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 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 $α$: 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 $0 \leq x \leq 1/2$. Our discussion is easily extended to more general types of input noise as well as more general response functions.