Variances based on data with dead time between the measurement

The accepted definition of frequency stability in the time domain is the two-sample variance (or Allan variance). It is based on the measurement of average frequencies over adjacent time intervals, with no "dead timen between the intervals. The primary advantages of the Allan variance are that (1) it is convergent for many actual noise types for which the conventional variance is divergent; (2) it can distinguish between many important and different spectral noise types; (3) the two-sample approach relates to many practical implementations, for example, the rms change of an oscillator's frequency from one period to the next; and (4) Allan variances can be easily estimated a t integer multiples of the sample interval. In 1974 a table of bias functions which related variance estimates with various config-urations of number of samples and dead time to the two-sample (or Allan) variance was The tables were based on noises with pure power-law power spectral densities. Often situations recur which unavoidably have distributed dead time between mea-surements, but still the conventional variances are not convergent. Some of these applications are outside of the time and frequency field. Also, the dead times are often distributed throughout a given average, and this distributed dead time is not treated in the 1974 tables. This paper reviews the bias functions B1(N, r,p), and B2(r,p) and introduces a new bias function, B3(2, r,p), to handle the commonly occurring cases of the effect of dis-tributed dead time on the computed variances. Some convenient and easy to interpret asymptotic limits are reported. The actual tables will be published elsewhere in book form