did it take to make this blog post?

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jezzamon
Oh man, it took me a while. I started in August and have been chipping away at
it on the side, until now.

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zero_kool
Amazing work! Thank you :)

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fdsak
Epic ...thanks

