
An animated introduction to the Fourier Transform [video] - e0m
https://www.youtube.com/watch?v=spUNpyF58BY
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knolan
3blue1brown’s videos are excellent. They build intuition in a calm and
friendly way with an appropriate amount of useful animation. This is how we
make mathematics accessible.

I’m currently considering moving back into academia and there are a lot of
topics in my field that I know students often struggle with that would be
greatly helped by some simple animations. Fortunately I’m pretty competent
with blender and I relish the idea of developing something worthwhile.

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jestinjoy1
Do you have any links showing how to do that with blender?

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knolan
Anything in particular? I was thinking of things like viscosity and stress
analysis for fluid mechanics. It would be easy enough to animate a stress
tensor and show how each term behaves when prodded. Similarity the basic
concepts behind laminar boundary layers would be equally straight forward.

Mimicking 3b1b’s style would be trickier since he uses a lot of 2D plots. Of
course you can run python directly from blender so you never know.

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magnat
Source code of the video:
[https://github.com/3b1b/manim/blob/master/active_projects/fo...](https://github.com/3b1b/manim/blob/master/active_projects/fourier.py)

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Patient0
He touches on it - but I’d love to see an intuitive explanation of why the
response of each frequency to the input function is linearly independent. i.e
the fact that Fourier transform of the sum is equal to the sum of the Fourier
transforms. This is “why it works” - it’s what makes the frequency space an
orthonormal basis - but it’s never been intuitively obvious to me. Otherwise,
there would be more than one way of decomposing a function into a
superposition. e.g. what would be useful is to give an example of a set of
functions which are not linearly independent.

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chestervonwinch
The orthogonality is essentially follows from (1) integer frequency complex
sinusoids have an average value of zero over [0,2π], and (2) if you multiply
two distinct integer frequency complex sinusoids, you get another integer
frequency complex sinusoid. I'm not sure that this is any more intuitive.

> what would be useful is to give an example of a set of functions which are
> not linearly independent.

See [1,2] for example, which (I believe) has applications in compressed
sensing and dictionary learning.

[1]:
[https://en.wikipedia.org/wiki/Frame_(linear_algebra)](https://en.wikipedia.org/wiki/Frame_\(linear_algebra\))

[2]:
[https://en.wikipedia.org/wiki/Overcompleteness](https://en.wikipedia.org/wiki/Overcompleteness)

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mlevental
>The orthogonality is essentially follows from (1) integer frequency complex
sinusoids have an average value of zero over [0,2π], and (2) if you multiply
two distinct integer frequency complex sinusoids, you get another integer
frequency complex sinusoid. I'm not sure that this is any more intuitive.

i think these kinds of explanations are hilariously pointless. and i don't
mean to disparage because you're just trying to answer op's question but all
you've done is restated the proof in english - i.e. of course it follows from
that because what you've just said is the inner product of basis functions is
0. well yes of course that's definition of orthogonal.

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nayuki
BetterExplained (Kalid Azad) has a good written article that covers the
Fourier transform in a similar manner to the 3Blue1Brown video:
[https://betterexplained.com/articles/an-interactive-guide-
to...](https://betterexplained.com/articles/an-interactive-guide-to-the-
fourier-transform/)

I have an article explaining step by step how to implement code for the
discrete version of the Fourier transform: [https://www.nayuki.io/page/how-to-
implement-the-discrete-fou...](https://www.nayuki.io/page/how-to-implement-
the-discrete-fourier-transform)

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adamnemecek
I’ll just leave this here

[http://tomlr.free.fr/Math%E9matiques/Math%20Complete/Analysi...](http://tomlr.free.fr/Math%E9matiques/Math%20Complete/Analysis/Mathematics%20of%20the%20Discrete%20Fourier%20Transform.pdf)

Mathematics of the discrete Fourier Transform by Julius O. Smith. (O stands
for Orange I hope)

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jacobolus
Poking around the web turns up
[https://amzn.com/097456074X/](https://amzn.com/097456074X/)
[https://ccrma.stanford.edu/~jos/pubs.html](https://ccrma.stanford.edu/~jos/pubs.html)

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nsb1
I really wish this stuff existed when I was learning about FFTs - this video
describes the theory far better and in far less time than my broken-english
college professors ever could.

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madez
Sound waves don't add up linearly. However, it is a good enough idealization
for many uses.

Fourier analysis is also approachable from the discrete setting of finite
vectors instead of functions, where the fourier analysis is just an orthogonal
(orthonomal when sanely defined) linear function, i.e. it acts by matrix
multiplication and is represented as that matrix.

This appropriately extended to the continous setting leads to the fourier
transform on functions, and also gives intuition why the fourier transform
uses integrals.

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kranner
This one is related and (I think) quite good:

[https://www.youtube.com/watch?v=r18Gi8lSkfM](https://www.youtube.com/watch?v=r18Gi8lSkfM)

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nwatson
I think it's much easier and more direct to visualize the time-domain as
superposition of helical components and the transform as an exploration of
what happens when you twist the "cylinder" with varying "intensities". You
avoid the vague center-of-mass spike depicted here and start from the get-go
with the terms of the transform.

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jacobolus
Perhaps you can explain what you mean by “exploration of what happens” and
“terms of the transformation”? That’s pretty vague as a description of a
visualization.

Maybe you’re talking about a visualizing a discrete Fourier transform?

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nwatson
Yep, it's vague, sorry, I'll need to try my hand at a video.

~~~
probinso
the source code brown brown blue uses for animations is open source

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probinso
In fMRI data, we refer to frequency space of volumetric image data as K-Space.

I would like a general term for frequency space of a signal, without the use
of the word `frequency` . This is because `frequency` is also used when
describing histograms in general image processing, and is in general an
overloaded term.

Any established words or phrases in the corpus? any tips?

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whatshisface
K-Space is sufficiently general, because "k" is defined as the wavenumber (2pi
over wavelength). That's simply related to frequency in nearly every case
unless your medium is interstellar hydrogen or shockwaves in air or something.
If you need to talk about frequency specifically you could talk about the
period (inversely proportional) or the angular frequency (factor of 2pi).

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ablaba
This fourier transform simulation example from shadertoy is good.
[https://www.shadertoy.com/view/ltKSWD](https://www.shadertoy.com/view/ltKSWD)

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ablaba
This fourier transform simulation example from shadertoy is good.

[https://www.shadertoy.com/view/ltKSWD](https://www.shadertoy.com/view/ltKSWD)

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codegladiator
The link kills my MacBook Pro. Makes the system unresponsive.

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wendyjreichert
How do you animate something like this?

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meseznik
He wrote his own tools in Python to achieve this (repo:
[https://github.com/3b1b/manim](https://github.com/3b1b/manim)).

FAQ: [http://www.3blue1brown.com/about/](http://www.3blue1brown.com/about/)

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wendyjreichert
Thanks!

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tambourine_man
[https://news.ycombinator.com/item?id=16244908](https://news.ycombinator.com/item?id=16244908)

