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1∕fβ noise for scale-invariant processes: how long you wait matters
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Journal Article
1∕fβ noise for scale-invariant processes: how long you wait matters
2017
Overview
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.
Publisher
Springer Berlin Heidelberg,Springer Nature B.V
