Universal spectral densities: white and flicker noises

Abstract

In this paper we introduce and analyze a model of a random collection of random oscillators. The model has a random number of oscillators, the oscillators have random amplitudes and random frequencies, and the model’s output is the aggregate of its oscillators’ outputs. Also, the model has two time scales: a ‘human’ time scale, over which the spectral density of the model’s output is measured; and a ‘cosmic’ or a ‘geological’ time scale, over which the model’s random parameters slowly evolve. Analyzing the model we establish that, with respect to the evolution of the oscillators’ frequencies: (i) general random-walk dynamics universally yield white noise, i.e. flat spectral densities; (ii) general geometric random-walk dynamics universally yield 1/f noise, i.e. harmonic spectral densities; and (iii) general Gaussian geometric random-walk dynamics universally yield white noise and flicker noise, i.e. inverse power-law spectral densities.