NIST Technical Note 1337: Characterization of Clocks and Oscillators

of the frequency measured in this manner over the interval r. The estimate of frequency,obtained by averaging the phase or time data, is degr~ded by about 10 percent from that obtainedby using just the end points [3]. This is analog~us to the algorithm for calculating modair) which, in this case, underestimates the uncerta~ty in measuring frequency by J2. Forwhite phase modulation (PM) noise, a = 2, the optimu~ estimator for frequency is obtained byaveraging the time or phase data over the interval r. i This is analogous to the algorithm forcalculating mod ay( r) which yields the one-sigma unbertainty in the estimate for frequencymeasured in this manner. This estimate for frequency ~s In better than that provided byay(r).Based on these considerations, it is our opinion that mod a y(r) can be profitably usedImuch more often than it is now. The presence of significant high-frequency FM or PM noise inthe measurement system, in an oscillator, or in an oscillator slaved to a frequency reference, isvery common. The use of mod CTy(r) in such circums~ances allows one to more quickly assesssystematic errors and long-term frequency stability. In lother words, a much more precise valuefor the frequency or the time of a signal (for a given m~surement interval) can be derived usingmod air) when n is large.iThe primary reasons for using ay( r) are that it ~s well known, it is simple to calculate, itis the most efficient estimator for FM noise (a ~ 0), :tnd it has a unique value for all r. Theadvantages of mod aye r) are cited in the above paragrkph. There are some situations where astudy of both air) and mod air) can be even more Gevealing than either one. The disadvantagesof using mod av(r) are that it is more complex tp calculate and thus requires more computertime and it has 'not been commonly used in the lit~rature, so interpretation of the results ismore difficult to reconcile with published information. Another concern sometimes raised is thatmod a/ r) does not have a unique value in regions ~ominated by FM noise (a ~ 0). Withrapidly increasing computer speeds, the computational d~sadvantage is disappearing. The correctedand expanded information presented in figure 2 ahd table 2 addresses the concern aboutuniqueness. :The primary disadvantage of using a/ r) is that ~he results can be too conservative. Thatis, if the level of high-frequency FM noise is high, the~ the results are biased high, and it cantake much longer (often orders of magnitude longer) to! characterize the underlying low-frequencyperformance of the signal under test. :References to Section A.6I[1] Walls, F. L, John Gary, Abbie O'Gallagher, [Roland Sweet and Linda Sweet, "TimeDomain Frquency Stability Calculated from the! Frequency Domain Description: Use ofthe SIGINT Software Package to Calculate T~e Domain Frequency Stability from theFrequency Domain," NISTIR 89-3916, 1989.I[2] Bernier, LG., Theoretical Analysis of the Modified Allan Variance, Proceedings of the41st Frequency Control Symposium, IEEE Cataldgue No. 87CH2427-3, 116, 1987.I[3] Allan, D.W. and J.E. Gray, Comments on the dctober 1970 Metrologia Paper "The U.S.Naval Observatory Time Reference and Perfoimance of a Sample of Atomic Clocks,"International Journal of ScientifiC Metrology, 7, 7~, 1971.!IN-12