NIST Technical Note 1337: Characterization of Clocks and Oscillators

The UxCt) function for flicker noise PH isextremely complicated and has not been developed,but one can arrive at an empirical value for it.The Ux(t) function is derivable for the otherpower law spectral processes. Table 2 gives therelationships between the time domain measureHod Gy2 (t) and its power law spectral counterpart,given in Eq. 1. Also listed in the right handcolumn of Table 2 are the asymptotic valuesof R(n):Ho1n Typ.M....TABLE 2at")Mod a/(t) C~nt n' •3 f hWhIte PPI +2 hacth2'~1.038 -+ 31n(w t)Flic:t.r PM +1hE~iricalhI'(20»"­R(n)""'ft. FN 0 hO'~ fJcac:t 0.5Fl1c:ker FM -1 h. Z1n(2) • R(n) [_pci r1 CI.'; Exa.c:t o. G74IA"atlablea.ndo. Val k FH -2 h_ ' ( 2n tt • R(n) £~it'icAl; Exact 0.824ZAllailabl.It is clear from Table 2 that Mod 0 y2(t) isvery useful for white PM and flicker noise PH, butfor a < +1 the conventional Allan variance, Oy2(t),gives both an easier-to-interpret and an easier-to·calculate measure of stability.It is interesting to make a graph of a versusIJ for both the ordinary Allan variance and themodified Allan variance. Shown in Fig. 2.is sucha graph. This graph allows one to determine powerlaw spectra for non-interger as well as intergervalues of C1. The dashed 1ine for the modifiedAllan variance has been intentionally moved to the1eft in Fig. 2 because for small va 1ues of n thevalue of ~ will appear to be slightly more negativethat for O/(t), even though for large n, theyboth approach the same slope (i.e., the samevalues of IJ). In fact, in the asymtotic 1imit,the equation relating IJ and C1 for the modifiedAllan variance isa = -IJ -I, for -3 < a < +3 . (17)'" See Appendix Note # 34*The value of IJ = -4 for a = +3 was verified empiricallywith simulated data, and it appears thatfor a > +3, IJ remains at -4.A direct application for using the modifiedAllan variance recently arose in the analysis ofatomic clock data as received from a GlobalPositioning System (GPS) sate11 ite. We wereinterested in knowing the short-term characteristicsof the newly developed, high-accuracy NBS/GPSreceiver, as well as the propagation fluctuations.Fig. 3 shows both 0 2(t) and Mod ayy2(t) for comparison.Using Mod 0 2(1), ywe can tell that thefundamental limiting noise process involved ih thesystem is white noise PH with the exciting resultthat averaging for four minutes can allow one toascertain time difference to better than onenanosecond excluding other systematic effects.ConclusionWe have developed a supplemental measure, the"Modified Allan Variance" (Mod 0/(1»), which hasvery useful properties when analyzing oscillatoror signal stability in the presence of white noisephase modulation or flicker noise phase modulation.It also works reasonably well as a stabilitymeasure for other commonly occuring noise processesin precision oscillators.We would recommend that for most time domainanalysis, 0 2(1) should be the first choice. Ifya 2(1) depends on t as t- 1 , then the modi fi edyAllan variance can be used as a substitute to helpremove the ambiguity as to the noise processes.~cknow1edgmentsThe authors are deeply grateful for stimulatingdiscussions with Dr. James J. Snyder, andfor the useful comments of Dr. Robert Kamper andMr. David A. Howe.References1. James A. Barnes, Andrew R. Chi, Leonard S.Cutler, Daniel J. Healey, David B. Leeson,Tomas E. HcGunical, James A. MUllen, Jr.,Warren L. Smith,' Richard L. Sydnor, Robert F.C. Vessot, and Gernot M. R. Winkler, Proc.IEEE Trans. Instrum. Meas. 1M-20, 105 (1971);also pub1 ished as NBS Tech. Note 394 (1971).2. David W. Allan, Proc. IEEE 54, 221 (1966).3. R. F. C. Vessot, Proc. 1EEE'54, 199 (1966).4. James J. Snyder, Proc. 35th Annual Symp. onFreq. Control (1981), to be pub1 ished.473TN-257