A general mechanism for the `$1/f$' noise

TL;DR

This paper presents a general mechanism showing how nonlinear transformations of Gaussian noise with a $1/f^{eta}$ spectrum can produce output signals with tunable spectral indices, explaining the widespread occurrence of $1/f$ noise.

Contribution

It demonstrates that nonlinear responses to Gaussian noise can continuously modify the spectral index, offering a universal explanation for $1/f$ noise phenomena.

Findings

Nonlinear devices can alter the spectral index of input noise.

The output spectrum's index can be tuned by the nonlinearity.

This mechanism explains the ubiquity of $1/f$ noise in nature.

Abstract

We consider the response of a memoryless nonlinear device that converts an input signal $\xi(t)$ into an output $\eta(t)$ that only depends on the value of the input at the same time, $t$. For input Gaussian noise with power spectrum $1/f^{\alpha}$, the nonlinearity modifies the spectral index of the output to give a spectrum that varies as $1/f^{\alpha'}$ with $\alpha' \neq \alpha$. We show that the value of $\alpha'$ depends on the nonlinear transformation and can be tuned continuously. This provides a general mechanism for the ubiquitous `$1/f$' noise found in nature.