Coupled nonlinear stochastic differential equations generating arbitrary distributed observable with 1/f noise

TL;DR

This paper introduces a system of two coupled nonlinear stochastic differential equations that generate 1/f noise with customizable steady-state distributions, broadening the modeling capabilities of 1/f noise sources.

Contribution

It generalizes existing models by coupling two equations, enabling 1/f noise with arbitrary steady-state distributions, derived from scaling properties.

Findings

Achieves 1/f spectrum over a wide frequency range

Allows arbitrary steady-state probability densities

Provides a flexible framework for modeling 1/f noise

Abstract

Nonlinear stochastic differential equations provide one of the mathematical models yielding 1/f noise. However, the drawback of a single equation as a source of 1/f noise is the necessity of power-law steady-state probability density of the signal. In this paper we generalize this model and propose a system of two coupled nonlinear stochastic differential equations. The equations are derived from the scaling properties necessary for the achievement of 1/f noise. The first equation describes the changes of the signal, whereas the second equation represents a fluctuating rate of change. The proposed coupled stochastic differential equations allows us to obtain 1/f spectrum in a wide range of frequencies together with the almost arbitrary steady-state density of the signal.