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

Abstract

Discovering the mechanism underlying the ubiquity of $"1/f^{\alpha }"$ noise has been a long--standing 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 {\it memoryless nonlinear response} suffices to explain the observed non--trivial values of $\alpha$: a random input noisy signal $S(t)$ with a power spectrum varying as $1/f^{{\alpha }^{\prime }}$, when fed to an element with such a response function $R$ gives an output $R(S(t))$ that can have a power spectrum $1/f^{\alpha }$ with $\alpha <{\alpha }^{\prime }$. As an illustrative example, we show that an input Brownian noise (${\alpha }^{\prime }=2$) acting on a device with a sigmoidal response function $R(S)= \sgn(S)|S|^x$, with $x<1$, produces an output with $\alpha =3/2+x$, for $0\le x\le 1/2$. Our discussion is easily extended to more general types of input noise as well as more general response functions.

5 pages, 5 figures

References in corpus (1)

• Power spectrum of mass and activity fluctuations in a sandpile

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• $1/f$ noise from the sequence of nonoverlapping rectangular pulses
• Energy fluctuations in one dimensional Zhang sandpile model