Title: Nonstationary Power Spectrum and Aging 1/ ƒ β Noise
Author: Nava Leibovich
The power spectrum of a stationary process is calculated using the Wiener – Khinchin theorem which gives the connection between the power spectrum and the correlation function of the observed process. In many experiments the spectrum exhibits 1/fβ noise, i.e. the spectral density is observed at low frequencies as S(ω) ∼ ω−β where 0 < β < 2. This power-law behavior seems unphysical since the total energy diverges, thus the noise cannot be described with the stationary Wiener – Khinchin theorem. We have found that the 1/ƒβ noise possess nonstationarity, i.e. the spectrum is time-dependent. The time dependence of the 1/ƒβ resolves the so called “1/ƒ paradox”.
The nonstationary 1/fβ noise is backed by only two noteworthy experimental evidences. The spectrum of intermittent quantum dots was measured showing that the 1/fβ noise ages as the measurement time is increased, indicating a nonstationary behavior. Furthermore this aging 1/fβ behavior was measured in the interface ﬂuctuations in the (1+1)-dimensional Kardar-Parisi-Zhang universality class . However, in many other processes, the 1/fβ noise does not exhibit experimental evidences of nonstationarity and the famous paradox apparently remains open. The tension between the requirement of time-dependent 1/fβ noise and the experimental evidences which support the stationarity is reduced in three levels: (i) we have shown that an unbounded process may present an appearance of time-independent 1/fβ noise, while for bounded process the 1/fβ noise ages , (ii) in macroscopic measurements the spectrum appears stationary while in the single-particle measurements the aging 1/fβ is recovered , and (iii) if the ﬁxed waiting time is much longer than the measurement time the 1/fβ does not depend on the measurement time