47.2

APPLICATION OF CLOCK ERROR PREDICTIONFOR DIGITAL COMMUNICATIONS

HARRIS A. STOVER

Deferme Gommurications Ergineerirg Center, 1860 Wiehle Ave., les ton, VA 22091

ABSTRACT

Survivability of digital defensecommunications can be enhanced by a free-runningclock mode of operation for use when the normaltiming modes are not available. Usefulness of a

free-running mode of timing depends on stabilityand accuracy. The cost of using cesium clocksthroughout the network could be very large, but

much less expensive quartz-crystal and rubidiumclocks could be used for many applications iferrors during a free-running period (following a

calibration period) can be predicted and removedwith sufficient accuracy for a long enoughperiod of time. This paper discusses one

relatively simple recursive method for makingsuch predictions.

INTRODUCTION

As the Defense Communications System (DCS)becomes more digital--digital transmission,digital switching, digital control--theimportance of system timing increases. Signalsfrom many sources, intended for manydestinations, have their time slots interleavedin time division multiplexers for transmissionover the same internodal link. Digital switchesare used which employ time slot interchange.Timing is much more important in militarycommunications systems than in civilian systemsbecause communications must survive even whenthe network is subjected to massive

destruction. Encoding not frequentlyencountered in civilian networks is used; and

encryption is frequently used, both on a

node-to-node basis and on an end-to-end basis.

Spread spectrum signals are used to overcome

jamming. All of these different functionsrequire synchronization.

In a switched all-digital world-widemilitary communications system, a bitoriginating anywhere in the world must be

available to fill an assigned time slot whenneeded at every node through which it passes.Storage buffers can temporarily store bits toaccommodate variations in signal transit time

and also small clock errors, but overflow can

cause an interruption of traffic while several

types of equipment (e.g., encryption, spreadspectrum, switching) at several differentlocations are sequentially resynchronized.

Studies by DCA and its contractors recommendthe use of a time reference distributed throughthe network to all major nodes, slaving of minornodes to major nodes, and a free-running clockmode for temporary use when other modes are notavailable. Stability and accuracy contributegreatly to the ability of a node to reenter thenetwork quickly following an outage during whichits clock must free-run. If clock errors in afree-running period following one of measurementand calibration, could be predicted and removedwith sufficient accuracy for a long enoughperiod of time, replacement of expensive cesiumclocks with less expensive clocks might bepossible in many locations. Such prediction andremoval of errors could improve the performanceof clocks whether or not lower cost clocks wouldbe used.

Other features (ref. 1) needed for asurvivable timing system will make possible theaccurate measurement of the errors of clocks innormally operating nodes of the network, i.e.,those nodes with at least one normally operatingpath to the network master or to the master ofany portion of a fragmented network. Thesemeasurements, which would be made even whenprediction techniques were not used, could alsobe used for calibration of the predictionprocess.

DIFFERENCE EQUATIONS AND DIGITAL FILTERS

Many time varying signals, systems, orprocesses can be represented by a differenceequation of the form shown in eq. 1 where ytrepresents the output signal or an observedvalue of the system at time t, and xtrepresents the input signal or an excitation ofthe system or process at time t. Clock errorscan be represented by such a difference equation.

QYtk=E

k=0

M

ak t-k k1k=l

bkyt-k (1)

731

CH 1 909-1/83/0000-0731$1 .00 1 9831EEE

The Z-transform provides a convenientnotation for use with difference equations. Avery useful property of the Z-transform is thatmultiplying the Z-transform of a sequence byz-n produces the Z-transform of the samesequence with every member shifted backward nsteps. Because of this property, z-n will beused here as an operator which shifts a term ofa sequence n steps backward in time, and thenotation z-nyt will be used interchangeablywith Yt-n where that proves advantageous.Taking the Z-transform of eq. 1 gives eq. 2which can be written in the more convenient formof eq. 3 where H(z) as defined by eq. 4, iscalled the transfer function of the filter.

Y(z) = akz ]kJx(z) - [kE bkzklY(z) (2)

Yz=O zk=l

Y(z) =X(z)H(z) (3)

H(z) =

Qakz

k=O

M

1 +

k=l

b zkk

If bk = 0 for all k in eq. 4, the transferfunction is a polynomial in z-1 and representsa nonrecursive filter called a Finite ImpulseResponse (FIR) filter. Otherwise, the transferfunction is a rational function in z-1 andrepresents a recursive filter called an InfiniteImpulse Response (IIR) filter.

AUTOREGRESSIVE AND MOVING AVERAGE MODELS

When eq. 1 is used to describe a randomprocess, each xt represents a random shock,and each Yt represents a correspondingobserved value of the process. If ak = 0 forAl k # 0. only the current value of xtappears in the equation, the process is calledan autoregressive process of order M because itrepresents a regression of the variable Yt onthe M previous values of itself. If bk = 0for all k, the process is called a movingaverage process of order Q because it representsa weighted average of the present value of xtand the Q previous values. A process with bothautoregressive and moving average terms iscalled an autoregressive moving average process.

INTEGRATED MODELS

For stability, the poles of the transferfunction, H(z), must not be outside the unitcircle in the complex Z-plane, and digitalfilters are normally designed with poles wellinside the unit circle. If H(z) has poles atunity, equation (2) can be written as equation(5) where (1 - z-l)d represents d poles ofH(z) at unity, Cp(z) is a factor representingthe remaining p poles of H(z), and Bq(z)

represents the q zeros of H (z). Since

Cp(z)(l - z-l)dy(z) = Bq(z)X(z) (5)

z-iyt represents yt-1, (1 - z-1) ytrepresents yt - yt-l, i.e., (1 - z-1) Y(z)represents the differences between consecutivevalues of yt. If this is multiplied by(1 - z 1) again, it represents differencesbetween those first order differences, i.e.,second order differences. If Y(z) is multipliedby (1 - z-l)d, it represents "d"th orderdifferences of yt. If the steps in time wereinfinitesimal instead of finite, it would be adifferential equation instead of a differenceequation, and there would be a correspondingintegral equation. Therefore, those systems orprocesses that have poles of the transferfunction H(z) at unity are called integrated.These integrated models are important becausemany time series exhibit a nonstationarybehavior, but the series obtained by taking thefirst, second, or higher order difference ofthese series are frequently stationary.

THE ARIMA MODEL

A model of the form given by eq. 5 is calledan Autoregressive Integrated Moving Average(ARIMA) model of order p, d, q (ref. 3). Itsautoregressive part, with the roots at unityremoved, is of order p; its integration part,represented by the roots of the autoregressivepart at unity, is of order d; and its movingaverage part is of order q. From the abovediscussion, it can be seen that the ARIMA modelcan be considered to represent the output, yt,from a linear filter whose input is white noise,xt. This is a useful concept for predictingclock errors. If d = 0 and all of the poles ofH(z) are inside the unit circle, then for awhite noise input, xt, with zero mean, theoutput, yt, will be stationary, i.e., itsstatistics will be independent of time. If d =

1, i.e., there is one pole of H(z) at unity,then for a white noise input, xt, with zeromean, the output, yt, can be nonstationary andits level can change. Similarly if d = 2, boththe level and slope are free to change, etc.

SOLUTIONS OF DIFFERENCE EQUATIONS

The ARIMA model is a difference equationmodel. Difference equations have a generalsolution that is the sum of two parts: acomplementary function, and a particularsolution. The complementary function is thesolution of the equation when all inputs, xt,are zero. Therefore only the p + dautoregressive terms play a part in thecomplementary function. When predicting futurevalues of yt (i.e., yt+f), the future valuesof xt (i.e., xt+f) will be unknown and willhave to be assumed to be zero; therefore, thecomplementary function is used for long termprediction. A distinct real pole of H(z) willcontribute an exponential term to thecomplementary function. A pair of complex poles

732

inside the unit circle contributes a damped sinewave. Multiple equal poles contribute theproduct of a polynomial and an exponential. Ifd equal poles are unity, as the d unity poles inan integrated model, the poles contribute to a

polynomial of degree d-l.

RELATING THE ARIMA MODEL TO CLOCK PREDICTION

If the error is predictable and notconstant, it is not stationary. A method isneeded for separating the predictablenonstationary part of the process from thestationary unpredictable part of the process.As stated earlier, many time series obtained bytaking the first, second, or higher differenceof these series are frequently stationary. Ifdifferences are taken between consecutivereadings of the clock errors, and differencesare then taken between these differences., etc..,until a set of differences is obtained that isstationary, that stationary set of differencesshould represent the unpredictable part of theerror. As shown later, the initial conditionsand the number of differences required forstationarity along with one or more parametersof the equation define the predictable part.

THE EFFECT OF AUTOREGRESSIVE TERMS IN CLOCKPREDICTION

During the period of time for which errorsare being predicted (the future), there is nopossibility of determining the random inputs,xt, pso we must assume that they are zero.Therefore, prediction uses the complementaryfunction starting with satisfactory initialconditions. Expecting that any clock errorsexisting prior to the start of prediction willexponentially decrease during the predictionperiod is unreasonable. Therefore, from theabove discussion of difference equations, thereshould be no distinct real roots of theautoregressive part within the unit circle.Unless the clock is subjected to some form ofcyclic environmental conditions, errors couldnot be expected to have sine wave terms, eitherdamped or undamped, so there should be nocomplex or imaginary roots. Clock errors mightbest be approximated by some form of simplepolynomial.

An ARIMA model for clock prediction shouldhave no autoregressive factors other than rootsat unity. It should be ARIMA (0, d, q). If d =1, the complementary function would be a zero

order polynomial which would predict a constantclock error. If d = 2, the complementaryfunction would be a first order polynomial whichwould predict a linear change in clock error

added to the initial clock error, i.e., an

initial phase error plus a constant frequencyerror. If d = 3, the complementary functionwould be a second degree polynomial, and itwould predict a linear change in the rate ofchange in the clock error, i.e., a linear changein frequency added to the initial frequencyerror and initial phase error.

Since quartz-crystal clocks and rubidiumclocks are known to have frequency drift, itwould seem that an ARIMA (0, 3, q) model wouldbe appropriate for these. On the other hand,cesium beam clocks have very little frequencydrift so that a (0, 2, q) model should be moreappropriate for them.

THE EFFECT OF MOVING AVERAGE TERMS ON CLOCKPREDICTION

If the nature of the prediction isdetermined entirely by the complementaryfunction, which in turn is determined entirelyby the autoregressive part, where does themoving average part participate? The answer is,in calibrating the initial conditions for theprediction, based upon the past performance ofthe clock. The moving average terms determinehow the prediction based on autoregressive termsis fitted to past measured error data. Fordiscussing this, let Jt+f be the predictionfor the clock error that will exist fmeasurement periods into the future. Sincethere is no way of knowing what futuremeasurements will be, the values of xt must beassumed to be zero after time t, and the valuesof Yt after time t must be predicted valuesrather than measured values. For purposes ofdiscussion, an ARIMA (0, 3, 3) model, as givenby eq. 6, will be assumed.

-1 3 -l -2 -3(l-z~ (a+a z +a2z + a3z )xt (6)

Substituting t+f for t and expanding theautoregressive part gives eg. 7.

Yt+f 3yt+f-1 3yt+f-2 Yt+f-3 axt+f +a xt+f-l+a2Xt+ft2+ a3xt+f-3 (7)

Letting xs = 0 and ys = Y for all valuesYS ~~~~Aof the subscripts, s, greater than t, where ysis a predicted value, gives the followingsequence of equations:A (

A AYt+2 3y t+ 3yt+ Yt_+ a2xt+ a3xt-A A Ayt+3 = 3yt+2- 3yt+l+ Yt a3xtA A - A AYtf=3tf13yt-3y +t+- for all f > 3

(9)

(10)

(11)

Note that all predictions beyond Yt+3 areformed completely from previous predictions andno measured data are included directly. Fromthis point forward the initial conditions forthe predictions have been determined, and theform determined by the complementary function isfollowed. The values of al, a2, and a3are imRortant in determining the values ofAyti+l Yt+2, and yt+3. If a3 is zero,the predicted errors 1or all f> 2 are dependenton the values of yt, yt+l, and 't+2- Ifboth a2 and a3 are zero, the predictederrors for all f t 1 are dependent on the valuesof yt-l, yt, and yt+l- If al, a2, and

733

a3 are all zero, the predicted errors for allf >0 are dependent on the yt-2, yt-l, and

yt. If all values of a are zero, anymeasurements made prior to Yt-2 have noinfluence on the predictions.

In order to see how the "a"s bring theprevious measurements into the predictionprocess, consider the simple moving averagemodel yt = ao + alxt-l with a0=1 andal=a as shownr in eq. 12. Remember, only theactual clock errors, Yt, are observed. Thereis no direct observation of the randomdisturbances, xt. These disturbances must beindirectly inferred from the measurements.

Yt= Xt+ axt-S t t t-i

Solving eq. 12 for xt gives eq. 13.

x = y - ax

Letting t be t-l in eq. 13 gives eq. 14.

xt-1 = Yt_l axt-2Substituting eq. 14 into eq. 13 gives eq. 15.

The ARIMA (0, 3, 1) model also has very littleerror introduced in the value of xt fromassuming xt-n = 0, if n is large and a isrelatively small. Again, the value of "a"determines the amount of effect past values of"y" have on the evaluation of xt.

DETERMINING INITIAL VALUES FOR THE PREDICTION

If all values of xt were known and themodel were absolutely correct, the ARIMA modelwould be completely deterministic and theprediction would be perfect. If the model wereabsolutely correct and the values of all xtwere determined with absolute precision up toand including the last measurement, theprediction would start with the correct initialconditions; it would be correct except for thoseunknown random perturbations caused by "xt"sthat occur after the beginning of the predictionperiod. One object of using the ARIMA modelmight be to evaluate the unpredictable part ofthe clock performance. The rms value of the"xt"s up to the beginning of the predictionperiod provides such an evaluation, and the"xt"s are also important for determininginitial conditions for the prediction period.

(12)

(13)

(14)

x = yt-a(yt - axtt t _-l -2 (15)

Subtracting 1 from each subscript in eq. 14 andsubstituting the resulting value for xt.2 intoeq. 15 gives eq. 16.

- a(yt_- a(y2- axt3t (16)

Continuing this procedure gives eqs. 17 and 18.

xt =Yt- (Y_l a(t-2 a(t-3 at-4)

2 3 ... n= - ayt 1+ ~-a yt-3 +(-a)

(17)

(18)

In each of these equations, all terms except thelast contain a "y", while the last term alwayshas an "x" instead of a "y". If n is large anda is small, the effect of assuming that xt.n =

0 can be very small because it is multiplied by(-a)n. The value of xt could be obtained byassuming some value of xt-n to be zero andfinding the appropriate weighted average of allthe measured "ly"s that occur after it. Thevalues of "a" determine the rate of decay of theinfluence of older measurements in evaluatingthe latest xt.

Substituting for each yt the correspondingexpression for the corresponding part of anARIMA (0, 3, 1) model, the value of xt can beevaluated for that model as in eq. 19.

2 3 2xt Yt (+3) -,+( +3a+3)y~t2-(a +3a 3+yt3

+ (a4 + 3a3 + 3a2 + a)y +***+t-4a)+(1I)n(an+3an I+3an-2+an3 )ytn+(-a)ntn

(19)

Although several methods could be used forevaluating the "xt"s and determining initialconditions for prediction, only a simplerecursive technique that is practical to includein an operational military communications systemwill be considered here.

For each measurement, the difference betweenthe measurement and the prediction made justprior to the measurement is the random impulse,xt, which is included in the actualmeasurement but could not be included in theprediction. Therefore, subtracting thepredicted value fromAeach new measurement givesan inferred xt = Yt-Yt. The recursiveprocess can be started at the initialmeasurement, t = 1, by assuming that xl = 0.When neither actual measurements nor predictionsare available, they are assumed to be zero. Theaccuracy of the estimates of xt will increaseas additional measurements are used. Thisprocess is illustrated in the sequence of eqs.20 to 30 for the ARIMA (0, 3, 3) model of eq. 7.

AYl+l 3y1

AY2 = y2 yl-lAy2+1 3y2- 3,

A= Y3- Y2+1

Ay4-y31

y4+1 3y4 3'A

x5 = y5- y4+1

(20)

(21)

(22)>Yl+ aIx2(23)

Y2+ Yi+ a1x3+ a2x2 (24;)

(25)(26)

(27)

IY3+Y2+aIx4+ a2x3+ a3x2

734

A xy-3,+ ax+axy51 = 3Y3y4+ y3+a5 2x4 3x3

AXt=Y- Y(t_l)+l

A 3y3y _+y+a,xt+ax a+xYt+l =3y-3 t l+t 2 lt2t-l+ 3 t-2

(28)

(29)

(30)

If the values of the "a"s are not too large,the accuracy of each inferred "xt" willincrease rapidly as more measurements are made.As the accuracy of xt increases, s2 does theprecision of the initial estimate yt+l' Notethat eq. 30 is the same as eq. 8. Therefore, ifall values of yt and xt are assumed to bezero prior to the first measurement, eqs. 29 and30 provide a very simple procedure forevaluating the first prediction, Yt+l, withthe precision of prediction increasing withadditional measurements. BX assuming that xs= 0 for all s > t and ys Ys for all s > t,eq. 30 also provides a simple recursive methodof continuing the prediction of clock errorsfollowing the last clock error measurement.

SELECTING PARAMETERS FOR THE ARIMA MODEL

The most likely model for quartz-crystal andrubidium clocks is an ARIMA (0, 3, q) model andthe most likely model for cesium clocks is anARIMA (0, 2, q) model. To determine the valueof q, (ref. 3) suggests taking p differencesuntil a stationary moving average sequence isobtained. The autocorrelation function,p , ofa moving average process of order-q has a finitevalue for q<'p but is zero for q > p. Thusthe value of q can be determined from theautocorrelation function. For clock errors, a

low value of q, perhaps 1, is expected to apply.

(Ref. 3) gives some formal procedures fordetermining the optimum values for the fak"s.However, the recursive procedures of eqs. 29 and30 can be used to provide one method,particularly if q = 1. It makes use of sum ofthe squares of the "xt"s from eq. 29. Sincethe "xt"s represent the unpredictable part ofthe data, the optimum set of parameters wouldseem to be that set which minimizes the sum ofthe squares of the "xt"s. If several valuesof each ak are used in the ARIMA model withmeasured error data for a particular clock, andthe rms values of the "xt"s are computed foreach set of values, this information can beplotted on graphs to obtain a good estimate ofthe optimum values for the "ak"s'

RELATED STUDY OF CLOCK PREDICTABILITY

UTC(USNO,MC) is the time scale generated bythe Naval Observatory master clock. It is basedon an ensemble of cesium clocks and is steeredto be within a few microseconds of UTC (BIH),the time scale generated by the BureauInternational de l'Heure. In order to improveUTC (USNO,MC) the Naval Observatory studied theARIMA model for use in a time-scale algorithm(refs. 4-6).

Based on this previous work, a preliminary"Evaluation of Predictability of Quartz-CrystalOscillators and Other Devices," (ref. 7) wasperformed by the Naval Observatory undersponsorship of the Defense CommunicationsAgency. An ARIMA (0, 2, 1) model was used forall clocks (whether quartz-crystal, rubidium, orcesium) with moving average parameters of ao =1 and al = - 0.75. No attempt was made tooptimize parameters.

This preliminary study of clockpredictability indicated that clock errors inhigh quality quartz-crystal oscillators can bepredicted with considerable accuracy during afree-running period following a calibrationperiod. There appears to be a high probabilitythat by using these predictions to remove errorsduring the free-running period, sufficientaccuracy can be obtained long enough to permitthe use of these clocks to replace much moreexpensive cesium clocks for some applications.Since rubidium clocks are much more predictable,there is an even higher probability that theycould be used to replace cesium clocks in manyapplications. Improvement was also shown incesium clocks.

The Naval Observatory is presently takingmeasurements on a number of clocks, making someof the measurements more frequently than in thepreliminary study to permit checking forpossible aliasing. These data will be morecompletely analyzed than those for the previousstudy, and an attempt will be made to optimizethe parameters for each of the three types ofclocks.

APPLICATION TO A COMMUNICATIONS SYSTEM

Figure 1 shows a timing system functionalblock diagram for one node typical of those thatshould be provided at all major nodes of aswitched digital military communications systemin order to provide the needed survivability.Several of the different functions might beprovided by a single microprocessor.

In such a system, which could use clockprediction very effectively, the Basic Clock isleft undisturbed so that it will maintain itsintrinsic stability. A Phase Shifter on itsoutput provides the control and correctionnecessary for coordination of all networkclocks. The output of the Phase Shifter is thenodal clock signal which is held withinacceptable phase (or time) tolerances of thoseat all other nodes of the network. Alldifferent frequencies which are needed in thecommunications node are produced by theFrequency Synthesizer to be coherent with theoutput of the phase shifter.

For survivability in a military network, themaster clock for the network must beautomatically selected from the clocks at allmajor nodes, and similarly the paths fordistributing the timing reference through the

735

COMMUNICATIONS TIMING SYSTEM AT A TYPICAL NODEFIGURE 1

network must be automatically selected. A newmaster or new paths through the network must beautomatically selected when necessary to allowfor damage or loss of portions of the network.Even the selection process must be distributedthroughout the network so that every node canautomatically make its own decisions, with nocentralized decisions required anywhere in thenetwork timing system. Then, should any networkfacility be lost or the network undergo massivedestruction, both the selection process and theremaining portions of the timing system willstill work properly. If the network becomesfragmented, each fragment will automaticallyselect its own master and the paths for timingdistribution. This function is performed by the

Timing System Self Organization block. Forfurther explanation, see (refs. 1, 8-9).

At major communications nodes, the NodalClock Error Measurement optimally combinesinformation obtained from different paths toprovide an accurate measurement of the nodalclock error (error in the output of the PhaseShifter) relative to the network master clock.The Filter between the Nodal Clock ErrorMeasurement and the Phase Shifter preventsmeasured errors from causing rapid changes inthe phase of the nodal clock signal which mightcause significant problems. For a very stableimplementation, with very long time constants inthe clock error correction circuits, local clock

736

errors will exist for extended periods of timebefore correction. This measured butuncorrected error in the nodal clock signal ispart of the information supplied to neighboringnodes by the Timing Information to Neighborsfunction. Some of these neighbors can use thisinformation in determining their own clockerrors relative to the master clock, and it alsoserves to prevent the propagation of timingerrors through the network. Two pieces ofinformation are used for Timing System SelfOrganization, and provide the node withinformation as to its position in the timinghierarchy relative to each of its neighbors.Another allows signal transit time to be removedfrom a comparison of clocks at neighboring nodesso that there is a direct comparison between thetwo clocks. This information is part of theinformation used to provide a very accurateNodal Clock Error Measurement (refs. 1, 8-9).

Since this system provides a very accuratemeasurement of the error in the nodal clocksignal, this information can be combined withthe amount of phase shift produced by the PhaseShifter to determine the error in the BasicClock. During normal operation this Basic Clockerror can be processed for ARIMA calibration bythe Clock Error Prediction block using themethods discussed in previous sections. Whenthe node must use the free-running mode, theClock Error Prediction function predicts theerrors that will occur in the Basic Clock sothat they can be removed by the Phase Shifter.Only the unpredictable error should remain.

CONCLUSIONS

Military digital communications networks aresubject to disruption, damage, and destructionthat usually do not occur to civilian networks.They also employ encryption, spread spectrumtransmission, and other techniques requiringtiming to a greater extent than civiliannetworks. Survivability of uninterruptedcommunications is dependent on system timing.Such survivability can be enhanced by providingeach node with a free-running clock mode for usewhen other modes are not available. Theusefulness of a free running mode depends on thestability and accuracy of the nodal clocksignal. By applying the methods of (refs. 1,8-9) an accurate measurement of local clockerror will be available at each major nodeduring normal operation of the timing system.These measured errors can be used to calibratethe ARIMA model discussed here so that errorscan be accurately predicted when the nodaltiming system must free-run. The predictederrors can be removed with sufficient accuracyto considerably enhance the normal capability ofthe clocks. In many applications this mightpermit the replacement of expensive cesiumclocks with much less expensive quartz-crystalor rubidium clocks.

The author would like to thank the followingpeople for their helpful comments: W.P. McKee,T. McCrickard, G. Winkler, R. Podell, and A.Trainer.

REFERENCES

1. H.A. Stover, "Network Timing/Synchronizationfor Defense Communications," IEEETransactions on Communications, pp.1234-1244 August 1980.

2. A.V. Oppenheim and R.W. Schafer, DigitalSignal Processing, Prentice Hall Inc.,Engelwood Cliffs, N.J. 1975.

3. Box and Jenkins, Time Series Analysis:Forecasting and Control, Holden-Day, 1970.

4. D.B. Percival, "A Heuristic Model ofLong-Term Atomic Clock Behavior,"Proceedings 30th Annual Symposium onFrequency Control, pp. 414-419, June 1976.

5. D.B. Percival, "Prediction Error Analysis ofAtomic Frequency Standards," Proceedings31st Annual Symposium on Frequency Control,pp. 319-326, June 1977.

6. D.B. Percival, "The U.S. Naval ObservatoryClock Time Scales, IEEE Transactions onInstrumentation and Measurement, pp.376-385, December 1978.

7. L.G. Charron and R.T. Clarke, III,"Evaluation of Predictability ofQuartz-Crystal Oscillators and OtherDevices," USNO-TS-1-81 (AD-A096380), January1981.

8. H.A. Stover, "Improved Time ReferenceDistribution for a Synchronous DigitalCommunications Network," in Proceedings ofEighth Annual Precise Time and Time IntervalInterval (PTTI) Applications and PlanningMeeting, U.S. Naval Research Laboratories,Washington, D.C. Nov 30-Dec 2, 1976,pp. 147-166.

9. H.A. Stover, "An Improved Time ReferenceDistribution Technique," U.S. Patent 4, 142,069.

737