Phase noise in oscillators: A unifying theory and numerical methods for characterization

Phase noise is a topic of theoretical and practical interest in electronic circuits, as well as in other fields, such as optics. Although progress has been made in understanding the phenomenon, there still remain significant gaps, both in its fundamental theory and in numerical techniques for its ch aracterization. In this paper, we develop a solid foundation for phase nois e that is valid for any oscillator, regardless of operating mechanism. We e stablish novel results about the dynamics of stable nonlinear oscillators i n the presence of perturbations, both deterministic and random. We obtain a n exact nonlinear equation for phase error, which we solve without approxim ations for random perturbations. This leads us to a precise characterizatio n of timing jitter and spectral dispersion, for computing which we develop efficient numerical methods. We demonstrate our techniques on a variety of practical electrical oscillators and obtain good matches with measurements, even at frequencies close to the carrier, where previous techniques break down. Our methods are more than three orders of magnitude faster than the b rute-force Monte Carlo approach, which is the only previously available tec hnique that can predict phase noise correctly.