We consider a class of nonlinear stochastic differential equations, giving the power-law behavior of the power spectral density in any desirably wide range of frequency. Such equations were obtained starting from the point process models of $1/f^{\beta }$ noise. In this article the power-law behavior of spectrum is derived directly from the stochastic differential equations, without using the point process models. The analysis reveals that the power spectrum may be represented as a sum of the Lorentzian spectra. Such a derivation provides additional justification of equations, expands the class of equations generating $1/f^{\beta }$ noise, and provides further insights into the origin of $1/f^{\beta }$ noise.

• Received 20 October 2009

DOI:https://doi.org/10.1103/PhysRevE.81.031105