Degrees of freedom of the estimate of the two-sample variance in the continuous sampling method

The two-sample variance, the frequency stability measure in the time domain, is defined in terms of the infinite time average. The estimate of the variance obtained from a finite data set must be accompanied by a confidence interval. Theoretical equations are derived for the variance or degrees of freedom in the chi-square distribution for the continuous sampling method to make more efficient use of a finite set of sampled time data. The results are plotted for degrees of freedom and show that there is considerable improvement in the phase modulation (PM) noise compared with the results for τ-overlap sampling, because of an increased number of τ-averaged frequency samples to be obtained from the time data. For white, flicker, and random-walk frequency modulation (FM) noise the improvements converge to about 100, 30, and 4%, respectively. The reasonableness of the assumption of stationarity of the random process is discussed