[PD] Colored (fractal, 1/f^a) noise generator for PD (fwd)

[Mathieu Bouchard]

this was supposed to go to the list.

---------- Forwarded message ----------
Date: Mon, 21 Aug 2006 01:10:13 +0200
From: jose manuel berenguer <jmbeal at telefonica.net>
To: Mathieu Bouchard <matju at artengine.ca>
Subject: Re: [PD] Colored (fractal, 1/f^a) noise generator for PD

El 20/08/2006, a las 20:58, Mathieu Bouchard escribió:
> On Sun, 20 Aug 2006, Charles Henry wrote:
>
>> 1/f^a noise is a fractal because it is self similar under
>> different size windows. 1/(kf)^a = 1/k^a * 1/f^a, so the magnitude
>> of spectrum viewed over a different window is just scaled by a
>> certain amplitude.  The same applies to the time domain as well
>> (but it has to be interpreted as a probability function).
>
> Then by this standard, the 1/x function is self-similar, and so are
> all hyperbolas. That is, as long as similarity is defined as modulo
> the group of diagonal matrices conjugated by rotation matrices.
>

this is true. but fractals are compact (closed and bounded) that
shows self-similarity and fractional dimension. f(x) = 1/x is self-
similar, but its dimension is 1 and it is not a compact set.


> Isn't the definition of fractal requiring some kind of noninteger
> Hausdorff dimension?

yes, indeed.  and this is the case of that noises. don't look at
theoretical spectra in frequency domain but signals in time
domain ... their dimension is fractional and they show self-
similarity.  I haven't now time to show they are compact sets, but,
for shure, a proof could be found in Mandelbrot, Barnsley, Peitgen,
etc ...

jmb

jose manuel berenguer
jmbeal at telefonica.net  +34932857046 +34696538403. http://
www.sonoscop.net/jmb/
jmberenguer at sonoscop.net +34933064128. http://www.sonoscop.net/
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