4. Cellular Automata and Fractal Evolution

4.F. 1/f Noise

Most of the fractals we have seen, both mathematical and natural, are generated by some dynamical process. For example,
  IFS generate fractal by iterating a collection of transformations,
  fractal coastlines grow by long-term erosion from wave action, and so on.
Yet usually we see just the final product, not the process.
Cellular automata make the time-dependence more obvious: time records of some CA are fractals (with the gasket appearing frequently - no surprise here).
We discuss three different kinds of noise and describe a CA experiment searching for 1/f noise.
A first tool in analyzing noise is the power spectrum.
  The simplest is white noise: a monkey banging on a piano, each note is unrelated to what came before.
  Opposite this is Brownian noise: a cat chasing a moth across a piano, each note is based on the previous note, but the change in notes is unrelated to the previous change.
  Midway between these is 1/f noise.
Interestingly, Richard Voss found 1/f characteristics in almost all music.
Though there is no general agreement about the mechanism producing 1/f noise, there are some attempts at explanation.
Finally, here is an experiment testing the relation between 1/f noise and complex CA.