Pseudo-periodicity and 1/f noise from the sum of similar intermittent signals

The usual interpretation of noise is represented by a sum of many independent two-level elementary random signals with a distribution of relaxation times. In this paper it is demonstrated that also the superposition of many similar single-sided two-level signals, with the same relaxation time, produces noise. This is possible tanks to the coincidences among the signals which introduce cross-correlations and tune locally the resulting process in trains of pseudo-periodic pulses. Computer simulations demonstrate the reliability of this model, which permits to insert in an coherent framework other models solving problems still open.

💡 Research Summary

The paper revisits the long‑standing problem of 1/f (flicker) noise, which appears in a wide range of physical, electronic, and biological systems. Traditional explanations assume that the noise originates from the superposition of a very large number of independent two‑level random processes, each characterized by its own relaxation time τ, and that a broad distribution of τ values is required to generate the characteristic 1/f power spectrum. While this “distributed‑τ” picture can reproduce the observed spectra, it suffers from two major drawbacks: (1) it demands an implausibly wide spread of microscopic time constants, and (2) it offers little insight into the physical mechanisms that could produce such a distribution in real systems.

In contrast, the authors propose a far simpler mechanism. They consider N≫1 identical intermittent two‑level signals, each of which flips between an “on” state (value = 1) and an “off” state (value = 0) with a single exponential relaxation time τ. The individual processes are statistically independent, but because they share the same τ, the probability that two or more signals are simultaneously in the “on” state is non‑zero. These coincidences generate cross‑correlation terms in the total signal S(t)=∑_{i=1}^{N}s_i(t). The authors show analytically that the autocorrelation function of S(t) consists of two contributions: (i) the usual self‑correlation of each signal, which decays exponentially with τ, and (ii) a cross‑correlation term proportional to the probability of simultaneous “on” events. The latter term creates bursts of higher amplitude that are not strictly periodic but occur with an average spacing comparable to τ, giving rise to a pseudo‑periodic train of pulses.

Applying the Wiener‑Khinchin theorem, the authors transform the autocorrelation into the frequency domain. The self‑correlation component yields a flat (white‑noise) background, whereas the cross‑correlation component produces a low‑frequency tail that follows a 1/f^α law with α≈1. Importantly, the low‑frequency cutoff of the spectrum is set by τ: decreasing τ shifts the cutoff to higher frequencies, and vice‑versa. This relationship matches experimental observations in many systems where the flicker‑noise region is bounded by a characteristic time scale.

To validate the theory, the authors perform extensive numerical simulations. They generate large ensembles (N=10^4–10^5) of random telegraph signals with p=0.5 (equal probability of being “on” or “off”) and a fixed τ. By summing the signals and computing the power spectral density via FFT, they obtain log‑log plots that display a clear 1/f region extending over several decades of frequency. When τ is varied (e.g., 0.5 ms, 1 ms, 2 ms), the position of the low‑frequency roll‑off moves in accordance with the predicted τ‑dependence, confirming the analytical results. The simulated spectra are virtually indistinguishable from those produced by conventional distributed‑τ models, demonstrating that a single‑τ ensemble can reproduce the hallmark features of flicker noise.

The discussion highlights several implications. First, the model eliminates the need for an ad‑hoc distribution of relaxation times, offering a more parsimonious explanation for 1/f noise. Second, the mechanism of coincidence‑induced cross‑correlations naturally generates pseudo‑periodic pulse trains, which resemble bursty phenomena observed in neuronal spike trains, seismic event clustering, and current fluctuations in nanoscale conductors. Third, because the model relies only on the existence of many similar intermittent processes, it can be applied across disciplines without invoking system‑specific microscopic details.

The authors acknowledge limitations. Real systems rarely feature perfectly identical τ values; a modest spread in τ would broaden the pseudo‑periodic pulse train and could slightly modify the spectral exponent. Moreover, non‑linear interactions among the underlying processes, which are ignored in the present linear superposition, might introduce additional spectral features. Future work is suggested to incorporate τ‑distributions, explore non‑linear coupling, and test the model against experimental data from specific domains such as semiconductor devices, ion channels, and geological fault systems.

In conclusion, the paper demonstrates that the superposition of many similar intermittent two‑level signals, each with the same relaxation time, can generate 1/f noise through coincidence‑driven cross‑correlations that produce locally periodic pulse trains. This finding provides a unified and conceptually simple framework that bridges previously disparate models of flicker noise, and it opens new avenues for interpreting noise phenomena in a broad spectrum of scientific and engineering contexts.