$1/f^{\alpha }$ spectra in elementary cellular automata and fractal signals

Abstract

We systematically compute the power spectra of the one-dimensional elementary cellular automata introduced by Wolfram. On the one hand our analysis reveals that one automaton displays $1/f$ spectra though considered as trivial, and on the other hand that various automata classified as chaotic/complex display no $1/f$ spectra. We model the results generalizing the recently investigated Sierpinski signal to a class of fractal signals that are tailored to produce $1/f^{\alpha }$ spectra. From the widespread occurrence of (elementary) cellular automata patterns in chemistry, physics and computer sciences, there are various candidates to show spectra similar to our results.

arXiv comment4 pages (3 figs included)

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• Theory of Branching and Annihilating Random Walks
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• Sierpinski signal generates $1/f^{\alpha }$ spectra

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• Quantum walks with spatiotemporal fractal disorder
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