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 memoryless nonlinear response suffices to explain the observed nontrivial 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.