https://blog.revolutionanalytics.com/2014/01/the-fourier-transform-explained-in-one-sentence.html

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January 03, 2014

The Fourier Transform, explained in one sentence

If, like me, you struggled to understand the Fourier Transformation
when you first learned about it, this succinct one-sentence
colour-coded explanation from Stuart Riffle probably comes several
years too late:

The fourier transform, explained in one color-coded sentence

Stuart provides a more detailed explanation here. This is the formula
for the Discrete Fourier Transform, which converts sampled signals
(like a digital sound recording) into the frequency domain (what
tones are represented in the sound, and at what energies?). It's the
mathematical engine behind a lot of the technology you use today,
including mp3 files, file compression, and even how your old AM radio
stays in tune.

The daunting formula involves imaginary numbers and complex
summations, but Stuart's idea is simple. Imagine an enormous speaker,
mounted on a pole, playing a repeating sound. The speaker is so
large, you can see the cone move back and forth with the sound. Mark
a point on the cone, and now rotate the pole. Trace the point from an
above-ground view, if the resulting squiggly curve is off-center,
then there is frequency corresponding the pole's rotational frequency
is represented in the sound. This animated illustration (click to see
it in action) illustrates the process:

FFT animation

The upper signal is make up of three frequencies ("notes"), but only
the bottom-right squiggle is generated by a rotational frequency
matching one of the component frequencies of the signal.

By the way, no-one uses that formula to actually calculate the
Discrete Fourier Transform -- use the Fast Fourier Transform instead,
as implemented by the fft function in R. As the name suggests, it's
much faster.

AltDevBlog: Understanding the Fourier Transform (note: updated link
20 Oct 2015 with active mirror)

Posted by David Smith at 13:30 in R, random | Permalink

Comments

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Very interesting article, thank you. Please take a moment to rephrase
the following key statement, if you would: "...then there is
frequency corresponding the pole's rotational frequency is
represented in the sound."

Posted by: C. Griffith | January 04, 2014 at 10:09

polar form e^ith is equal to the rectangular form costh+isinth and
corresponds to the coordinates (costh,sinth) such that
e^i0 = 1 = (1,0)
e^it/4 = i = (0,1)
e^it/2 = -1 = (-1,0)
e^it3/4 = -i = (0,-1)
e^it = 1 = (1,0)

Posted by: MasterG | January 04, 2014 at 16:33

May I suggest a minor exception to your claim about FFT: most modern
languages, R included, use some variation of the "pure" 2^N
Cooley-Tukey FFT algorithm as appropriate to support factors of 3, 5,
etc. in the length of the dataset, and even default to the "raw" DFT
for other data lengths (unless specifically suppressed by the user).
And, of course, the FFT is in fact that equation, just with gobs of
like terms grouped together. :-)

Posted by: Carl Witthoft | January 06, 2014 at 08:40

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