Phase Noise and Noise Induced Frequency Shift in Stochastic Nonlinear Oscillators

Phase noise plays an important role in the performances of electronic oscillators. Traditional approaches describe the phase noise problem as a purely diffusive process. In this paper we develop a novel phase model reduction technique for phase noise analysis of nonlinear oscillators subject to stochastic inputs. We obtain analytical equations for both the phase deviation and the probability density function of the phase deviation. We show that, in general, the phase reduced models include non Markovian terms. Under the Markovian assumption, we demonstrate that the effect of white noise is to generate both phase diffusion and a frequency shift, i.e. phase noise is best described as a convection-diffusion process. The analysis of a solvable model shows the accuracy of our theory, and that it gives better predictions than traditional phase models.