Fluctuations of1/fNoise and the Low-Frequency Cutoff Paradox

Abstract

Recent experiments on blinking quantum dots, weak turbulence in liquid crystals, and nanoelectrodes reveal the fundamental connection between 1/f noise and power law intermittency. The nonstationarity of the process implies that the power spectrum is random--a manifestation of weak ergodicity breaking. Here, we obtain the universal distribution of the power spectrum, which can be used to identify intermittency as the source of the noise. We solve in this case an outstanding paradox on the nonintegrability of 1/f noise and the violation of Parseval's theorem. We explain why there is no physical low-frequency cutoff and therefore why it cannot be found in experiments.

Class of perfect1/fnoise and the low-frequency cutoff paradox

“…This wide variety arises because, as we will show, it is enough to consider models derived from multiplicative stochastic equations with exponent μ > 1. They are complementary to the model presented in [2]. In this sense it would be of interest from an experimental point of view to distinguish between both types of processes.…”

“…Both conditions only hold for the class of stationary noise (SN) with α = 0, α s < 0 [3]. But, however, for α s 0 there exists the possibility of being stationary in a weak sense (α = 0) as in the case of the model introduced in [2], which in [3] is generically called the class of stationary (weak sense) fractal curves (SF). Obviously, this class shows integrable spectra (α = 0) and explains the cutoff paradox, although the spectra shift with size as A(T ) ∼ T −2α s .…”

“…The renewal model introduced in [2] was suggested by recent experiments where a kind of intermittent behavior was observed. Intermittent behavior can be easily modeled by chaotic maps [10].…”

“…Series with 1/f β spectra can be generated by using renewal point processes [6] although for β > 1 only the stationary case (with a cutoff in the inter-pulse generation) was considered. Here we take the same model used in [3], which is a generalization of the one used in [2]. It consists of a series of pulses with a renewal process for the interpulse time {τ i } whose time probability follows a power law,…”

“…The so-called low-frequency cutoff paradox appears because there are many experiments, mainly with electronic devices [1], whose spectral density exhibits a 1/f β shape with β 1 but without any apparent cutoff, even at very low frequencies. In a recent Letter [2] a solution to this paradox has been postulated by including the possibility of having spectra whose amplitudes vary with the size of the time series as A(T )/f β . A simple model of dichotomous noise with renewal times serves to support this possibility.…”