A general mechanism for the `$1/f$' 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 $1/f^{\alpha}$, 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.