1∕fβ noise for scale-invariant processes: how long you wait matters★
Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University,
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Received in final form: 4 September 2017
Published online: 27 November 2017
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 [tw, tw + tm], namely the observation started after a waiting time tw, and tm 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 tm and tw. Furthermore we provide a general relation between the non-analytical behavior of the scale-invariant correlation function and the aging 1∕fβ noise. We illustrate our general results with two-state renewal models with sojourn times’ distributions having a broad tail.
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2017