Characterization of Frequency Stability: A Transfer Function Approach and Its Application to Measurements via Filtering of Phase Noise

Frequency stability of high-quality signal sources is characterized in the Fourier frequency domain by the spectral density Sy(f) of the fractional instantaneous frequency deviation y(t), and in the time domain by the Allan variance ¿y2(¿). Two well-known types of measuring apparatus used to evaluate these parameters are analog spectrum analyzers and digital electronic counters, respectively. A detailed analysis of the structure of the relation between ¿y2(¿) and Sy(f) shows that it is possible to define a variance, i.e., a time-domain measure, by its transfer function in the Fourier frequency domain, even when no corresponding measurement sequence exists in the time domain. Two different kinds of variance are then defined, which possess different properties for white and flicker phase noises. One of these variances is an estimate of the Allan variance. These variances may be measured by a suitable filtering of phase noise at the output of a phase detector.