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

[Charles Henry]

On 8/20/06, Mathieu Bouchard <matju at artengine.ca> wrote:
> If a signal is random then how can I expect any part of it to be similar
> to any other part of it?
>
> And then, if I look at more general tendencies using bigger signal blocks
> or averaging a lot of signal blocks together, how do I NOT approach a
> theoretical model from probability theory?

If you average signal blocks together, they should approach zero.  How
not to approach it from probability?  I have no idea.

The only reason I know jack about this problem is that I just had a
course on fractional Brownian motion and stochastic calculus this
year.  And still.... it's a friggin devil to wrap your mind around :)

The way these noises are self similar is in terms of their auto-covariance.
The different values of the Hurst exponent, h, determine the exponent,
a, in 1/f^a noise.
A single stochastic process unfolds in a single way (it's a single
deterministic function) out of a large number of possiblities (one big
function space).

A single fractional Brownian motion is a probabilistic function of
time that has 0 mean (relative to the point we start at) at all points
in time, and a variance of t^(2h).

The auto-covariance is a measure of how the function correlates with
itself with different amounts of lag.  For different values of time, s
and t, the auto-covariance of this type of "random" signal is
|t-s|^(2h).
So, in the time domain, it is correlated with itself by using a
probabilistic interpretation. I may be wrong....I had substantial
difficulty with the subject, and I'm certainly interested in finding
out how this stuff really works.

Chuck