NIST Technical Note 1337: Characterization of Clocks and Oscillators

is ~he preferred measure, since, unambiguously, it ft et + r:always cen be measured.y(t·)dt'.tob. Iime-pomal0:For fairly simple models, regression analysis canIn the ~ime domain, frequency inscability shall beprovide efficient escimaees of the TIE (Draper anddefined by the cwo-sample deviation ~,(r) which isSmieh. 1966: CcrR, 1986). In general, ehere arethe square root of the cwo-sample variance o,1(r),many estimators possible for any staciseical quanti~y.Ideally, ve would like an efficiene andThis variance, c,J(r). has no dead-time between thefrequency samples and is also called the Allanunbiased es~imator. Using the time domain measurevariance. For che sampling time r. ve WTlte:a,2(r) defined in (b), the follOWing estima~e ofvhere~ + r; --;- fie y(~)d~ =~x k+lThe symbol < > denotes an infinite time average.In practice, the requirement of infinite timeaverage is never fulfilled; the use of ~he foregoingcerms shall be permitted for finite timeaverages. The ~ and ~+ are time residual measurementsmade aE 7K ana ~ 1 = ~+r. k=1.2.3 ...••and l/r is the nominal fixed sampting ra~e whichgives zero dead time between frequency measuremencs."Residuai" implies the kno~ syseematiceffects have been removed.If dead cime exists becween the frequency departuremeasurements and this is ignored in the computationof c,(r), resul~ing instabilicy values vill bebiased (except for vhite frequency noise). Some ofthe biases have been seudied and some correctiontables published [Barnes, 1969; Lesage. 1983;Barnes and Allan, 19881. Therefore. the term a,(r)shall not be used to describe such biased measurements.Raeher. if biased instability measures aremade, the info~ation in the references should beused to repore an unbiased eseima~e.If che inieial sampiing race is specified as l/ro ,then it has been shovn~hat. in general, we mayobeain a more efficienc estimaee of a,(r) usingvha~ is called "overlapping eseimaees." Thisescimaee is obeained by computingN-2mLi=lwhere N is che number of original time residualmeasuremenes spaced by r.(N=~+l, where M is ~henumber of originai fr~quency measur~ments of sampleei:e r.) and T " mT•.From ~he above equaeion, We see thae a,1(r) actsiike a second-difference operator on the timedeviation residuals--providing a stationary measureof the scochascic behavior eVen for nonscacionaryprocesses. Additio~al variances, which may ~e usedco describe frequency instabilities, are deflned inA?pendix I I.c. ~L9ck-Time Prediceionthe scandard deviation (R~S) of TIE and ies associacedsyseemacic depar~ure due to 4 linearfrequency drifc (or ics uncertainty) can be U3ed topredict a probable ti=e interval error of a clocksynchronized at t"t.=O and left free running thereafter:1l(t)2]1tR.'!STtE "t [2 t2 + a 2 + a 2 (r t) + (--2-)est 4 Yo y twhere "a" is the normalized linear frequency drifeper unic of cime (aging) or the uncertainty in thedrift estimace, a, the two-sample devia~ion ofthe initial freque~cy adjustment, a,(r) the cwosampledeViation describing ehe random frequencyinseabiliey of the clock ae t=T, and x(t.) i5 theinieial synchroniza~ion uncertainey. The ehirdterm in the brackees provides an opeimum and unbiasedescimate (under the condition of an optimum(R~) prediction meehod) in the cases of whitenoise FM and/or random walk FM. The third term istoo optimistic, by about a faccor of 1.4, forflicker noise FM, and coo pessimistic, by about afactor of 3, for whice noise P~.This estimaee is a useful and fairly simple approximaeion.In general. a more complete erroranalysis becomes difficult; if carried out. such ananalysis needs to inclUde the methods of time prediction,the uncertainties of the clock parameters,using ehe confidence limits of measurements definedbelow. che detailed clock noise models, syseematiceffaces. etc.4. Confidence Limits of ~easurementsAn eseimaee for Oy(T) can be made from a finiee daea setwieh Mmeasuremenes of Yj as follows:M-la ( r)I ---l.- - 2y 2(~-l) L (Yj +1- Yj)j=lor, if ehe data are time rudings It; :M-l1I~a/r) L (X j+ 2- 2xj + l + x j2r2(~-1))j=l>,2IojIThe 68 percene confidence in~erval (or error bar), I.,for Gaussian noise of a particular value a,(r) obtainedfrOm a finite number of .ampl~s can be estimated asfo!.lo"s:The variation of the time difference be~~een a realclock And an ideal uniform ~ime scale, also knovnas ei~e intervai error, TIE, observed over a timelnter-I.11 starting a: time t. And ending at t.+tshall be defined as:420wher.. :to~alnumber of data poln~s used in the@'f1':lala~e. Ian ineeger as defined in A?pendix r,z "1 = 0.99,TN-140