Memoryless nonlinear response: A simple mechanism for the 1/f noise

Abstract

-Discovering the mechanism underlying the ubiquity of "1/f α " noise has been a longstanding problem. The wide range of systems in which the fluctuations show the implied long-time correlations suggests the existence of some simple and general mechanism that is independent of the details of any specific system. We argue here that a memoryless nonlinear response suffices to explain the observed non-trivial values of α: a random input noisy signal S(t) with a power spectrum varying as 1/f α ′ , when fed to an element with such a response function R gives an output R(S(t)) that can have a power spectrum 1/f α with α < α ′ . As an illustrative example, we show that an input Brownian noise (α ′ = 2) acting on a device with a sigmoidal response function R(S) = sgn(S)|S| x , with x < 1, produces an output with α = 3/2 + x, for 0 ≤ x ≤ 1/2. Our discussion is easily extended to more general types of input noise as well as more general response functions.Although first observed almost ninety years ago and subsequently found to occur in very diverse systems, the origins of the ubiquitous low frequency noise with power spectrum 1/f α where α ≈ 1-called flicker noise or pink noise-has been a long-standing conundrum [1]. Some examples are the fluctuations in voltage across a resistor or other components in electronic equipment [1], river discharges [2, 3], traffic flow [4], the frictional force in sliding friction under wear conditions [5], and the acoustic power in music or speech [6]. Many signals involving response to physical stimuli in living systems, such as fluctuations in the response time of motor-response to a periodic stimulus in humans [7], in the time series of errors of replication of spatial or temporal intervals by memory in humans [8], or in human colour vision [9] have also been found to have 1/f α spectrum. While it seems unlikely that these very diverse types of processes could share a single underlying mechanism, it also seems reasonable that the number of mathematical mechanisms underlying this behaviour would not be very large. Considerable effort has therefore been devoted to discovering these [1].In this Letter, we point out that memoryless nonlinear response (MNR) is sufficient to explain the observed nontrivial values of α. Despite its simplicity, this very general mechanism does not appear to have been explicitly discussed in the literature so far. When the response-time of the system is much shorter than the time-scale of fluctuations under discussion, we may model the system as a nonlinear device which when subjected to the input signal produces a response that is a nonlinear function of the instantaneous value of the input noise. Thus if S(t) is some noise process, we obtain another process R(t), whose value at any time t is related to the value of S(t) at the same time by the transformation: R(t) = h(S(t)), where h is a simple nonlinear function. The process R(t) can have a different value of the spectral power exponent, and we expect that this gives rise to the observed nontrivial val...

Energy fluctuations in one dimensional Zhang sandpile model

“…Astonishingly diverse systems exhibit temporal fluctuations characterized by a lowfrequency 1/f α -type power spectral density (PSD), with the spectral exponent α lying between 0 and 2 [1][2][3][4][5][6][7]. Typical examples range from voltage fluctuations across a resistor to rhythm variations in music [8].…”