NIST Technical Note 1337: Characterization of Clocks and Oscillators

214SAMUEL R. STEINTABLE 12·3Variance of the Relative Difference between the SampleAllan Variance and the True Value (C./N)·Noise type CI C.White phase 2 3.88Flicker phase 1 3.88White frequency 0 2.99Flicker freq uency -I 2.31Random-walk frequency -2 2.25• N is the number of phase measurements. The result isaccurate to beller than 10 ~~ for N larger than 10.The quantity A(N) has mean zero. For N larger than 10, the variance of A isapproximately0- 2 (.1.) = CJN. (11-41)Table 11-3 gives the constant C a for the five major noise types.Using the same set of data it is also possible to estimate the Allan variancesfor integer multiples of the base sampling interval t = mto. Now thepossibilities for overlapping sample Allan variances are even greater. For adata set of N phase points, one can obtain a maximum of exactly N - 2msample Allan variances for r = mro. Of course only (N - 1)2m of these aregenerally independent. Still, the use of all of the data is well justified since theconfidence of the estimate is always improved by so doing. Consider the caseof an experiment extending for several weeks in duration with the aim ofgetting estimates of the Allan variance for r values equal to a week or more.As always, the purpose is to estimate the "'true" Allan variance as well aspossible, that is, with as tight an uncertainty as possible. Thus, one wants touse the data as efficiently as possible. The most efficient use is to average allpossible sample Allan variances ofa given r value that one can compute fromthe data. This procedure is illustrated in Fig. 12-14.In order to calculate confidence intervals for a sample variance, it isnecessary to know the number of degrees of freedom. This has been done byboth analytical and Monte Carlo techniques, and empirical equations havebeen found that are accurate to 1% for white phase, white frequency, andrandom-walk frequency modulation. The tolerance is somewhat larger forflicker frequency and phase modulation (Howe et al., 1981). The empiricalequations for the degrees of freedom are given in Table 12-4. Table 12-5 givesthe degrees of freedom for selected values of N, the tot31 number of phasevalues, and m, the number of intervals averaged. Figure 12-15 illustrates thenumber of degrees of freedom for all noise processes as a function of t for thecase of 101 total phase measurements.IN-84