
Visualizing the Discrete Fourier Transform - rndn
http://blog.revolutionanalytics.com/2015/09/because-its-friday-visualizing-ffts.html#
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chestervonwinch
Rather than funny analogies about signals spinning on poles, I think it's
relatively easy to understand what the DFT is with just two things that many
people already know: 1) knowledge of the complex exponential definition of
Fourier series [1], and 2) how to approximate integrals with the left
rectangle rule [2].

Take your continuous signal and represent it with a Fourier series. Since the
Fourier series is a linear decomposition into integer frequency sinusoids, the
coefficients of the series tell you the amount of each frequency contained in
the signal. The DFT gives you an approximation of these.

The coefficients of the Fourier series of a function are integrals.
Approximate these integrals with a left Riemann sum. Integrals turn into sums
... sums turn into a linear system ... the linear system turns into a matrix
... bingobango there's the DFT matrix [3].

[1]:
[http://users.wpi.edu/~goulet/Matlab/overlap/efs.html](http://users.wpi.edu/~goulet/Matlab/overlap/efs.html)

[2]:
[https://en.wikipedia.org/wiki/Riemann_sum#Left_Riemann_Sum](https://en.wikipedia.org/wiki/Riemann_sum#Left_Riemann_Sum)

[3]:
[https://en.wikipedia.org/wiki/DFT_matrix](https://en.wikipedia.org/wiki/DFT_matrix)

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blt
I like the linear algebra interpretation, where the DFT is a basis change, and
the only 'magic' is believing/proving that the Fourier basis is orthogonal.
(It's an easy proof.)

I still don't understand how the FFT works though...

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4ad
If you understand DFT, it should be very easy to understand FFT. It's the same
thing but with an infinite-dimensional vector space. The fact that it's a
vector space is the important thing. The infinite dimensionality is relatively
a minor detail.

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ablatner
No that's not right. The FFT is not infinite dimensional. It's the same
dimension as whatever length FFT you take. The FFT is just a set of algorithms
for computing the DFT. In general they work by recursively decomposing an
N-length DFT into shorter length DFT's.

The DTFT on the other hand is "infinite" dimensional in that it takes an
infinite time series as input and outputs a continuous spectrum with period of
2pi.

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4ad
Sorry, This is embarrassing, I must have not drank my coffee... Of course you
are right.

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JshWright
On a similar note, The Engineer Guy's walkthrough of Albert Michelson's
Harmonic Analyzer gave me a much better understanding of Fourier analysis in
general.

[https://www.youtube.com/playlist?list=PL0INsTTU1k2UYO9Mck-i5...](https://www.youtube.com/playlist?list=PL0INsTTU1k2UYO9Mck-i5HNqGNW5AeEwq)

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gruez
Why not link to the original site?
[http://www.billconnelly.net/?p=276](http://www.billconnelly.net/?p=276)

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lobo_tuerto
Some admin should fix the link, the current article don't contribute anything
important over the original.

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Thriptic
I found this video to be an incredibly useful explanation of the Fourier
Transform:
[https://www.youtube.com/watch?v=FjmwwDHT98c](https://www.youtube.com/watch?v=FjmwwDHT98c)

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saintx
I love the way they used color and plain English to describe that function. It
would be awesome if a general app for this sort of color coded simple
translation were available to help kids learn about mathematics. It'd have to
be more inclusive for people with dichromacy and anamalous trichromacy, but
there could be settings in the app to compensate for that.

