NIST Technical Note 1337: Characterization of Clocks and Oscillators

110 lEER TR.~NSAcrlO:'IS ON I:'lSTRl:ME:iHTlON .~:iD MEASl:REMENT, MAY 1971get) = e(t) + net) (12)where e(t) is some deterministic function of time and net),the noise, is a nondeterministic function of time. We willdefine e(t) to be the systematic tTend to the function get).A problem of significance here is to determine when andin what sense e(t) is measurable.2) Specific Case-LineaT Drift: As an example, if weconsider a typical quartz crystal oscillator whose fractionalfrequency deviation is yeo, we may letdget) = dt y(t). (13)With these conditions, c{t) is the drift rate of the oscillator(e.g., Urlo/day) and '1 (t) is related to the frequency"noise" of the oscillator by a time derivative.One sees that the time average of get) becomes1 J,.+t 1 f,·+tT I. get) dt = c, + If.. net) dt (14)where c(t) = CJ. is assumed to be the constant drift rateof the oscillator. In order for Cl to be an observable,it is natural to expect the average of the noise term tovanish, that is, converge to zero.It is instructive to assume [8], [10] that in additionto a linear drift, the oscillator is perturbed by a flickernoise, i.e.,0< f S fl(15). f> flwhere h_ 1 is a constant (see Section V-A-2) and thus,S.(f) = {(21f/h_d , (16)0,for the oscillator we are considering. With these assumptions,it is seen thatand thatI J,.+rlim -T net) dt = K(O) = 0 (17)T'-_ t.~~ {Variance [f {.+r net) dtJ} = 0 (18)where «U) is the fourier transform of net). Since &(0)= 0, «(0) must also vanish both in probability and inmean square. Thus, not only does net) average to zero,but one may obtain arbitrarily good confidence on theresult by longer averages.Having shown that one can reliably estimate the driftrate Ct of this (common) oscillator, it is instructive toattempt to fit a straight line to the frequency aging.That is, letand thusget) = yet) (19)g(t) = Co + c, (t - to) + n'(t) (20)where Co is the frequency intercept at t = to and Cl isthe drift rate previously determined. A problem ariseshere becauseandS.,(1) S.(1) (21)~~ {variance [f t'+T n'(t) dtJ} = '" (22)for the noise model we have assumed. This follows fromthe fact that the (infinite N) variance of a flicker noiseprocess is infinite [7], [8], [10]. Thus, ~ cannot bemeasured with any realistic precision, at least, in anabsolute sense.We may interpret these results as follows. After experimentingwith the oscillator for a period of time onecan fit an empirical equation to y (t) of the formyet) = Co + tel + n'(t),where n' (t) is nondetenninistic. At some later time it ispossible to reevaluate the coefficients ~ and Cl' Accordingto what has been said, the drift rate Cl should bereproducible to within the confidence estimates of theexperiment regardless of when it is reevaluated. For Co,however, this is not true. In fact, the more one attemptsto evaluate Co, the larger the fluctuations are in theresult.Depending on the spectral density of the noise term,it may be possible to predict future measurements of~ and to place realistic confidence limits on the prediction[11]. For the case considered here, however, theseconfidence limits tend to infinity when the predictioninterval is increased. Thus, in a certain sense, Co i~"measurable" but it is not in statistical control (to usethe language of the quality control engineer r91).V. TRANSLATIONS AMONG FREQUENCY STABILITYMEASURESA. Frequeru;y Domain to Time Domain1) General: It is of value to define r = T/T; that is,T is the ratio of the time interval between successivemeasurements to the duration of the averaging period.Cutler has shown (see Appendix I) that(u:(N, T, T» *= N 1~ df Sen [sin' (r/T)} {I _ sin' (rrfNr)}.(N - 1) 0 • (1ff")' N' sin' (rrfr)(23)Equation (23) in principle allows one to calculate thetime-domain stability (u:(N, T, r» from the frequencydomainstability S.(j).2) Specific Model: A model that has been found useful[8], [1O]-[13J consists of a set of five independentnoise processes z.. (t), n = -2, -1,0,1,2, such that... See Appendix Note 11 19TN-151