Sierpinski signal generates $1/f^\alpha$ spectra

TL;DR

This paper analytically studies the Sierpinski automaton's binary pattern as a time series, revealing a unique power spectrum with a 1/f^α decay, offering insights into natural systems exhibiting similar spectral properties.

Contribution

It provides an analytical calculation of the power spectrum of the Sierpinski automaton's signal, linking simple fractal models to 1/f^α spectral behavior in natural phenomena.

Findings

Power spectrum exhibits a power law decay with exponent ~1.15

The Sierpinski signal has a rugged fine structure in its spectrum

Model can represent spectral features of natural systems like reactions and patterns

Abstract

We investigate the row sum of the binary pattern generated by the Sierpinski automaton: Interpreted as a time series we calculate the power spectrum of this Sierpinski signal analytically and obtain a unique rugged fine structure with underlying power law decay with an exponent of approximately 1.15. Despite the simplicity of the model, it can serve as a model for $1/f^\alpha$ spectra in a certain class of experimental and natural systems like catalytic reactions and mollusc patterns.