NIST Technical Note 1337: Characterization of Clocks and Oscillators

12 FREQUENCY AND TIME MEASUREMENTI~I----.------;-----+­211TIMEFIG.12-11 Calculation of two average frequencies.vl and Yz by measuring the phase of anoscillator x(/) at times I,. t 2 • and r l .completely symmetric, with negative values being as likely as positive ones.The resulting distribution is called a chi-squared distribution, and it has one"degree of freedom" since the distribution was obtained by considering thesquares of individual (i.e., one independent sample), normally distributedvariables (Jenkins and Watts, 1968).In contrast, from five phase values four consecutive frequency values canbe calculated, as shown in Fig. 12-12. It is possible to take the first pair andcalculate a sample Allan variance. A second sample Allan variance can becalculated from the second pair (i.e., the third and fourth frequencymeasurements). The average of these two sample Allan variances provides animproved estimate of the true Allan variance, and one would expect it to havea tighter confidence interval than in the previous example. This could beexpressed with the aid of the chi-squared distribution with two degrees offreedom.However, there is another option. One could also consider the sampleAllan variance obtained from the second and third frequency measurements,that is, the middle sample variance. This last sample Allan variance is notindependent ofthe other two, since it is made up ofparts ofeach ofthe others.But this does not mean that it cannot be used to improve the estimate of thetrue Allan variance. It does mean that the new average of three sample Allanvariances is not distributed as chi squared with three degrees offreedom. The~i~IIIIII II, I. I, I. I.TIMEYz. Yl' and Y. from five phaseFIG.12·12 Calculation of four frequency values ji"Feasurements at limes C, • C,. Cl. I•• and Is. The sample variance formed from YI and Y2 and theone formed from Y3 and j. are independent. The sample variance fonned from Y2 and YJ is nolindependent of the other two but does contain some additional information useful in estimating~he true sample variance.IN-81