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

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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.

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Authors and Affiliations

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

    Nava Leibovich & Eli Barkai

Authors
  1. Nava Leibovich
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  2. Eli Barkai
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Correspondence to Nava Leibovich.

Additional information

Contribution to the Topical Issue “Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook”, edited by Ryszard Kutner and Jaume Masoliver.

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Leibovich, N., Barkai, E. 1∕ f β noise for scale-invariant processes: how long you wait matters. Eur. Phys. J. B 90, 229 (2017). https://doi.org/10.1140/epjb/e2017-80398-6

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  • DOI: https://doi.org/10.1140/epjb/e2017-80398-6

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