NIST Technical Note 1337: Characterization of Clocks and Oscillators

522 LESAGE AND AUDOIN01., 1971J:S,I']'P(/)Ii> (I)x(1) =-- and y(/) =-- (6)2'11"11 02'11"v owhere x(t) and y(/) are the fractional phase andfrequency fluctuations, respectively. The quantityx(t) represents the fluctuation in the time definedby the generator considered as a clock.At first, we will make the following assumptions:I. The quantities x(!) and y(!) are random functionsof time with zero mean values, which impliesthat systematic trends are removed [Barnes el 01.,1971 J . They might be due to ageing or to imperfectdecoupling from environmental changes such astemperature, pressure, acceleration, or voltage.Characterization of drifts will be considered insection 8.2. The statistical properties of the stable frequencygenerators are described by a model whichis stationary of order 2. This point has been fullydiscussed in the literature [Barnes et 01., 1971;Boileau and Picinbono, 1976; Barnes, 1976]. Thisassumption allows one to derive useful results andto define simple data processing for the characterizationof frequency stability.Actual experimental practice shows that, besideslong-term frequency drifts, the frequency of ahigh-quality frequency source can be perturbed bya superposition of independent noise processes,which can be adequately represented by randomfluctuations having the following 'one-sided powerspectral density of fractional frequency fluctuations:Sy (f) = 2:2 11..1'" (7)Q--2Sy (f) is depicted in Figure I. Its dimensions areHz-I. Lower values of (I( may be present in thespectral density of frequency fluctuations. Theyhave not been clearly identified yet because ofexperimental difflculties related to very long termdata acquisition and to control of experimentalconditions for long times. Moreover, the relatednoise processes may be difficult to distinguish fromsystematic drifts.Finite duration of measurements introduces alow-frequency cutoff which prevents one fromobtaining in{ormation at Fourier frequencies smaller«.. -1Fig. I. Asymptotic log-log plot os Sy(f) for commonly encounterednoise processes.than 1/9, approximately, where 9 is the total durationof the measurement [Cutler and Searle, 1966].Alternatively, this made it possible to invok.e physicalarguments to remove some possible mathematicaldifficulties related to the divergence of S" (f)as/- 0 for (I( < U.Furthermore, high pass fI1tering is always presentin the measuring instruments or in the frequencygenerator to be characterized. It insures convergenceconditions at the higher-frequency side ofthe power spectra for a > O.The spectral density of fractional phase fluctuationsis also often considered. From (6), one canwrite, at least formally,The dimensions of S,,{f) are S2 Hz-I. Similarly,the spectral density of phase fluctuations ",(t) issuch asIt is expressed in (rad)2 Hz-I.The quantity 2(f) [Halford et al., 1973] issometimes considered to characterize phase fluctuations.If phase fluctuations at frequencies >/are small compared with I rad, one has, .(8)(9).Y(f) = }S..(f) (10)where S (f) is the spectral density of phase fluctuationsof..the frequency generator considered.. Thedefmition of 2{f) implies a connection with theradio frequency spectrum, and its use is not recommended.Since the class of noise processes for which y(l)is stationary is broader than that for which theIN-I72