Sierpinski signal generates 1/f^α spectra

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.