Eur. Phys. J. B (2017) 90: 229
https://doi.org/10.1140/epjb/e2017-80398-6

Regular Article

1∕f^β noise for scale-invariant processes: how long you wait matters^★

Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 5290002, Israel

^a e-mail: pitangabetty@yahoo.com

Received: 5 July 2017
Received in final form: 4 September 2017
Published online: 27 November 2017

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 functions 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^β noise. We illustrate our general results with two-state renewal models with sojourn times’ distributions having a broad tail.