Point process model of 1/f noise versus a sum of Lorentzians

TL;DR

This paper introduces a simple point process model for 1/f^β noise that does not require a wide distribution of relaxation times, providing explicit spectral expressions and connecting to existing models.

Contribution

The paper presents a novel point process model for 1/f^β noise that relies on a single relaxation rate and multiplicative noise, differing from traditional sum-of-Lorentzians models.

Findings

Model produces 1/f^β spectra over wide frequency ranges.

Explicit formulas for power spectra are derived.

The model exhibits power-law distribution of signal intensity.

Abstract

We present a simple point process model of $1/f^{\beta}$ noise, covering different values of the exponent $\beta$. The signal of the model consists of pulses or events. The interpulse, interevent, interarrival, recurrence or waiting times of the signal are described by the general Langevin equation with the multiplicative noise and stochastically diffuse in some interval resulting in the power-law distribution. Our model is free from the requirement of a wide distribution of relaxation times and from the power-law forms of the pulses. It contains only one relaxation rate and yields $1/f^ {\beta}$ spectra in a wide range of frequency. We obtain explicit expressions for the power spectra and present numerical illustrations of the model. Further we analyze the relation of the point process model of $1/f$ noise with the Bernamont-Surdin-McWhorter model, representing the signals as a sum of the uncorrelated components. We show that the point process model is complementary to the model based on the sum of signals with a wide-range distribution of the relaxation times. In contrast to the Gaussian distribution of the signal intensity of the sum of the uncorrelated components, the point process exhibits asymptotically a power-law distribution of the signal intensity. The developed multiplicative point process model of $1/f^{\beta}$ noise may be used for modeling and analysis of stochastic processes in different systems with the power-law distribution of the intensity of pulsing signals.