NIST Technical Note 1337: Characterization of Clocks and Oscillators

Thus. with 9~ probability the calculated samplevariance, S2, satisfies the inequality:3.94 ~ (d.f~~.s2 < 18.3 (5.5)and this inequality can be rearranged in the form1. 64 ~ 0 2 < 7. 61 (5.6)or, taking square roots:1. 28 ~ " .: 2.76 (5.7)Now someone might object to the form ofeq (5.7) since it seems to be saying thatthe true sigma falls within two limits with 90%probabil ity.Of course, this ;s either true ornot and is not subject to a probabilistic interpretation.Actually eq (5.7) is based onthe idea that the true sigma is not known and weestimate it with the square root of a samplevariance, S2.This sample variance is a randomvariable and is properly the subject of probability,and its value (Which happened to be 3.0 inthe example) will conform to eq (5.7) ninetimes out of ten.Typically, the sample variance is calculatedfrom a data sample using the relation:NS2 = N=l 1: (x - x)2 (5.8)n=1 nwhere it is implicitly assumed that the xn's arerandom and uncorre1ated (i.e., white) and where xis the sample mean calculated from the same dataset. If all of this is true, then S2 is chi-squaredistributed and has N-l degrees of freedom.Thus, for the case of white x ' and a convenntional sample variance (i.e., eq (5.8», thenumber of degrees of freedom are given by theequation:d. f. N-l (5.9)The prob 1em of interest here is to obtai n thecorresponding equations for Allan Variances usingoverlapping estimates on various types of noise(i. e. , white FM, fl iclc.er FM, etc.).Yoshimura)Other authors (Lesage and AUdoin, andhave considered the question of thevariance of the Allan Variances without regard tothe distributions.re1ated prob1emresults.This is, of course, a closelyand use will be made of thei rThese authors considered a more restrictiveset of overlapping estimates than will beconsidered here, however.VI.MAXIMAL USE OF THE DATA AND DETERMINATION OFTHE DEGREES OF FREEDOM.6.1 Use of DatacomparedCons i der the case of two asci 11 ators bei ngin phase and exactly N values of thephase di fference are obtai ned.Assume that thedata are taken at equally spaced intervals, to'Fromthese N phase values, one can obtain N-lconsecuti ve values of average frequency and fromthese one can compute N-2 individual, sample AllanVariances (not all independent) for t = to' TheseN-2 values can be averaged to obtain an estimateof the Allan Variance at t = to' The variance ofthis variance has been calculated by the aboveci ted authors.Using the same set of data, it is also possibleto estimate the Allan Variances for integermultiples of the base sampling interval, t =nt 'oNow the possibilities for overlapping sample AllanVariances are even greater.For a phase data setof N points one can obtain exactly N-2n sampleAllan Variances for t =nt ' Of course only aofracti on of these are genera11 y independent.Still the use of ALL of the data ;s well justified(see fig. 6.1).Consider the case af' an experiment extendingfor several weeks in duration with the aim ofgetting estimates of the Allan Variance for tauva1ues equal to a week or more. As always thepurpose is to estimate rel iably the "true" AllanVariance as well as possib1e--that is, with astight an uncertainty as possible. Thus one wants15TN-28