NIST Technical Note 1337: Characterization of Clocks and Oscillators

NIST Technical Note 1318, 1990.VARIANCES BASED ON DATA WITH DEAD TIME BETWEEN THE MEASUREMENTSJames A. BarnesAustron, Inc.Boulder, Colorado 80301andDavid W. AllanTime and Frequency DivisionNational Institute of Standards and TechnologyBoulder, Colorado 80303The accepted definition of frequency stability in the time domain isthe two-sample variance (or Allan variance). It is based on themeasurement of average frequencies over adjacent time intervals,with no "dead time" between the intervals. The primary advantagesof the Allan variance are that (1) it is convergent for manyencountered noise models for which the conventional variance isdivergent; (2) it can distinguish between many important anddifferent spectral noise types; (3) the two-sample approach relatesto many practical implementations; for example, the rms change of anoscillator's frequency from one period to the next; and (4) Allanvariances can be easily estimated at integer multiples of the sampleinterval.In 1974 a table of bias functions which related variance estimateswith various configurations of number of samples and dead time tothe Allan variance was published [1]. The tables were based onnoises with pure power-law spectral densities.Often situations occur that unavoidably have dead time betweenmeasurements, but still the conventional variances are notconvergent. Some of these applications are outside of the time-andfrequencyfield. Also, the dead times are often distributedthroughout a given average, and this distributed dead time is nottreated in the 1974 tables.This paper reviews the bias functions Bl(N,r,~), and B2(r,~) andintroduces a new bias function, B3(2,M,r,~), to handle the commonlyoccurring cases of the effect of distributed dead time on thecomputed variances. Some convenient and easy-to-interpretasymptotic limits are reported. A set of tables for the biasfunctions are included at the end of this paper.Key words: Allan variance; bias functions; data sampling and deadtime; dead time between the measurement; definition of frequencystability; distributed dead time; two-sample variance1TN-296