NIST Technical Note 1337: Characterization of Clocks and Oscillators

12 FREQUENCY AND TIME MEASUREMENT 2Q1x(t)I II IIt-r-I'I II,T-ITIMEFIG.12-3 lIiklsuremenr process for the computation of the sample ,·ariance. The phasedifference between two oscillators is plotted on the ordinate. The measurement yields a set offrequencies averaged over equal intervals t separated by dead time T - t.from which it follows that(12-20)The quality r is often referred to as the sampling time or the averaging time.Eq uations (12-19) and (12-20) are not the only way to define mean freq uency.but they are the simplest. Other definitions lead to alternative measures ofstability that may have desirable properties.Suppose that one has measured the time or frequency fluctuations betweena pair of precision oscillators and a stability analysis is desired. The process isillustrated in Fig. 12-3. These are N values of the fractional frequency y;.Each one is measured over a time t, and measurements are repeated afterintervals of time T If the measurement repetition time exceeds the averagingtime, then there is a dead time equal to T - t between each frequencymeasurement, during which there is no information available.There are many ways to analyze these data. A fairly general approach is theN-sample variance defined by the relation(12-21)where the angle brackets denote the infinite time average. Frequently, Eq.(12-21) does not converge as N ...... x, since some noise processes in oscillatorsdiverge rapidly at low Fourier frequencies. This implies that the precisionwith which one estimates the variance does not improve simply as the samplesize is increased. For this reason, the two-sample variance with no dead timeis preferred. Also called the Allan variance, it converges for all the majornoise types observed in precision oscillators. It may be written as(12-22)TN-71