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arXiv · 1707.01226

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

Nava Leibovich·Eli Barkai
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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 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.

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Nava Leibovich, Eli Barkai. 2017-07-05. $1/f^{\beta}$ noise for scale-invariant processes: How long you wait matters. https://doi.org/10.1140/epjb%2Fe2017-80398-6

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