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

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

Single-trajectory spectral analysis of scaled Brownian motion

“…where μ and σ 2 are defined in equations (11) and (12), respectively. Result (15) shows that the PDF of the singletrajectory PSD for SBM is fully defined through its first and second moment, and that it has exactly the same functional form as the results for Brownian motion and FBM derived in [3,4]. As we have already remarked, this is a direct consequence of the Gaussian nature of the parental process X α (t) of SBM.…”

“…The calculation of the exact explicit forms of the properties defined in equations (10) to (12) represents the chief goal of our work.…”

“…The variance of the single-trajectory PSD is defined in equation (12). It can be calculated exactly for arbitrary α, f and T , and the details of the intermediate steps are presented in Appendix A.…”

“…This includes, for instance, the variation of the loudness of musical performances [5], the temporal evolution of climate data [6] and of the waiting-times between earthquakes [7], the retention times of chemical tracers in groundwater [8] and noises in graphene devices [9], fluorescence intermittency in nano-devices [10], current fluctuations in nanoscale electrodes [11], or ionic currents across nanopores [12]. The PSD was also calculated analytically for individual time series in a stochastic model describing blinking quantum dots [13], for non-stationary processes taking advantage of a generalised Wiener-Khinchine theorem [14,15], for the process of fractional Brownian motion (FBM) with random reset [16], the running maximum of a Brownian motion [17], as well as for diffusion in strongly disordered Sinai-type systems [18], to name but a few stray examples.…”

“…A T -dependent evolution of the PSD can in fact be rather peculiar and system dependent. For instance, the PSD may be ageing and its amplitude may decay with T , as it happens for nonstationary random signals [12], or conversely, it can exhibit an unbounded growth with T , a behaviour predicted analytically and observed experimentally for superdiffusive processes of fractional Brownian motion (FBM) type [17]. As a consequence, the standard textbook definition (2) of the PSD which emphasises the limit T → ∞, can become rather meaningless.…”