$1/f^{\beta}$ noise for scale-invariant processes: How long you wait matters

TL;DR

This paper investigates how the measurement timing affects the observed $1/f^{eta}$ noise in nonstationary signals, introducing a generalized aging Wiener-Khinchin theorem and analyzing scale-invariant processes with broad-tailed waiting times.

Contribution

It introduces a generalized aging Wiener-Khinchin theorem linking spectrum and correlation functions for arbitrary measurement windows in nonstationary signals.

Findings

Derived a relation between non-analytical correlation functions and aging $1/f^{eta}$ noise.

Established a connection between measurement timing and observed noise characteristics.

Illustrated results with two-state renewal models with broad-tailed waiting times.

Abstract

We study the power spectrum which is estimated from a nonstationary signal. In particular we examine the case when the signal is observed in a measurement time window $[t_w,t_w+t_m]$, namely the observation started after a waiting time $t_w$, and $t_m$ is the measurement duration. We introduce a generalized aging Wiener-Khinchin theorem which relates between the spectrum and the time- and ensemble-averaged correlation function for arbitrary $t_m$ and $t_w$. Furthermore we provide a general relation between the non-analytical behavior of the scale-invariant correlation function and the aging $1/f^{\beta}$ noise. We illustrate our general results with two-state renewal models with sojourn times' distributions having a broad tail.