NIST Technical Note 1337: Characterization of Clocks and Oscillators

where aZ(N,T,r) is the expected sample variance given in eq (1) and based on Nmeasurements at intervals T and averaged over a time rand r = Tlr. In words,B1(N,r,p) is the ratio of the expected variance for N measurements to theexpected variance for two samples (everything else held constant). Thevariances on the right in eq (3) depend implicitly on the noise type eventhough p or 0 are not shown as independent variables. The noise-typeparameter, p, is shown as an independent variable for all of the biasfunctions in this paper, because the values of the ratio of these variancesexplicitly depend on p as will be derived later in the paper. Allan showedthat if Nand r are held constant, then the 0, p relationship shown in figure2 is the same; that is, we can still infer the spectral type from the rdependence using the equation 0 = -p-1, -2 ~ p < 2 [3].4. The Bias Function B 2 (r,p)The bias function Bz(r,p) is defined in [1] by the relation,a Z (2, T, r) a Z B (2,T,r)z (r, p) (4)a Z (2, r, r) a; (r)In words, Bz(r,p) is the ratio of the expected two-sample variance with deadtime to that without dead time (with N = 2 and r the same for both variances).A plot of the Bz(r,p) function is shown in figure 3. The bias functions B 1and B z represent biases relative to N = 2 rather than infinity; that is, theratio of the N sample variance (with or without dead time) to the Allanvariance and the ratio of the two-sample dead-time variance to the Allanvariance respectively.5. The Bias Function B 3(N,M,r,p)Consider the case where a great many measurements are available with dead timebetween each pair of measurements (To> ro). The measurements are averagedover the time interval ro' the spacing between the beginning of onemeasurement to the next is To, and it may not be convenient to retake thedata. We might want to estimate the Allan variance at, say, multiples M of5TN-300