NIST Technical Note 1337: Characterization of Clocks and Oscillators

is determi ned by the measurement system.If thetime difference or the time fluctuations areavailable then the frequency or the fractionalfrequency fl uctuations may be ca 1cu latt!tl frOli oneperiod of sampl ing to the next over ttle datalength as indiciated in figure 4.1. Supposefurther there are M values of the fractionalfrequency Yi'these data.Now there are many ways to analyzeHistorically, people have typicallyused the standard deviation equation shown infigure 4.1, a td d (t),s . ev.where y is the averagefractional frequency over the data set and issubtracted from each value of Yj before squaring,summing and dividing by the number of values minusone, (M-1), and taki ng the square root to get thestandard deviation.At NBS, we have studied whathappens to the standard deviation when the dataset may be characteri zed by power 1aw spectrawhich are more dispersive than classical whitenoise frequency fluctuations.In otne'T' words, ifthe fluctuations are characterized by flickernoise or any other non-white-noise frequencydeviations, what ha~pensto the standard deviationfor that data set? One can show that the standarddeviation is a function of the number of datapoints in the set; it is also a function of thedead time and of the measurement system bandwidth.For example, using flicker noise frequency modulationas a model, as the number of data pointsincreases, the standard deviation monotonicallyincreases without limit. Some statistical measureshave been deve loped whi ch do not depend upon thedata length and which are readily usable forcharacterizing the random fluctuations in precisionoscillators. An IEEE subcommittee on frequencystability hasrecommended what has come to beknown as the "A11 an vari ance" taken from the setof useful variances developed, and an experimentalestimation of the square root of the Allan varianceis shown as the bo~tom right equation infi gure 4.1. Thi 5 equation is very easy to imp 1ementexperimentally as one simply need add up thesquares of the differences between adjacent valuesof Yi' divide by the number of them and by two, and'take the square root. One then has the quantitywhich the IEEE subcommittee has recommended for