NIST Technical Note 1337: Characterization of Clocks and Oscillators

534 LESAGE AND AUDOINof measurement results or by mtering with the helpof an analog fIlter [Graslambert, 1976].Such a technique of linear fIltering has been usedto show that good quartz crystal oscillators exhibitflicker noise of frequency for Fourier frequenciesas low as 10- 3 Hz [Lesage and Audain, 1975b].Furthermore, it is well suited to the design ofautomated measurement setups [Peregrina andRicci, 1976; Groslambert, 1977].The best use of experimental time domain datafor selective fIltering has been considered by Boileau[1976].8.2. High-pass filteringIf frequency fluctuations y(t) are mtered in anideal high-pass fIlter with transfer function Go (f. 'f) such that//.its output z(t) is such that0;,: r~ Go(f,fI)Sy(f)d!= r'0 Sy(f)df (44)JoJhEquation (44) shows that the derivative of U; is-Sy(f), and spectral analysis, and thereforecharacterization offrequency stability, are possible,in principle, by high-pass fIltering.Possible realization of the high-pass fIlter bytechniques of digital data processing have beenspecified, such as the method offInite-time varianceand the method of fInite-time frequency control.Processing of fInite-time data is aimed to properlydeal with the nonintegrable singularity of the powerspectral density at v = 0 [Boileau, 1975; Boileauand Picinbono, 1976]. The method is well suitedto the analysis of drifts or slow frequency changes.Practical use of this method has not been reportedyet.8.3. Use of the sample spectral densityIt has been shown in section 8.1. that spectralanalysis from the Hadamard variance or its modifiedforms requires a series of measurements at timeinterval T in order to specify the spectral densityat frequency 1/2T. Another point of view has beenconsidered [Boileau and Lecourtier, 1977]. Froma set of measurements of Y., sampled at frequency11T, it allows one to obtain an estimation of thespectral density for discrete values of the Fourierfrequency.9. STRUCTURE FUNCTIONS OF OSCILLATORFRACTIONAL PHASE AND FREQUENCYFLUCTUATIONSInterest in the variance of nth-order differenceof phase fluctuations was recognized early in thefIeld of time keeping (see for instance, Barnes[1966]). This can be easily understood from (43),which shows that an effIcient ftltering of lowfrequencycomponents of frequency fluctuationsis then introduced. It allows one to deal properlywith frequency drifts, which will now be considered,and poles of Sy(f) of order 2(n - 1) at the origin.It is equivalent to saying that the nth differenceofphase fluctuations allows one to consider randomprocesses with stationary nth-order phase increments.This question has been formalized by Lindseyand Chie [1976, 1971], who introduce structurefunctions of oscillator phase fluctuations. The nthorder structure function of phase fluctuations isnothing else but the variance of the nth differenceof phase fluctuations, as considered in section 8.Then, by defInition, the nth-order structure functionof fractional phase fluctuations is given byD~)(T) = EO"..1 •.•x(t.)] 2} (45)where E {.} means expectation value. The fractionalphase (or the clock reading) at time tIc is x(t,,).We assume T = T.Let us consider an oscillator, the phase If" (t)of which is of the following form except for anadditive constant:(46)where 0" is a random variable modeling the kthorderfrequency drift and If'(t) represents randomphase fluctuations. We then haveandtIcx'(t) =L d H ~ + x(t) (47)"-2 k.I-ItIcy'(t) = L dIe - + y(t) (48)>_. k!where dIe = O,,/21Tv o is the normalized drift coeffI­TN-184