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

[Martin Peach]

Charles Henry wrote:
> 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?
>
> ...
> 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.
>
Yes, it's not instantaneously self-similar, it's statistically 
self-similar and the autocorrelation and autocovariance both increase as 
a increases from zero in 1/(f^a).
When a = 0 you have white noise and the signal is randomly correlated 
with itself, but as you add low-frequency power to the signal, the 
correlation at any given lag will have to increase because of the low 
frequencies present.
Beyond red noise there is also 'long-tailed' noise (for example: 
http://www.pnas.org/cgi/content/full/102/13/4771 or the stock market) 
where there is more correlation than expected at greater lag, or 
clumping or non-gaussian distribution of values. This doesn't seem to be 
covered by the 1/(f^a) concept.

Martin