NIST Technical Note 1337: Characterization of Clocks and Oscillators

652 IEEE TRANSACTIONS ON ULTRASONICS, FERROElECTRICS, AND FREQUENCY CONTROl. VOL. UFFC-)4. NO.6. NOVEMBER 1987TABLE 111aTypical Noise TypesNameOptimumPredictionx( T p ) nns"Time Error:Asymptotic Form2Io- I- 2while-noise PMflicker-noise PMwhite-noise FMflicker-noise FMrandom-walk FMTp • a,( Tp)/,fj- TI' • a, ( T1') ../"In-Tp/7.: 2--:-,n-T0T p ' a.• (T p )T p • a.(T p)/,J"1;"2T p • a.• (T p )constant~1~f7T~!,T p-"T p is the prediction interval.TIME AND FREQUENCY ESTIMATION AND PREDICTIONUsing a,(T), O',(T), S,(f), or S4>(f), one can characterizetypical power law processes. Once characterized,this opens the opportunity for determining optimum estimatesof values by employing the statistical theorem thatthe optimum estimate of a white-noise process is the simplemean.For example, consider the very common and very importantcase of white-noise FM typically found on the signalsfrom cesium standards, rubidium standards, and passivehydrogen masers_ The optimum estimate of thefrequency is the simple mean frequency, which is equivalentto (x" - XI)/MTo. It is still all too common withinour discipline to see our colleagues erroneously determiningthe frequency for these kinds of oscillators by calculatingthe slope from a linear least-squares fit to the timedeviations and quoting the standard deviation around thatfit as a measure of the clock performance. There are threeproblems in proceeding this way. First, the frequency estimateis not optimum in a mean-square-error sense. It isequivalent to throwing away about 20 percent of the dataand thereby increasing the cost in the case of a calibration.Second, the standard deviation diverges as the squareroot of the data length. Third, the standard deviation issignificantly dependent on the filter form, e.g., linear leastsquares, as well as the clock deviations. On the otherhand, such a filter is sometimes useful for assessing outliers.The optimum "end-point" method outlined earlierhas the risk that if either of the points is abnormal, (i .e"the model fails), the result will of course be adverselyeffected. Therefore such a filter is useful to assess whetherthere are outliers-paying special attention to the endpoints. Also, if the measurement noise exceeds the combinednoise in the clocks, then the end points will be adverselyaffected. The key message is that the end-pointmethod for estimating frequency is only optimum if thei:~~: ~:r:)r:;~~~eI~~; ~:~~h is easy to determine fromI There are other useful, and maybe not so obvious, optimumestimators appropriate for time-difference data sets.1) Given white-noise PM, the best time estimate is thesimple mean of the time deviations; the frequencyestimate then is the slope from a linear least-squaresfit to the time deviations, and the frequency drift Dis determined from a quadratic least-squares fit tothe time deviations per (l).2) Given white-noise FM, the optimum estimate of thetime is the last value; the optimum-frequency estimateis outlined in the previous paragraph, and theoptimum-frequency-drift estimate is derived from alinear least-squares fit to the frequency.3) Given random-walk FM, the current optimum timeestimate is the last value plus the last slope (clockrate) times the time since the last value; the opti·mum-frequency estimate is obtained from the lastslope of the time deviations; and the optimum-frequency-driftestimate is calculated from the meansecond difference of the time deviations. Cautionneeds to be exercised here, for typicaIly there willbe higher frequency component noise in a real datastream, such as white-noise FM, along with random-walkFM, and this can significantly contaminatethe drift estimate from a mean second difference.If random-walk FM is the predominant longtermpower-law process, which is often the case,then the effect of high-frequency noise can be reducedby calculating the second difference from thefirst, middle, and end-time deviation points of thedata set.The flicker-noise cases are significantly more complicated,though filters can be designed to approximate optimumestimation [14]-[ 16]. As the data length increaseswithout limit, time is not defined for flicker-noise PM,and frequency is not defined for fl icker-noise FM. Thishas some philosophical implications for the definitions oftime and frequency. unless some low-frequency cutofflimits exist. If significant frequency drift exists in the data,it should be optimally subtracted from the data or it willbias the long-term values of 0', ( T):DT0'.(7) = C' (14). ,,2Once the power-law spectra are deduced for a pair ofoscillators or clocks, then one can also develop an optimumpredictor. Table III gives both the optimum predictionuncertainty values for the various relevant purepower-law spectra as well as their asymptotic forms. Specialforecasting techniques must be used for optimal predictionwhen combinations of these processes are present[17]. To illustrate how these concepts relate to real devices,Fig. 10 shows a Oy( 7) diagram for some interestingTN-126