NIST Technical Note 1337: Characterization of Clocks and Oscillators

TIME AND FREQUENCY535cient and x(t) and yet) have the same meaning asin the preceding sections. The notations x' and y Irefer to an oscillator with drift.It can then be shown that we haveD~a)(T) = '1'2.'1 E [d~_,J"" sm:bl(-lT/T)+ 2:b1 S if) df t = n (49)0" (2'lTf) 2and~1.. Sin:bl('lTfT)D(a)(T) = 2 210 S if) df t> n (50)• 0" (2'lTf) 2If one applies (49) to the case of an oscillatorwithout drift, one can easily show that the followingequations are satisfied:and(51)0;('1") = (l/2T2)D~2)(T) (52)This is indeed not surprising because apart frommore or less complicated mathematical formalismthe definitions of the considered variances andstructure functions are closely related, as has beenemphasized here.Relations between sample variance and structurefunctions have been given by Lindsey and Chie[1976], whereas the relation between structurefunctions and several different approaches of frequencystability characterization has been analyzedby Rutman [1917, 1978].For the generally accepted noise model definedby (7) the 'I' dependence of higher-orde.r structurefunctions is the same as the two-sample variance,as shown in Table 4 [Lindsey and Chie, 1978b].As stated above, structure functions of order nallow one to consider spectral densities which varyasr at the origin with a ~ -2(n - I). For instance,for n = 3 it is possible to characterize frequencyfluctuations of an oscillator with a power spectraldensity of fractional frequency fluctuations givenby S)f) = ~~_-4 hJo. This oscillator exhibitsstationary third-order increments of phase fluctuations.In the presence of a frequency drift describedby a polynomial of degree I - I, structure functionsof degree 11 < I are meaningless: their computationyields a time-dependent result. For n = I the lthstructure function shows a long-term 'I' dependenceproportional to 2J T . This dependence disappears forn > t. Although a power spectral density of theformj-(2J-I) would also give the structure functionsa variation of the form '1'2/, this variation does notdepend on n, provided that the function is meaningful.It is then possible, at least in principle, toidentify frequency drifts and to specify their order.This is illustrated in Figure 23 according to Lindseyand Chie (l978b]. However, there are not yetexperimental proofs that such a characterizationis achievable in practice.10. POWER SPECTRAL DENSITY OF STABLEFREQUENCY SOURCESThe power emitted by a source of time-dependentvoltage vet) given by (I) is S)v)dvin the frequencyrange [v, v + dv], where S)v) is the power spectraldensity of the source. The dimensions of Sv(v) areV 2 Hz-I. The main interest of power spectraldensity, in frequency metrology, is related to highorderfrequency multiplication. We will only introducethe subject by giving the relations betweenSv(v) and S'f{f) and stating present problems inthe field.TABLE 4.Structure functions of orders 1,2, 3, and 4 for fraclional phase fluctualions of commonly encounlered noise processesfor 2Trf. T :> IS,,(f)5 ~h i-;- lid.2Tr 2 >J.TrI 3 5 35II.! - h, In (Trf.T) -2 h,m frrf.T) -;- h, In (Trf. T) -h,In(Trf.T)2Tr 2 2Tr Tr 2Tr 2-}lI o T hoT IOheT411_,T 2 1n2 20.7 "_,T 1From Lillds~ and Cllie (l978bI4_Tr 1 11T'3 -2TN-I8S