Statistical study of phase fluctuations and oscillator stability

The stability of amplitude-stabilized oscillators is limited by microscopic or macroscopic random phase fluctuations. The authors study the relation between phase and instantaneous frequency and deduce that in general the phase is a process with stationary increments. Some properties of such processes are presented, and in particular the cases where the phase cannot be stationary is discussed. To describe the frequency stability a new method of analysis, the method of finite-time frequency control, is presented and compared with some others previously presented, in particular the Allan-variance procedure. The advantage of the method appears in its definition of short- and long-term stability, its low-frequency spectral analysis, and its determination of the precision obtained in finite-time measurements.