A general mechanism for the `$1\mathrm{/}f$' noise

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 $1\mathrm{/}{f}^{\alpha }$, the nonlinearity modifies the spectral index of the output to give a spectrum that varies as $1\mathrm{/}{f}^{{\alpha }^{\mathrm{\prime }}}$ with $\alpha’ \neq \alpha$\mathrm{.}Weshowthatthevalueof$\alpha’$ depends on the nonlinear transformation and can be tuned continuously. This provides a general mechanism for the ubiquitous `$1\mathrm{/}f$’ noise found in nature.