Frequency-Counted Measurements and Phase Locking to Noisy Oscillators

The phase-error variance of a phase-locked loop is dependent on the stability of its voltage-control oscillator and that of the source oscillator being tracked. Statistics relative to oscillator stability are commonly gathered by counted-frequency techniques and are so specified in manufacturers' data. In order thus to predict the performance of a loop using such an oscillator, it is necessary to know how to relate counter data to loop error. This paper presents such a method based on a simple but realistic model of oscillator noises and shows that the mean sample variance of the counted-frequency method converges rather curiously and slowly to the actual variance. The model for the flicker component of the noise is physically realistic (finite power) and allows one to find the range of validity for the usual formal calculations for the sample variance of the counted frequency. In addition, insight is gained into the relation between the sample variance and the actual finite variance of the realistic model. The effect of oscillator instability on a first-order loop is longterm steady-state phase-error drift and short-term zero-mean fluctuation about this steady state. For the second-order loop, the steady-state drift disappears.