REVIEW ARTICLE Introduction to time and frequency … metrology, including a discussion of some of the types of measurement hardware in common use and the statistical machinery that

REVIEW ARTICLE

Introduction to time and frequency metrologyJudah Levinea)

JILA and Time and Frequency Division, NIST and the University of Colorado, Boulder, Colorado 80309

~Received 24 July 1998; accepted for publication 22 February 1999!

In this article, I will review the definition of time and time interval, and I will describe some of thedevices that are used to realize these definitions. I will then introduce the principles of time andfrequency metrology, including a discussion of some of the types of measurement hardware incommon use and the statistical machinery that is used to analyze these data. I will also introducevarious techniques of distributing time and frequency information, with special emphasis on theglobal positioning system satellites. I will then discuss the advantages of clock ensembles and aprototype time-scale algorithm. I will conclude with a discussion of how clocks are synchronized toremote servers using noisy and poorly characterized transmission channels.@S0034-6748~99!00106-9#

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 70, NUMBER 6 JUNE 1999

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I. INTRODUCTION

Clocks and oscillators play important roles in manyeas of science and technology, ranging from tests of genrelativity to the synchronization of communication systeand electric power grids. Hundreds of different types of dvices exist, ranging from the cheap~but remarkably accurate!crystal oscillators used in wrist watches to one-of-a-kind pmary frequency standards that are both seven or more orof magnitude more accurate and almost the same numbpowers of ten more expensive than a wrist watch. In spitethis great diversity in cost and performance, almost all ofdevices can be described using basically the same statismachinery. The same is true for methods of distributing tiand frequency signals—the same principles are used to trmit time over dial-up telephone lines with an uncertaintyabout 1 ms and to transmit time using satellites with ancertainty of about 1 ns.

In this article I will begin by describing the general chaacteristics of clocks and oscillators. I will then discuss htime and frequency are defined, how these quantitiesmeasured, and how practical standards for these quanare realized and evaluated. This characterization, andanalysis of time and frequency data in general, is comcated because the spectrum of the variation is almost nwhite and the conventional tools of statistical analysistherefore usually not appropriate or effective.

I will then discuss the basic principles that govern meods for distributing time and frequency information. Thchannels used for this purpose are often noisy, and this nis usually neither white nor very well behaved statisticallyis often possible to separate the degradations due to the nin the channel from the fluctuations in original data. Athough the details of how this can be done vary among

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different applications, we will see that all of them canunderstood in terms of a few basic principles. I will discuthe details of specific applications only insofar as is neeto illustrate these basic ideas.

I will not discuss the very large number of optical frequency standards that have been developed. Such dewould certainly qualify as standards from the stability poof view, but their outputs must be handled by optical raththan by electronic means at the current time, and the resing technology is very different as a result. In addition, itextremely difficult to use these devices as clocks in the ussense of that word—counting an optical frequency has bdone using various heterodyne methods, but these meaments are far from routine.

II. CHARACTERISTICS OF OSCILLATORS ANDCLOCKS

An oscillator is comprised of two components: a genetor that produces periodic signals and a discriminator tcontrols the output frequency. In many configurations,discriminator is actively oscillating and the output frequenof the device is set by a resonance in its response. Penduclocks and quartz–crystal oscillators are of this type—both cases the frequency is determined by a mechanical rnance which must be driven by an external power sourceother configurations, the discriminator is passive—frequency-selective property is interrogated by a variabfrequency oscillator whose frequency is then locked topeak of the discriminator response function using some foof feedback loop.

Atomic frequency standards fall into both categories. Lsers are usually~but not always! in the first category, whilecesium and rubidium devices are almost always in the s
ond. Hydrogen masers can be either active or passive—bothdesigns have advantages, as we will see below.
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Either type of oscillator can be used to construct a cloThe periodic output is converted to a series of pulses~usinga mechanical escapement or an electrical zero-crossingtector, for example!, and the resulting ‘‘ticks’’ are counted insome way. While the frequency of an oscillator is a functiof its mechanical or physical properties, the initial readinga clock is totally arbitrary—it has to be set based on soexternal definition. This definition cannot be specified frothe technical point of view—for historical and practical resons the definition that is used is based on the motion ofearth.1

The vast majority of oscillators currently in use are sbilized using quartz crystals. Even the quartz crystals usethe frequency reference in inexpensive wrist watcheshave a frequency accuracy of 1 ppm and a stability a faof 10 or more better than this. Substantially better perfmance can be achieved using active temperature conThey are usually the devices of choice when lowest crobust design, and long life are the most important considations. They are not well suited for applications~such asprimary frequency standards! where frequency accuracy olong-term frequency stability are important. There are treasons for this. The mechanical resonance frequencypends on the details of the construction of the artifact antherefore hard to replicate. In addition, the resonancequency is usually affected by fluctuations in temperatureother environmental parameters so that its long-term stabis relatively poor. A number of very clever schemes habeen developed to mitigate these problems, but none of tcan totally eliminate the sensitivity to environmental pertbations and the stochastic frequency aging that are the cacteristics of an artifact standard.2

Atomic frequency standards use an atomic or molecutransition as the discriminator. In the passive configuratithe atoms are prepared in one of the states of the clock tsition and are illuminated by the output from a separatecillator. The frequency of the oscillator is locked to the mamum in the rate of the clock transition. An atomic frequenstandard therefore requires a ‘‘physics package’’ that ppares the atoms in the appropriate state and detects thethat they have made the clock transition following the intaction with the oscillator. It also requires an ‘‘electronipackage’’ that consists of the oscillator, the feedbackcuitry to control its frequency, and various synthesizersfrequency dividers to convert the clock frequency to standoutput frequencies such as 5 MHz or 1 Hz. Since the tration frequencies are determined solely by the atomic strture in principle, these standards would seem to be freemany of the problems that limit the accuracy of artifact stadards. This is true to a great extent, but the details ofinteraction between the atoms and the probing frequencythe interaction between the physics and electronics packaffect the output frequency to some degree, and blurdistinction between a pure atomic frequency standard

2568 Rev. Sci. Instrum., Vol. 70, No. 6, June 1999

one whose frequency is determined by a mechanical artifa


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III. DEFINITION OF TIME INTERVAL OR FREQUENCY

Time interval is one of the base units of the InternationSystem of Units~the SI system!. The current definition of thesecond was adopted by the 13th General ConferenceWeights and Measures~CGPM! in 1967:

The second is the duration of 9 192 631 770 periods ofthe radiation corresponding to the transition betweenthe two hyperfine levels of the cesium 133 atom.3

This definition of the base unit can be used to deriveunit of frequency, the hertz. Multiples of the second, suchthe minute, hour, and day are also recognized for use inSI system. Although these common units are definedterms of the base unit just as precisely as the hertz is, thenot enjoy the same status in the formal definition of tinternational system of units.4

The definition of the SI second above implies that itrealized using an unperturbed atom in free space and thaobserver is at rest with respect to the atom. Such an obsemeasures the ‘‘proper’’ time of the clock, that is the resulta direct observation of the device independent of any cventions with respect to coordinates or reference frames

Practical timekeeping involves comparing clocks at dferent locations using time signals, and these processes iduce the need for coordinate frames. International atotime ~TAI ! is defined in terms of the SI second as realizedthe rotating geoid,5 and TAI is therefore a coordinate~ratherthan a proper! time scale.

This distinction has important practical consequencesperfect cesium clock in orbit around the earth will appearobservers on the earth as having a frequency offset withspect to TAI. This difference must be accounted for usingusual corrections for the gravitational redshift, the first- asecond-order Doppler shifts, etc. These frequency cortions are important in the use of the global positioning stem ~GPS! satellites, as we will discuss below.

A. Realization of TAI

TAI is computed retrospectively by the International Breau of Weights and Measures~BIPM!, using data suppliedby a world-wide network of timing laboratories. Each paticipating laboratory reports the time differences, measuevery five days~at 0 h onthose modified Julian day numbewhich end in 4 or 9!, between each of its clocks and ilaboratory time scale, designated UTC~lab!. To make it pos-sible to compare data from different laboratories, each laratory also provides the time differences between UTC~lab!and GPS time, measured using an algorithm defined byBIPM and at times specified in the BIPM tracking schedul~The GPS satellites are used only as transfer standardsthe satellite clocks drop out of the data.! Approximately 50laboratories transmit data from a total of about 250 clocksthe BIPM using these methods.

Using an algorithm called ALGOS, the BIPM computethe weighted average of these time-difference data to pduce an intermediate time scale called echelle atomique l~EAL!. The length of the second computed in this way

Judah Levine

ct.compared with data from primary frequency standards and acorrection is applied if needed so as to keep the frequency as


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close as possible to the SI second as realized by thesemary standards. The resulting steered scale is TAI. AsDecember, 1998, the fractional frequency differenf (EAL) 2 f (TAL) 57.13310213. At that same epoch, thfractional frequency offset between TAI and the SI secondrealized on the rotating geoid was estimated by the BIPMbe

d5~uTAI2u0!/u05~24610!310215.

The determination of the uncertainty for the estimate ofd iscomplex. It includes the uncertainties~usually the ‘‘type B’’uncertainty, which we define later! associated with the datfrom the different primary frequency standards; extrapoing these data to a common interval also requires a modethe stability of EAL. The details are presented in repopublished by the BIPM.6

B. Coordinated universal time and leap seconds

It is simple in principle to construct a timekeeping sytem and a calendar using TAI. When this scale was definits time was set to agree with UT2~a time scale derived fromthe meridian transit times of stars corrected for seasovariations! on 1 January 1958. Unfortunately, the lengththe day defined using 86 400 TAI seconds was shorter tthe length of the day defined astronomically~called UT1! byabout 0.03 ppm, and the times of the two scales begadiverge immediately. If left uncorrected, this divergenwould have continued to increase over time, and its rwould have slowly increased as well as the rotational ratethe earth continued to decrease.

Initially, this divergence was removed by introducinfrequency offsets between the frequency used to run atoclocks and the definition of the duration of the SI secoSmall time steps of 0.05 or 0.1 s were also introducedneeded.7 This steered time scale was called coordinated uversal time~UTC!—it was derived from the definition of theSI second, but was steered~that is, ‘‘coordinated’’! so that ittracked the astronomical time scale UT1.

This method of coordinating UTC turned out to be awward in practice. The frequency offsets required for coornation were on the order of 0.03 ppm, and it was difficultinsert these frequency offsets into real-world clocks. OnJanuary 1972, this system was replaced by the currenttem which uses integral leap seconds to realize the steeneeded so that atomic time will track UT1. The rate of UTis set to be exactly the same as the rate of TAI, anddivergence between UTC and astronomical time is remoby inserting ‘‘leap’’ seconds into UTC as needed. The leseconds are inserted so as to keep the absolute magnituthe difference between UTC and UT1 less than 0.9 s. Thleap seconds are usually added on the last day of JunDecember; the most recent one was inserted at the end oDecember 1998 and the time difference TAI-UTC becaexactly 32 s when that happened. The leap seconds arways positive, so that UTC is behind atomic time, and ttrend will continue for the foreseeable future.

Rev. Sci. Instrum., Vol. 70, No. 6, June 1999

Although this system of leap seconds is simple in concept, it has a number of practical difficulties. A leap secon


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is inserted as the last second of the day~usually on the lastday of either June or December!, after 23:59:59, whichwould normally have been the last second of the day.name is 23:59:60, and the next second is 00:00:00 of theday. The first problem is that it is difficult to define a timtag for an event that happens during the leap second.time does not even exist on conventional clocks, for eample, and a time tag of 23:59:60 might be rejected aformat error by a digital system. Furthermore, there isnatural way of naming the leap second in systems that douse the hh:mm:ss nomenclature. Many computer systemsinto this category because they keep time in units of seco~and fractions of a second! since some base date.

Calculating the length of a time interval that crossesleap second presents additional ambiguities. An interbased on atomic time or on some other periodic processdiffer from a simple application of a calculation based on tdifference in the UTC times, since the latter calculationfectively ‘‘forgets’’ that a leap second has happened onchas passed.

There is no simple way of realizing a time scale thatboth smooth and continuous and simultaneously tracks UThis is because the intervals between leap seconds areexactly constant and predictable, and there is no simplegorithm that could predict them automatically very far inthe future. Each leap second is announced several montadvance by the International Earth Rotation Service in Pa8

Thus, changing the SI definition of the second~which wouldproduce difficulties in the definitions of other fundamenconstants anyway! could reduce the frequency of leap seonds, but would not entirely remove the need for them. Tproblem would return in the future anyway as the lengththe astronomical day increased.

It is possible to avoid the difficulties associated with leseconds by using a time scale that does not have them,as TAI or GPS time. There are a number of practical diculties with this solution, such as the facts that most timlaboratories disseminate UTC rather than TAI, and that Gtime may not satisfy legal traceability requirements in sosituations. Practical realizations of this idea often requirecillary tables of when leap seconds were inserted into UT

IV. CESIUM STANDARDS

Cesium was chosen to define the second for a numbepractical reasons. The perturbations on atomic energy leare quite small in the low-density environment of a beaand atomic beams of cesium are particularly easy to prodand detect. The frequency of the hyperfine transition tdefines the second is relatively high and the linewidth canmade quite narrow by careful design. At the same timetransition frequency is still low enough that it can be mnipulated using standard microwave techniques and circu

In spite of these advantages, constructing a devicerealizes the SI definition of the second is a challenging aexpensive undertaking. Although the frequency offsets pduced by various perturbations~such as Stark shifts, Zeema

2569Judah Levine

-dshifts, Doppler shifts,...! are small in absolute terms, they arelarge compared to the frequency stability that can be


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achieved in a well-designed device. A good commercialsium frequency standard, for example, might exhibit frational frequency fluctuations of 2310214 for averagingtimes of about one day. The frequency of the same demight differ from the SI definition by 1310213 or more~sometimesmuch more!, and this frequency offset machange slowly with time as the device ages.~This frequencyoffset is what remains after corrections for the perturbatimentioned above have been applied. If no correctionsapplied, the fractional frequency offset is usually dominaby Zeeman effects, which can be as large as 1310210.!

Constructing a device whose accuracy is comparablits stability is a difficult and expensive business, and onlfew such primary frequency standards exist. It is usuallypossible to reduce the systematic offsets to sufficiently smvalues, and the residual offset must be measured andmoved by means of a complicated evaluation processoften takes several days to complete.9 The operation of thestandard may be interrupted during the evaluation, somany primary frequency standards cannot operate contously as clocks. In addition, the systematic offsets mchange slowly with time even in the best primary standaso that evaluating their magnitudes is a continuing~ratherthan a one-time! effort.

A simple oven can be used to produce a beam of cesatoms. The oven consists of a small chamber and a naslit through which the atoms diffuse. The operating tempeture varies somewhat, but is generally less than 100 °Cthat the thermal mean velocity of the cesium atoms is sowhat less than 300 m/s. Operating the oven at a higher tperature increases the flux of atoms in the beam and thfore the signal to noise ratio of the servo loop to controllocal oscillator, but the velocity of the atoms also increaas the square root of the temperature. This decreasesinteraction time and increases the linewidth by the sametor, but the more serious problem is usually that the collimtion of the beam is degraded. The decrease in interactime resulting from raising the temperature can be offsetusing only the low-frequency ‘‘tail’’ of the velocity distribution.

Conventional frequency standards use an inhomoneous magnetic field~the ‘‘A’’ magnet! as the state selectoThe atoms enter the A magnet off axis and are deflectedthe interaction between the inhomogeneous magneticand the magnetic dipole moment of the atom. The geomis arranged so that only atoms in theF53 state emerge fromthe magnet—atoms in the other state are blocked by achanical stop. It is not practical to make the magnetic fieldthe A magnet large enough to separate themF sub levels ofthe F53 state.

The atoms that emerge from the A magnet enter anteraction region~the ‘‘C’’ region!, where they are illumi-nated by microwave radiation in the presence of a consmagnetic field, which splits the hyperfine levels becausethe Zeeman effect. Although the transition between theF53 mF50 to F54 mF50 is offset somewhat as a resultthis field, the sensitivity to magnetic field inhomogeneities


reduced, and the clock transition is independent of the manetic field value in first order. The actual value of the field is


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usually measured by observing the frequencies of fiedependent transitions between othermF states.

The microwave radiation that induces the clock trantion is applied in two separate regions separated by a dspace~Ramsey configuration10!. The atoms experience twkinds of transitions: ‘‘Rabi’’ transitions due to a single inteaction with the microwave field, and ‘‘Ramsey’’ transitiondue to a coherent interaction in both regions. The Rabi trsitions have a shorter interaction time and therefore a lalinewidth. The magnitude of the magnetic field in the C rgion is usually chosen so that the Rabi transitions betwthe different sub levels will be well resolved. A typical valufor a conventional thermal-beam standard would be ab531026 T. It is possible to use a lower value of magnefield which does not completely resolve the Rabi transitioWhile this reduces the Zeeman correction, it complicatesanalysis of the Ramsey line shapes.

The length of this drift space and the velocity of thatoms determine the total interaction time, which in turn sthe fundamental linewidth of the device. The microwave fquency is adjusted until the transition rate in the atoms imaximum; the transitions are detected using a second inmogeneous magnetic field, which is created by the ‘‘Bmagnet. The gradient of the B field is usually configuredas to pass only those atoms that are in the final state ofclock transition~a ‘‘flop-in’’ configuration!. This configura-tion eliminates the shot noise in the detected beam duatoms that do not make the clock transition in the C regiHowever, it is more difficult to test and align, since thereno signal at all until everything is working properly.

The state-selection process in a conventional cesiumvice is passive—it simply throws away those atoms thatnot in the proper state. There are 16 magnetic sub levelthe ground state: 9 forF54 and 7 forF53 ~that is, 2F11 in both cases!. All of them have essentially the samenergy and are therefore roughly equally populated in a thmal beam. As a consequence, only 1/16 of the flux emergfrom the oven is in a singleF, mF state, and only this smalfraction of the atoms is actually used to stabilize the oscitor. This unfavorable ratio can be improved using active stselection—that is, by optically pumping the input beam soto put nearly all of the atoms into the initial state of the clotransition. A similar technique can be used to detect theoms that have undergone the clock transition in the intertion region. This design increases the number of atomsare used in the synchronization process and thereforesignal to noise ratio of the error voltage in the servo loop tcontrols the oscillator. In addition, it removes some of toffsets and problems associated with the fields due to thand B magnets. Although the optical pumping techniquthat are needed to realize this design have been knownmany years,11 realizing a standard using these principles wnot practical until recently when lasers of the appropriwavelengths became available.

Another technique for improving the performance of tdevice is to increase the interaction time in the C regionincreasing its length or by decreasing the velocity of t

Judah Levine

g-atoms. The ultimate version of this idea may be an ‘‘atomicfountain’’ in which atoms are shot upward through the inter-


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action region, reverse direction, and fall back down throuthe same region a second time. This idea was proposed mthan 40 years ago,12 but has only recently become feasiblea result of developments in laser cooling and trapping. Tfountain idea has a number of advantages over one-waysigns. The atoms are moving more slowly and the interactime can be longer as a result. In addition, the fact that atotraverse the interaction region in both directions can be uto cancel a number of systematic offsets that would othwise have to be estimated during the evaluation process

V. OTHER ATOMIC STANDARDS

Cesium is not the only atom that can be used as a reence for a frequency standard; transitions in rubidium ahydrogen are also commonly used. In order to relate thfrequencies to the SI definition, devices based on rubidor hydrogen must be calibrated with respect to primcesium-based devices if accuracy is important to their uAs we have discussed above, this is true for commercesium devices as well if the accuracy of the output fquency is to approach the frequency stability of the devIn spite of this similarity, there are real differences amothe devices.

Rubidium devices usually use low-pressure rubidiumpor in a cell filled with a buffer gas such as neon or heliurather than the atomic-beam configuration used for cesiThe clock transition is theF51, mF50 to F52, mF50transition in the2S1/2 ground state of87Rb, its frequency isabout 6.83 GHz. The two states of the clock transitionessentially equally populated in thermal equilibrium.

The atoms are optically pumped into theF52 state.There are a number of ways of doing this. One commmethod uses a discharge lamp containing87Rb whose light ispassed through an absorption cell containing85Rb. The lightemerging from the lamp could induce optical frequency trsitions from both theF51 andF52 states to higher excitelevels. However, there is a coincidence between transitin 85Rb and87Rb. The effect of this coincidence is that thlight which would have induced transitions originating frothe F52 state is preferentially absorbed in passing throuthe85Rb filter, whereas the component that interacts withF51 lower state is not. When the filtered light enters tcell, it preferentially excites theF51 state of the clock transition because the light that would interact with theF52state has been preferentially absorbed in the85Rb cell. Theexcited atoms decay back to the two ground sub levThose atoms which decay back to theF51 state are preferentially excited again, while those atoms that decay bactheF52 state tend to remain there. The result is to builda nonequilibrium distribution in these two sub levels. Ascesium, a magnetic field is applied to split the hyperfine slevels and to reduce the sensitivity to any residual inhomgeneities and fluctuations in the ambient field. When thecrowave clock frequency is applied, theF51, mF50 state ispartially repopulated by the induced transition betweentwo clock states, the absorption of the incident light from t


lamp increases again, and the transmitted intensity dropThis decrease in transmitted intensity~typically on the order


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of 1%! is used to lock the incident microwave frequencythe clock transition.

The beauty of this design is that the microwave trantion modulates an optical intensity, and changes in theible photon flux are much easier to detect than the sameof microwave photons would be. On the other hand, thecrease in the transmitted light through the cell must betected against the shot noise in the full background lifrom the lamp. Using a discriminator based on the visibflux from the lamp makes the device simpler and cheapbut it also makes for poorer stability and larger and movariable frequency offsets than would be found in a gocesium-based device. This is because of pressure shifts icells, light shifts due to the lamp, a large dependence ofclock frequency on the ambient magnetic field, and oteffects. The values of many of these parameters muschosen as a compromise between accuracy and stabilithigher-intensity lamp, for example, increases the signanoise ratio in the resonance-detection circuit, but alsocreases the light shift~the ac Stark effect on the clock transition!. A typical rubidium-stabilized device might havestability and an accuracy 100 times~or more! poorer than adevice using cesium in the standard beam configurationterms of the Allan deviation~to be defined later!, the perfor-mance is generally about 3310211/t1/2 for averaging times~that is, values oft! between about 1 and 104 s; the long-term frequency aging is typically on the order of310211/month. On the other hand, they are usually substtially cheaper, lighter, and smaller than standard cesiumvices.

Both the advantages and the limitations of rubidium dvices are due to the implementation in an optically pumpcell with a buffer gas rather than to any inherent differenbetween cesium and rubidium. A cesium-based deviceused a cell rather than a beam, for example, might hmany of the advantages of both systems, and such a dehas been under development for some time. Furthermorfountain based on rubidium would have a number of advtages over a cesium-based device. Although the clock tsition has a lower frequency in rubidium, the atom–atocollision cross section in the cold beam of a fountain is sstantially smaller than in cesium13 so that the beam flux canbe greater. The short-term stability will be improved asresult; a number of other systematic offsets would alsosmaller. ~Improving the short-term stability is a very valuable advantage, since it eases the requirements on the sity of the local oscillator which interrogates the atoms awhich acts as a flywheel between cycles of the fountain.!

As with cesium and rubidium, the clock transition inhydrogen maser is a hyperfine transition in the ground sof the atom. TheF50 andF51 hyperfine states are seprated using an inhomogeneous field as in a cesium devbut instead of using the beam geometry of a cesium-badevice, the atoms in the initial state of the clock transitienter a bulb that is inside of a microwave cavity.14 In apassive maser, the atoms in the cavity are probed witmicrowave signal at 1.42 GHz, and the interaction with t

2571Judah Levine

s.atoms produces a phase shift in the reflected signal that isdispersive about line center. In an active maser, the losses of


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The frequency of oscillation of an active maser is a funtion of both the cavity resonance frequency and the atotransition frequency, so that some means of stabilizingcavity is usually required. Collisions with the walls of thbulb also affect the frequency of the device. Various stragies are used to minimize these effects, including applyspecial ‘‘nonstick’’ coatings to the inside of the bulb anactively tuning the resonance frequency of the cavity soit is exactly on atomic line center.

In spite of these techniques, hydrogen masers usuhave frequency offsets that are large compared to the stity of the device.~The fractional frequency offset of a hydrogen maser is often of order 5310211, while the fractionalfrequency fluctuations at one day can be as small a310216.! In addition to being significant, these offsets usally change slowly with time in ways that are difficult tpredict. Nevertheless, the frequency stability over short pods~usually out to at least several days! can be much bettethan the best cesium device. Hydrogen masers are therethe devices of choice when the highest possible frequestability is required and cost is not an issue.

Finally, a number of other transitions have been usedstabilize oscillators. Some of the most stable devicesthose using a hyperfine transition in the ground state ofalkali-like ion,15 such as199Hg1.16 This ion has a structuresimilar to cesium, with a single 6s electron outside of aclosed shell. The clock transition is the hyperfine transitin the ground state analogous to the transition in cesiumhas a frequency of 40.5 GHz. Using a linear ion trapconfine the ions, frequency standards with a stability o310214/t1/2 ~wheret is the averaging time in seconds! havebeen reported15 and devices based on other ions are bedeveloped.17 These devices have the advantage that theteraction time can be much longer than in a conventiobeam configuration.

VI. QUARTZ CRYSTAL OSCILLATORS

Each of the devices we have discussed above is immented using a quartz–crystal oscillator whose frequencstabilized to the atomic transition using some form of feeback loop. The details of the loop design vary from odevice to another, but the loop will always have a finbandwidth and therefore a finite attack time. The interoscillator is essentially free running for times much shorthan this attack time, and the stability of all atomic-stabilizoscillators is therefore limited at short periods to the frerunning stability of the internal quartz oscillator.~Theboundary between ‘‘short’’ and ‘‘long’’ periods varies fromone device to another, of course. A typical value for macommercial devices is probably on the order of seconds!

An obvious simplification would be to use the baquartz–crystal oscillator and to dispense altogether withstabilizer based on the atomic transition. The stabilitysuch a device is clearly the same as an atomic-based sy


at sufficiently short times. The stability at longer times willdepend on how well the quartz crystal frequency referenc


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can be isolated from environmental perturbations, or hwell the effects of these perturbations can be estimatedremoved.2 Quartz–crystal oscillators can be surprisinggood in this respect. Even cheap wrist watches have osctors that may have a frequency accuracy of about 1 ppma stability 10 or 50 times better than this. The residual fquency fluctuations are largely due to fluctuations in the abient temperature, and stabilizing the temperature canprove the performance.

VII. MEASURING TOOLS AND METHODS

The oscillators that we have discussed above generproduce a sine-wave output at some convenient frequesuch as 5 MHz.~This frequency may also be divided dowinternally to produce output pulses at a rate of 1 Hz. Crysdesigned for wrist watches and some computer clocks ooperate at 32 768 Hz to simplify the design of these 1dividers.! A simple quartz–crystal oscillator might operadirectly at the desired output frequency; atomic standarelate these output signals to the frequency appropriate toreference transition by standard techniques of frequemultiplication and division. The measurement system thoperates at a single frequency independent of the typeoscillator that is being evaluated. The choice of this fquency involves the usual trade-off between resolutiwhich tends to increase as the frequency is made higher,the problems caused by delays and offsets within the msurement hardware, which tend to be less serious as thequency is made lower.

Measuring instruments generally have some kind of dcriminator at the front end—a circuit that defines an evenoccurring when the input signal crosses a specified referevoltage in a specified direction. Examples are 1 V with apositive slope in the case of a pulse, or 0 V with a positiveslope in the case of a sine-wave signal. The trigger poinchosen at~or near! a point of maximum slope, so as to minmize the variation in the trigger point due to the finite ritime of the wave form.

The simplest method of measuring the time differenbetween two clocks is to open a gate when an event is tgered by the first device and to close it on a subsequent efrom the second one.18 The gate could be closed on the venext event in the simplest case, or theNth following onecould be used, which would measure the average time inval over N events. The gate connects a known higfrequency oscillator to a counter, and the time interval btween the two events is thus measured in units of the peof this oscillator. The resolution of this method dependsthe frequency of this oscillator and the speed of the counwhile the accuracy depends on a number of parameterscluding the latency in the gate hardware and any variatiin the rise time of the input wave forms. The resolution cbe improved by adding an analog interpolator to the digcounter, and a number of commercial devices usemethod to achieve sub-nanosecond resolution withoutneed for a reference oscillator whose frequency is 1 GHz

Judah Levine

egreater.

In addition to these obvious limitations, time-difference


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measurements using fast pulses have additional probleReflections from imperfectly terminated cables may distthe edge of a sharp pulse, and long cables may have enshunt capacitance to round it by a significant amount.addition to distorting the wave forms and affecting the trger point of the discriminators, these reflections can altereffective load impedance seen by the oscillator and pull itfrequency. Isolation and driver amplifiers are usuallyquired to minimize the mutual interactions and complicareflections that can occur when several devices must benected to the same oscillator, and the delays through thamplifiers must be measured. These problems can bedressed by careful design, but it is quite difficult to constra direct time-difference measurement system whose msurement noise does not degrade the time stability of aquality oscillator, and other methods have been develofor this reason. Averaging a number of closely spaced timdifference measurements is usually not of much help becathese effects tend to be slowly varying systematic offswhich change only slowly in time.

Many measurement techniques are based on someof heterodyne system. The sine-wave output of the oscillaunder test can be mixed with a second reference oscillwhich has the same nominal frequency, and the much lodifference frequency can then be analyzed in a numbedifferent ways. If the reference oscillator is loosely lockedthe device under test, for example, then the variations inphase of the beat frequency can be used to study thefluctuations in the frequency of the device under test. Terror signal in the lock loop provides information on thlonger-period fluctuations. The distinction between ‘‘fasand ‘‘slow’’ would be set by the time constant of the locloop.

This technique can be used to compare two oscillaby mixing a third reference oscillator with each of them athen analyzing the two difference frequencies using the timinterval counter discussed above. In the version of this ideveloped at NIST,19 this third frequency is not an independent oscillator, but is derived from one of the input signusing a frequency synthesizer. The difference frequencya nominal value of 10 Hz in this case. The time intervcounter runs with an input frequency of 10 MHz and ctherefore resolve a time interval of 1026 of a cycle. This isequivalent to a time-interval measurement with a resolutof 0.2 ps at the 5 MHz input frequency.

All of these heterodyne methods share a common advtage: the effects of the inevitable time delays in the measment system are made less significant by performingmeasurement at a lower frequency where they make a msmaller fractional contribution to the periods of the signunder test. Furthermore, the resolution of the final timinterval counter is increased by the ratio of the input frequcies to the output difference frequencies~5 MHz to 10 Hz inthe NIST system!. This method does not obviate the needcareful design of the front-end electronics—if anything tincreased resolution of the back-end measurement syplaces a heavier burden on the high-frequency portions of


circuits and the transmission systems. As an example, tstability of the National Institute of Standards and Techno


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ogy ~NIST! system is only a few ps—about a factor of 1020 poorer than its resolution.

Heterodyne methods are well suited to evaluatingfrequency stability of an oscillator, but they often have prolems in measuring time differences because they usuhave an offset, which is an unknown number of cycles ofinput frequencies. Thus the time difference between tclocks measured using the NIST mixer system is offset wrespect to measurements made using a system based onHz pulse hardware by an arbitrary number of periods of thMHz input frequency~i.e., some multiple of 200 ns!. Tofurther complicate the problem, this offset can change ifpower is interrupted or if the system stops for any othreason.

The offset between the two measurement systems mbe measured initially, but it is not too difficult to recoverafter a power failure, since the step must be an exact multof 200 ns. Using the last known time difference and frquency offset, the current time can be predicted usinsimple linear extrapolation. This prediction is then compawith the current measurement, and the integer numbecycles is set~in the software of the measurement system! sothat the prediction and the measurement agree. This conis then used to correct all subsequent measurements.lack of closure in this method is proportional to the frquency dispersion of the clock multiplied by the time inteval since the last measurement cycle, and the procedureunambiguously determine the proper integer if this time dpersion is significantly less than 200 ns. This criterioneasily satisfied for a rubidium standard if the time intervalless than a few hours; the corresponding time intervalcesium devices is generally at least a day.

Given the difficulties of making measurements usingHz pulses, it is natural to consider abandoning them in faof phase measurements of sine waves. The principal reafor not doing this now is that almost all of the methods thare used to distribute time are designed around 1 Hz ticAs we will see below, extracting 1 Hz ticks from a GPsignal is a straightforward business in principle, while phacomparisons between a local 5 MHz oscillator and the saGPS signal have a number of awkward aspects and ambities. Very similar problems also make measurements oHz pulses the method of choice in two-way satellite timtransfer.

VIII. NOISE AND MEASURES OF STABILITY

A. Introduction to the problem

We now turn to looking at ways of characterizing thperformance of clocks and oscillators. The simple notionsGaussian random variables will turn out to be inadequatethis job, because many of the noise processes in clocksnot even close to Gaussian. In particular, the commontions of mean and standard deviation~which completelycharacterize a Gaussian distribution! must be applied withgreat care—while their values usually exist in a formal senthey do not have the nice properties that we usually think

2573Judah Levine

hel-when these parameters are used to characterize a true Gauss-ian variable. In particular, calculating either the mean or the


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standard deviation of one of these data set results inmates that are not stationary and that depend on the lengthe time series used for the analysis.

When we speak of the time or frequency of a clockthe following discussion, we almost always meanrelativevalues. Time is measured as the difference~in seconds! be-tween the device under discussion and a second idennoiseless one, and frequency is thefractional frequency dif-ference between the same two devices. If the nominalquency of the devices isv0 , and if the phase differencbetween them at some instant isf, then the fractional fre-quency difference,y, is given in terms of the time derivativof the phase by

y5f

2pv0. ~1!

Using this convention, frequencies are dimensionless quaties. We will also use the term frequency aging to denslow changes in the fractional frequency~where slow meanslong with respect to 1/y and certainly with respect to thmuch smaller 1/v0!. The unit of frequency aging is s21.

B. Separation of variance

Noiseless measuring systems and perfect clocksscarce commodities, and we must use real devices in asystem and extract from these data the performance paeters of each of the devices. One of the most importantpects of this job is the concept of separation of variancthat is identifying which part of the system is responsiblethe observed fluctuations in the data. This idea most oarises in two contexts:

~a! We are trying to evaluate the performance of a devor system and we would like to be able to separateperformance from the noise due to the system thatare using as a reference. Both systems can be phyclocks or pseudo clocks, such as time scales. Thisis easy if the reference is much less noisy thandevice under test; if this is not the case then the ‘‘thrcornered hat’’ method can be used:20

Suppose we have three devices whose individvariances ares i

2, s j2, and sk

2. We wish to estimatethese parameters, but we can only compare the devto each other using pair-wise comparisons. The msured variances of these pair-wise comparisons ares i j

2 ,s jk

2 , andski2 . If the variances of the three devices a

roughly equal, then we can estimate the three invidual variances by

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These estimates can be misleading if the variances a


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correlated—the variances estimated in this way ceven be negative in this case despite the fact that thimpossible on physical grounds.

~b! We are trying to evaluate~or synchronize! a deviceusing data that we receive from a distant calibratisource over a noisy channel. Even though the referedevice may be much more stable than the device untest, its data are degraded by the channel noise.

It would be very helpful to be able to separate tcontributions to the variance due to the channel andlocal clock in this configuration. This separation is important for two reasons. In the first place, it tellswhich part of the overall system is limiting its overaperformance, and where we should spend time andfort to improve things. In the second place, separatof variance plays a central role in designing methofor synchronizing clocks when the calibration data adegraded in this way.

This situation arises in a number of difference cotexts. We discuss later two common situations: sychronizing local clocks to GPS signals which are dgraded by selective availability and other effects asynchronizing clocks using time signals transmittover packet networks such as the Internet.

C. Measurement noise

All of the instruments that are used for time and frquency measurements have some kind of discriminator ainput—something that is triggered when the input wave focrosses some preset threshold. Any noise on the input sicombines with the internal noise of the circuit to producejitter in the trigger point. These effects will produce a flutuation in the measured time difference between two clothat has nothing to do with the clocks themselves—it aripurely in the measurement process and there is no cosponding jitter in the underlying frequency difference btween the two devices. It is especially important thatdevelop techniques that can identify the source of these fltuations in the data in applications where the goal is to schronize the local clock to a second standard. Adjustingfrequency of the local clock to compensate for this measutime jitter is almost always the wrong thing to do, as we wsee below.

If the input data are a sine wave with amplitudeA andfrequencyv, then additive noise at the input to the discrimnator of amplitudeVn will result in a time jitter whose am-plitude is approximately

Dt>Vn

Av. ~3!

We usually assume that in each discriminator the equivanoise voltage at the input is not correlated with the corsponding input signal, so that the time jitter in each chanhas no mean offset relative to the corresponding noise-trigger point. Consecutive measurements of the time diffence are then randomly distributed about a mean value

Judah Levine

rerepresents the actual time difference between the two clocks.If the instantaneous noise-induced offset in the trigger point


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at some epocht is e(t), then the independence between tsignal and the noise in each channel implies that

ê~ t !e~ t1t!&5ê2~ t !&d~t!. ~4!

Since the autocorrelation ofe(t) is a delta function, itspower spectrum is constant for all frequencies, and measment noise of this type is often called white phase noisethis reason.~Although white phase noise is often thoughtas a Gaussian variable, this is not necessarily true. Whileproperties described above arenecessaryfor the statisticaldistribution of the phase noise to be Gaussian, they aresufficientto guarantee this. Nevertheless, although the invidual noise-induced offsets may or may not be Gaussconsecutive estimates of their mean will exhibit a Gaussdistribution as a result of the central limit theorem of stattics.!

Note that Eq.~4! is only an approximation—we cannomaket either very small or very large and so the autocorlation is in fact bounded at small times by the finite banwidth of the measurement system and at long times bylength of the experiment. A real-world spectrum will therfore be not quite ‘‘white’’ for very short or very long measurement intervals but this is usually not important. Thenite response time of the overall system limits timportance of very high-frequency noise, and other sourof noise always become important at longer averaging tim

If the white phase noise is superimposed on a consfrequency offset between the two devices, then consecumeasured time differences will scatter uniformly about a lwhose slope is given by this offset frequency. Becausethis uniform scatter, the frequency difference betweentwo oscillators can be estimated using the usual least-squmachinery. Although the white phase noise will contributethe uncertainty of this estimate, it will not introduce a biasfrequency estimates computed from different blocks of dwill scatter uniformly about the true frequency offset btween the two devices. The statistical distribution of thefrequency estimates can be calculated using the stanmethods of propagation of error.

D. Frequency noise processes

Although white phase noise arises in the measuremprocess and is not due to the oscillator itself, there arenoise sources within the oscillator that we must consider.we discussed above, an atomic clock~or any other oscillatorfor that matter!, is comprised of two basic components:frequency reference~such as an atomic transition or a vibrtional mode in a piece of quartz! and an oscillator that interrogates this reference frequency and is locked to it by meof some kind of feedback loop. As with all circuits, there winevitably be some noise in this feedback loop~due to shotnoise in the number of atoms that are interrogated or Johnnoise in the circuit resistors!, and this noise will producejitter in the lock point of the oscillator. In the best of apossible cases, this noise will not be correlated with the


nals in the feedback loop. Using exactly the same argumeas above, this noise will produce jitter in the output fre


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quency of the device whose autocorrelation function isdelta function. By analogy with the argument above,would call this white frequency noise.

The frequency fluctuations in the oscillator result in timfluctuations as well. If we started the device at some time2Tin the past, and if the instantaneous frequency at any timy(t), then the time difference at some epocht between thisoscillator and a second noiseless but otherwise identicalvice would be given by

X~ t !5E2T

t

y~t!dt. ~5!

The integration of the instantaneous frequency offset to pduce a time difference has an important consequence forpower spectrum of the resulting time fluctuations. If the flutuations in y(t) are dominated by white frequency noisthen its spectral density is a constant at all frequencies~keep-ing in mind the comment above that the infinities impliedthis formal description are not a problem because of thenite limits both in time and in frequency of a real-worsituation!. That is, the Fourier expansion ofy(t), the fre-quency offset of the oscillator as a function of time, is givin terms of the Fourier frequencyV by

y~t!5E Y~V!eiVtdV, ~6!

where

uY~V!u5C ~7!

andC is a constant. Keep in mind that the variabley repre-sents the real fractional frequency difference betweennominally identical physical oscillators. Unfortunately, thquantity is not a constant. The expressiony(t) expresses thevariation in y in the time domain as a function of epocht,while Y(V) expresses this variation in the Fourier frequendomain as a function of Fourier frequencyV. Also note thatEq. ~7! cannot be literally correct for a true noise proceWe will discuss this point in more detail below.

Using Eqs.~5!, ~6!, and ~7!, we can see that the powespectral density ofX(t) is not white but rather varies as 1/V2

with an amplitude determined byC in Eq. ~7!. Thus there isa fundamental difference between white phase noisewhite frequency noise. The former results in time-differenfluctuations that still have a white spectrum, while the ingration implied by the relationship between time and fquency causes white frequency noise to have a ‘‘reFourier-frequency dependence when its effect on timdifference measurements is considered. This is an imporconclusion because it means that we can distinguish betwfrequency noise and phase noise by examining the shapthe Fourier spectrum of the time-difference data.~The formof the dependence on Fourier frequency distinguishestwo cases—not the magnitude of the spectral density itse!

Unlike the white phase noise case, if there is a determistic frequency offset between two devices whose noise sptra are characterized by white frequency noise, it is no lontrue that the time differences between the devices will sca

2575Judah Levine

nt-uniformly about a simple straight line determined from thisoffset. A least-squares fit of a straight line to these time


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differences will not result in a statistically robust estimatethis frequency offset for this reason.~We mean by this thathe mean difference between these estimates and the ulying deterministic frequency offset is zero,and that themean and variance are characteristics of the data set anindependent of the sample size or of which portion ofdata is used for computing the estimate.! However, consecu-tive frequency estimates~i.e., the first difference of the timedifferences normalized by the time interval between the!are distributed uniformly about the mean frequency offsand the average of these first differences will produce atistically robust estimate of the deterministic frequency offwith a well-defined variance.

These results illustrate a more general conclusinamely that there is no single optimum method for estiming clock parameters. The best strategy depends on thederlying noise type. Unfortunately, there is no simple opmum strategy for some types of noise~flicker processes, forexample!, and the best we can do is an approximate analwhose predictions are not statistically robust.

It is clear that these ideas could be extended to highorder derivatives. If we had an oscillator in which the frquencyaging, d(t), had uncorrelated fluctuations about~possibly nonzero! mean value, then, by analogy with E~5!, the frequency at any time would be given by

y~ t !5y~0!1E d~t!dt. ~8!

Using the same argument as above, this white frequencying would result in fluctuations in the frequency differenbetween this clock and an identical noiseless one that vaas 1/V2 relative to the spectral density of the aging itseThe time differences between the two devices would havspectrum that varied as 1/V4 relative to the aging, since thaging must be integrated once to compute the frequencya second time to compute the time difference. Again,could distinguish the different contributions by examinithe shape of the Fourier spectrum of the time difference

Using the same sort of argument as above, whennoise spectrum of the device is dominated by white fquency aging, we cannot make a statistically robust estimof the frequency of a device by averaging the consecufrequency estimates, since those estimates are no longetributed uniformly about a mean value. The best we cannow is to estimate a mean value for the frequency aging

The previous discussion used the power spectrum oftime differences as the benchmark, but it is also commonthe literature to find a discussion in which the relative depdencies on Fourier frequency are described with respecthe power spectrum of the frequency fluctuations. The retionship between frequency and phase specified in Eq.~5! isunchanged, of course—only the benchmark spectrum isferent. White phase noise would then be characterizedhaving a relative dependence proportional to the squarthe Fourier frequency in this case. A white frequency-agprocess would be described as having a relative dependon Fourier frequency of 1/V2, where both of these spectr


would be evaluated relative to the power spectrum of thfrequency fluctuations.


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Since the aging and the frequency specify the time elution of the frequency and the time, respectively, white fquency aging is sometimes called random-walk frequenoise in the literature and white frequency noise is somtimes called random-walk phase noise.

There is another way of thinking of the differenceamong the types of noise processes. If a time-difference msurement process can be characterized as being limitepure white phase noise, then there is an underlying timeference between the two devices. The measurements scabout the true time difference, but the distribution of tmeasurements~or at worst the mean of a group of them! canbe characterized by a simple Gaussian distribution with otwo parameters: a mean and a standard deviation. Weimprove our estimate of the mean time difference by avering more and more observations, and this improvementcontinue forever in principle. There is no optimum averagitime in this simple situation—the more data we are prepato average the better our estimate of the mean time differewill be. As we discussed above, a least-squares fit ostraight line to these time differences can be used to coma statistically robust estimate of the frequency differencetween the two devices.

The situation is fundamentally different for a measument in which one of the contributing clocks is dominatedzero-mean white frequency noise. Now it is the frequenthat can be characterized~at least approximately! by a singleparameter—the standard deviation. If the time differencetween two identical devices at some epoch isX(t), we wouldestimate the time difference a short time in the future us

X~ t1t!5X~ t !1y~ t !t, ~9!

wherey(t) is the instantaneous value of the white frequennoise, and

^y~ t !&50, ~10!

^y2~ t !&5e2. ~11!

Sincey(t) has a mean of 0 by assumption, the time diffeence at the next instant is distributed uniformly aboutcurrent value ofX, and the mean value ofX(t1t) is clearlyX(t). In other words, for a clock whose performancedominated by zero-mean white frequency noise, the omum prediction of the next measurement is exactly the crent measurement with no averaging. Note that this doesmean that our prediction is that

X~ t1t!5X~ t !, ~12!

but rather that

^X~ t1t!2X~ t !&50, ~13!

which is a much weaker statement because it does not mthat our prediction will be correct, but only that it will bunbiased on the average. This is, of course, the best thacan do under the circumstances. The frequencies in constive time intervals are uncorrelated with each other by d

Judah Levine

enition, and no amount of past history will help us to predictwhat will happen next. The point is that this is the opposite


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extreme from the discussion above for white phase nowhere the optimum estimate of the time difference wastained with infinite averaging of older data.

Clearly, there must be an intermediate case betwwhite phase noise and white frequency noise, where samount averaging would be the optimum strategy, anddomain is called the ‘‘flicker’’ domain. Physically speakinthe oscillator frequency has a finite memory in this domaAlthough the frequency of the oscillator is still distributeuniformly about a mean value of 0, consecutive values offrequency are not independent of each other and time difences over sufficiently short times are correlated. Bothfrequency and the time differences have a short-te‘‘smoothness’’ that is not characteristic of a true randovariable. Flicker phase noise is intermediate between wphase noise and white frequency noise, and we would thfore expect that the spectral density of the time differendata would vary as the inverse of the Fourier frequency rtive to the spectral density of the phase fluctuations theselves.

The same kind of discussion can be used to definflicker frequency noise that is midway between white fquency noise and white aging~or random walk of fre-quency!. The underlying physical effect is the same, excthat now it is the frequency aging that has short-period crelations. We could think of a flicker process as resultfrom a very large number of very small jumps—not muhappens in the short term because the individual jumpsvery small, but the integral of them eventually producesignificant effect. The memory of the process is then relato this integration time.

Data that are dominated by flicker-type processesdifficult to analyze. They appear deterministic over shortriods of time, and there is a temptation to try to treat themwhite noise combined with a deterministic signal—a stratethat fails once the coherence time is reached. On the ohand, they are not quite noise either—the correlationtween consecutive measurements provides useful infortion over relatively short time intervals, and short data scan be well characterized using standard statistical measHowever, the variance at longer periods is much larger tthe magnitude expected based on the short-period standeviation.

The finite-length averages that we have discussed carealized using a sliding window on the input data set. Thissimple in principle but requires that previous input datastored; an alternative way of realizing essentially the satransfer function is to use a recursive filter on the outvalues. This method is often used in time scales, since thalgorithms are commonly cast recursively. Both the recsive and nonrecursive methods could be used to realizeerages with more complicated transfer functions. Thmethods are often used in the analysis of data in the tdomain.21

E. Two-sample Allan variance


The ability to predict the future performance of a clockbased on past measurements can be quantified by consid


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ing the two-sample or Allan variance. Suppose we have msured the time difference between two clocks at timest1 andt25t11t. If we want to predict the time difference at a timt35t21t, then we would use the two measurements totimate the frequency difference between the two clocks ding the first interval

y125X~ t2!2X~ t1!

t~14!

and we would combine this estimate with the measuremat t2 to estimate the value at timet3 :

X~ t3!5X~ t2!1y12t52X~ t2!2X~ t1!, ~15!

where we have assumed~for lack of any better choice! thatthe frequency in the interval betweent2 andt3 is the same asthe value in the previous interval. If this is not the case,prediction error will be proportional to the difference btween the actual frequency in the interval betweent2 andt3 ,which is y23, and our estimate of it, which is that it was thsame as the average frequency over the previous interva

e5X~ t3!2X~ t3!'y232y12. ~16!

Expressed in terms of time difference measurements, thediction error would be proportional to

X~ t3!22X~ t2!1X~ t1!

t~17!

and the mean-square value of this quantity, usually denated assy

2(t), is called the two-sample~or Allan! variancefor an averaging time oft. ~The variance is actually defineas one half of this mean-square value so that it will be csistent with other estimators when the input data are chaterized by white phase noise.!

If the frequency of the oscillator is absolutely constaover these time intervals, then the expressions in Eqs.~16!and ~17! are nonzero only because of the white phase nothat contributes to the measurements ofX @and indirectlythrough Eq.~14! to our estimates ofy#. The numerators ofEqs.~14! or ~17! are independent oft in this case so that theAllan variance for measurements dominated by white phnoise will vary as 1/t2.

If the frequency of the oscillator varies with time, thethe numerator in Eq.~14! will have a dependence ont. Atleast whent is sufficiently small, this dependence must cotain only non-negative powers oft, because it must be tha

X~ t1t!2X~ t !→0 as t→0. ~18!

The Allan variance must therefore decrease more slowly t1/t2 in this case and may actually increase witht if theexpansion of the numerator in Eq.~17! contains a sufficientlylarge power oft.

The Allan variance measures predictability in the rathnarrow sense of Eq.~15!. However, it does not make a cleadistinction between oscillators that have stochastic frequenoise and those that may have deterministic frequency vation ~a constant frequency aging, for example!, even though

2577Judah Levine

er-such a deterministic parameter might be included in a modi-fied version of Eq.~15!. It also does not give any information


averaging time,and oscillators.

2578 Rev. Sci. Judah Levine

Downloaded 18 S

TABLE I. Summary of noise types.

Name Slope of spectral density Slope of two-sample variance

Phase Frequency Simple Modified Time

White phase noise 0 V2 22 23 21Flicker phase noise 1/V V 22 22 0White frequency noise 1/V2 0 21 21 1

~Random-walk phase!Flicker frequency noise 1/V3 1/V 0 0 2White frequency aging 1/V4 1/V2 1 1 3

Instrum., Vol. 70, No. 6, June 1999

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about frequency accuracy—arbitrarily large frequency osets will make no contribution to Eqs.~16! or ~17!.

A constant frequency offset will result in time differences that increase linearly with time, and a constantquency aging results in a quadratic dependence of thedifferences. It is tempting to remove either~or both! of theseeffects using standard least-squares methods before coming the Allan variance. As we have discussed above, lesquares estimators may produce biased estimates if theused on data containing appreciable variance due to randwalk or flicker processes; the resulting Allan variance emates are likely to be too small as a result.

F. Relationship between the Fourier decompositionand the Allan variance

The Allan variance and Fourier decomposition viewsthe noise process are always valid in principle, but theyparticularly useful if the measurements are dominated bsingle noise process, since both parameters have particusimple forms in these special cases. Specifically, if the slof the power spectral density as a function of frequency olog–log plot has a slope ofa, that is if the spectral density othe frequency fluctuations is such that

Sy~V!5haVa, ~19!

whereha is a constant scale factor anda is an integer suchthat

22<a<1, ~20!

then the two-sample ordinary Allan variance we havefined above@as one half of the mean square of Eq.~17!#, hasa slope ofm when plotted as a function of averaging timea log–log plot. That is,

sy2~t!5bmtm ~21!

with

m52a21. ~22!

Measuring the slope of the Allan variance as a functionaveraging time therefore determines the slope of the spedensity and the noise type. This is a substantial advantbecause computing the Allan variance is usually a much spler job than computing the full power spectrum. This prcedure is most useful when the spectrum is dominated bsingle noise type in each range of Fourier frequency

~Random-walk frequency!

which is quite often the case for many clockBoth the Allan variance and the power spe

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tral density can be approximated by a series of straight-segments on a log–log plot in this case, and the approxition that only a single type of noise dominates the spectrin each one of the corresponding Fourier frequenaveraging time domains is well justified in practice. We haalready mentioned the five most common noise processeclock data, and their properties are summarized in Table

This relationship between Allan variance, spectral desity, and noise type has three important limitations. The fiis that the slope of the simple Allan variance that we hadefined above cannot distinguish between flicker phmodulation and white phase modulation—in both casesslope is 1/t2. This defect can be remedied by definingmodified Allan variance22 ~MOD AVAR ! which replaces thetime differences in Eq.~17! @or the frequencies of Eq.~16!#with averages over a number of adjacent intervals. The pcipal advantage of this modification is that it provides a wof distinguishing between white phase noise and flicphase noise, as is shown in Table I. A relative of the mofied Allan variance is the time variance, usually call‘‘TVAR’’ and designatedsx

2(t).23 It is an estimate of thetime dispersion resulting from the corresponding frequenvariance, and it is normalized so as to be consistent withconventional definition of the variance when the datadominated by white phase noise. It is defined by

sx25

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It has the same statistical properties as the modified Avariance, but it is somewhat easier to interpret in applicatiwhere time dispersion rather than frequency dispersion iprimary concern. Using TVAR, it is also somewhat easieridentify the onset of the domain in which white frequennoise dominates the spectrum by simply looking at the pof the variance as a function of averaging time. This is bcause it is easier to see the point at which the slope of TVchanges from 0 to11 than it is to identify the point at whichthe slope of MOD AVAR changes from22 to 21 ~see TableI!.

The second limitation is that the relationship betweenAllan variance and the slope of the power spectral dendepends on the fact that the input data are really noise annot have any coherent variation. The most common exction to this assumption is the case of deterministic freque
sc-aging—the relatively constant drift in the frequency of hy-drogen masers, for example, or a nearly diurnal frequency

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fluctuation in quartz–crystal oscillators driven by changesthe ambient temperature. Both the Allan variance andspectral density function are well defined in these cases,the simple relationships we have described are no lonvalid. The spectral density function of a periodic signal is,course, not a smoothly varying function but a peak atcorresponding frequency, while the Allan variance oscillaat the period of the perturbation and no longer has a simmonotonic slope either.

The third limitation is that the variance is a statisticparameter, and it does not give any information on wocase performance. This problem is not unique to the Alvariance, of course—all statistical estimators share this fing. It is possible, however, to define an uncertainty inAllan variance estimate;24,25 while this does not speak tworst-case performance, it does provide some informaon the width of the distribution function—that is on thspread that might be expected from measurements on ansemble of identical oscillators. It is also important to remeber that both mod avar and tvar are computed using anerage over adjacent frequency estimates, and the resumagnitudes are not the same as what would be obtainedthe same device measured without that averaging.

G. Uncertainty of the Allan variance

The uncertainty in the Allan variance estimate forgiven averaging time is proportional to the number of diffeences that contribute to it. If the entire data set spans aT, for example, then an estimate using a time difference otwill be comprised of aboutT/2t estimates. This numbeclearly decreases with averaging time, so that the uncertain each estimate must increase as the averaging timecreases. In the limit we will have only one estimate foraveraging time oft5T/2, and we will have almost no information at all about its uncertainty as a result. To make mters worse, consecutive estimates of the frequency hacorrelation that depends on the noise type, so that simminded estimates of the number of degrees of freedombe misleading even for small averaging times.

The summation of Eq.~17! that is required to computethe Allan variance can be considered to be a convolutionthe input data setX(t j ) with the series 1,22, 1, 0,... . If theseries of time differences are dominated by noise proce~rather than deterministic effects! then it is likely that theseries will be at least approximately stationary. If this is tcase, then we can replace the ordinary convolution witcircular one,26 and this change removes the difficulty in thcalculation for large time differences. If the data are not stionary initially, then it may be possible to pre-whiten theusing digital filter techniques or by removing a low-ordpolynomial first.27

H. Uncertainty using Fourier decomposition

The uncertainties in Fourier frequency estimates of noprocesses are discussed in the literature.28 Although Eq.~7!~and other Fourier estimates of noisy series! accurately char-


acterizes the power spectral density of white noise in thmean, there are significant uncertainties in the estimates.


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other words, the Fourier decomposition of a white noise pcess is not a smooth straight line but is itself a random vable whose statistical properties can be derived fromoriginal time series.~Although more sophisticated analyseare usually used, the spectral densities are linear combtions of the time series data, and the standard methodpropagation of error can be applied to estimate the variaof the spectral density.!

It is common practice to reduce the variance in the Frier coefficients by averaging a number of adjacent valuThe simplest method is a straight average which effectivweights the adjacent points equally, but more complica‘‘triangular’’ functions ~such as 1/6, 2/3, 1/6 or 0.23, 0.50.23! are also used. These averages provide a trade-ostability against frequency resolution. The practice of avaging adjacent estimates using a sliding window is maematically equivalent to convolving the Fourier coefficienwith the square or triangular-shaped function derived frthe weights. This convolution in turn is equivalent to mulplying the original time series by a ‘‘windowing’’ functionthat varies smoothly from a maximum at the midpoint of tdata set to zero at the end points.

The choice of windowing function is particularly important when analyzing time series that have significant poat discrete frequencies, since the windowing process canfect both the magnitude and shape of these ‘‘bright lineMore generally, the windowing operation can cause ambiities when the width of the windowing function~in frequencyspace! contains statistically significant variation in the powspectral density of the data, since quantitative estimatethe spectral density will depend on the shape of the windin this case.29 Analyzing a data set that contains a numberdiscrete spectral components requires considerable carthe choice of a windowing function because a poor chomay result in frequency aliases—spurious peaks in the stral density corresponding to frequencies that are not represent in the time series. It is common practice to remsuch bright lines~using simple least squares, for examp!before estimating the spectral density. This process is ca‘‘pre-whitening.’’

I. Other noise estimators

A number of applications require an estimator that pvides some measure of the worst-case~rather than the average! performance of an oscillator. The most common appcation is the use of oscillators to synchronize tmultiplexors in communications systems. This hardwareused to combine data from several circuits into one highspeed data stream using time-division multiplexing~in whicheach input circuit is given a periodic time slice!. This com-bined stream is de-multiplexed at the other end of the lusing receiving hardware that routes the data to the appriate output line based on the same time-slice logic. A ctain amount of frequency difference between the hardwarthe end points can be accommodated by ‘‘slip’’ buffers, bthese have finite capacity, and themaximumtime dispersion

2579Judah Levine

eInand frequency difference between the two end stations mustbe controlled to prevent overflowing these buffers.30


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This requirement leads naturally to the conceptMTIE—the maximum peak-to-peak time-interval errorsome observation time,T.31 A number of different versionsof MTIE exist—in some cases the reference for the measment is the input frequency to the network element andother cases it is an external standard reference frequenc

In addition to specifyingT, measurements of MTIE usually also specify some observation period for the measuments of the time dispersion. This is often implemented asliding window of widtht,T. The value of MTIE for thecomplete data set is the maximum of the values for eacthe windows.

The variation of MTIE witht can provide some insighinto the character of the frequency offset between theclocks. A static frequency offset, for example, producevalue of MTIE that increases strictly linearly witht. Theconverse is not necessarily true, however. Random-walkcesses can produce almost the same effect.

Another estimator that is often used in telecommunitions work is ZTIE. This estimator uses averages of the fidifference of the time data@Eq. ~14!# instead of the seconddifferences@Eq. ~17!# as in the Allan variance. As withMTIE, the value of the estimator is the maximum valuethe statistic over the averaging window.

Both ZTIE and MTIE are sensitive to static frequenoffsets, whereas AVAR~in all its variations! is not. A clockwith an arbitrarily large and constant frequency offset is‘‘good’’ clock by AVAR standards—while its time dispersion will eventually become very large, its performancepredictable. The same clock is not nearly so useful by MTor ZTIE standards because its static frequency offsetproduce repeated overflows of the slip buffers when it is uas the reference for a communications switch. The factthese overflows are regular and predictable is besidepoint. In addition, MTIE is more sensitive to transients—will give a better estimate of the worst-case performanbecause of this, but it can also become saturated by a sevent that will mask its response to the remainder of the d

IX. REPORTING THE UNCERTAINTY OF AMEASUREMENT

In 1978, the International Committee on Weights aMeasures~CIPM! requested that the BIPM address the prolem of the lack of an international consensus on the expsion of uncertainties in measurements. The BIPM convea working group, which produces a number of recommentions. These were approved by the CIPM in 198132 and werereaffirmed in 1986.33 These recommendations have bepublished as the ‘‘Guide to the Expression of UncertaintyMeasurement.’’34

The guide divides uncertainties into two categories: tyA uncertainties, in which the method of evaluation uses stistical analysis of a series of observations, and type Bcertainties, in which the method of evaluation is by meaother than statistical analysis. Note that there is no dircorrespondence with the terms ‘‘random’’ and ‘‘systemati


errors.Type A uncertainties are usually specified using a var


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ance ~or its square root, the standard deviation! combinedwith the number of degrees of freedom in the estimateneeded, covariances are also specified. These termstheir conventional statistical definitions.

Type B uncertainties are also characterized by a varia~or its square root, the standard deviation!. The variances areapproximations to the corresponding statistical parametthe existence of which is assumed. If needed, covariancesincluded in the same way.

The combined uncertainty is computed in the usual was the simple sum of the variances, and the combined sdard deviation is the square of the sum of the squares ofcontributing standard deviations.~The guide recognizes thain some applications it may be appropriate to multiply tcombined uncertainty by an additional ‘‘coverage’’ factothis factor must be specified when it is used.!

The procedure of simply summing the variances to copute the combined uncertainty is justified if the contributioare statistically independent of each other—an assumpthat is easy to make but hard to prove. In principle, itpossible to deal with data where the observations arestatistically independent of each other, but this is difficultpractice because the correlation coefficients are usuallywell known. Furthermore~as we discussed above!, summingvariances~or averaging data! that are dominated by flicker orandom-walk noise processes usually does not producemates that are statistically robust. As a specific case in powhile adding more and better clocks to EAL certainly helmatters, the accuracy of TAI is still evaluated by using dfrom primary frequency standards.

If the combined uncertainty is computed as the simsum of variances of all types then identifying a particucontributor as being of type ‘‘A’’ or ‘‘B’’ does not affect thefinal result, since only the magnitude of each contributionsignificant. However, it is useful in that it makes it easierunderstand how the result was obtained. Many time andquency applications depend more on the stability~or predict-ability! of a device or on its frequency accuracy. In theapplications the question of type A versus type B is leimportant than the magnitude of the Allan variance~forstability/predictability! or MTIE ~for accuracy!.

X. TRANSMITTING TIME AND FREQUENCYINFORMATION

We now turn to a discussion of how time and frequeninformation is transmitted. Distribution systems basedrelatively simple radio broadcasts were adequate in the edays, because the clocks themselves were not very good.transit time of an electromagnetic signal is about 3ms/km; inthe ‘‘old days’’ errors in estimating this delay were usuanot significant compared to the time dispersion of the clothemselves.

Short-wave signals from radio stations such as WWand low frequency~60 kHz! signals from stations such aWWVB continue to be widely used for transmitting time anfrequency information.~A complete list of stations that trans

Judah Levine

i-mit time and frequency information is contained in the An-nual Report of the BIPM.35! However, the uncertainties in


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the one-way propagation delay of either signal cansignificant—perhaps as much as tens of millisecondsWWV and a few milliseconds for WWVB. Short-wave signals tend to travel as both a ground wave and a sky wawhich is reflected from the ionosphere~possibly multipletimes!. The interference between these two signals andvariations in the height of the ionospheric reflecting layresult in significant variation in the path delay. These prolems are less serious for lower-frequency signals, but ethese signals show a strong diurnal variation in the pathlay, which tends to vary most rapidly near sunrise and sset. This diurnal variation can be almost completely remoby averaging the observations for 24 h~leaving a muchsmaller uncertainty on the order of microseconds! but manyoscillators have too much flicker or random-walk frequennoise to support averaging for this long a time. Even wthis level of averaging, the uncertainties are too largemore demanding applications, and other techniques hbeen developed to meet these needs. Roughly speakingdesign goal for any distribution system is to transmit the dto a user without degrading the accuracy or stability ofsource. This goal is not easy to meet now, and it will greven more difficult in the future as clocks continue to improve.

Although time and frequency are closely related quaties, there are important differences between them, and tdifferences affect how they are distributed and what unctainties are associated with these processes. Time disttion is, of course, directly affected by the delay in the tramission channel between the clock and the user,uncertainties in this delay enter directly into the error budgA frequency, on the other hand, is a time interval~rather thanan absolute time! so that the uncertainty in its distributionlimited more by temporal fluctuations in the transmissidelay rather than by its absolute magnitude. In other wothe time delay of a channel must be accurately known if tchannel is to carry time information, while the delay neonly be stable if frequency information is to be transmitteThis distinction is not of purely academic interest; macommon channels have large delays that are difficult to msure but which change slowly enough with time that thmay be thought of as constant for appreciable periods; schannels are clearly better suited to frequency-based acations.

The simplest way of dealing with the transmission deis to ignore it, and many users with modest timing requiments do just that. Many users of the time signals broadby short-wave stations~NIST radio stations WWV andWWVH, for example!, fall into this group. A somewhamore sophisticated strategy is to approximate the actuallay using some average value—perhaps derived from ssimple parameter characterizing the distance betweenuser and the transmitter.36 Users of the signal from NISTradio station WWVB, for example, often do not measuretransmission delay, but rather model it based on averagemospheric parameters and the physical distance betwthemselves and the transmitter.~Many users of WWVB are


interested in frequency calibrations and do not model thdelay at all. Instead they use the fact that the variation in th


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delay is dominated by a diurnal effect that can be cancealmost perfectly by averaging the data for 24 h.37 This is animportant example of the difference in the requirementstime and frequency distribution. Although the uncertaintythe transit time of a WWVB signal might be as large as 1 mthe variation in the delay averaged over 24 h is probaalmost 1000 times smaller than this value.!

The relatively simple models used to correct transmsions from high-frequency radio stations~like WWV! are notvery accurate, and they are only adequate for users with vmodest timing accuracy requirements on the order of 1This is a result of the inadequacy of the models, howevand is not a fundamental limitation of the technique. Tramissions from the global positioning system~GPS! satellites,for example, can be corrected to much higher accuracy umuch more complex models of the satellite orbit andionospheric and atmospheric delays.38 In addition, multiple-wavelength methods~to be described below!, can also beused to estimate the delay through dispersive media.

The fundamental limitation to modeled corrections is oten due to the fact that the adequacy of the model cannoevaluated independently of the transmission itself. The Gsystem is a notable exception because of the redundancyvided by being able to observe several~almost always atleast 4 and sometimes as many as 12! satellites simulta-neously, and this technique is especially powerful when thare located at different azimuths with respect to the anten

When modeling is not sufficiently accurate to estimathe path delay, users must measure the delay with an untainty small enough for the task at hand. There are a numof methods that can be used for this task, and we willscribe them in the following sections. Many applications umore than one of them to estimate different componentsthe delay. The full accuracy of time transfer between a glopositioning system satellite and an earth station, for examcan only be realized by combining a model of the tropspheric delay with a measurement of the delay throughionosphere using multiple wavelength dispersion. Using Gsignals to transfer time between two earth stations often befits from common-view subtraction as well.

A. Delay measurements using portable clocks

The simplest method for measuring the time delay aloa path is to synchronize a clock to the transmitter and thecarry it to the far end of the path where its time is compato the signals sent from the transmitter over the channel tocalibrated. Many timing laboratories used this techniquethe past. This method is limited in practice by the finite frquency stability of the portable clock while it is being tranported along the path and by uncertainties in a numberelativistic corrections that may have to be applied. Thecorrections will probably be needed if the transport velocis high enough that time dilation must be considered, ifpath is long enough that the Sagnac correction is importor if points along the path are at different heights abovegeoid so that gravitational frequency shifts must be includWhile this method can be used to calibrate a static transm

2581Judah Levine

eesion delay, it is too slow and cumbersome to evaluate thetemporal fluctuations in the delay through real transmission


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Even though various electrical or electromagnetic disbution methods are usually faster and more convenient,portable clock method remains quite useful because it teto be affected by a very different set of systematic uncertaties than the other methods to be described below, and itprovide estimates of these uncertainties that are difficulachieve in any other way.39

B. Two-way time transfer

A second method for measuring delay is two-way timtransfer, in which signals are sent in both directions alongpath ~the traditional ‘‘Einstein’’ clock synchronizationmethod!. If the path is reciprocal, the one-way delay is esmated as one-half of the round-trip transit time. Real paare unlikely to be perfectly reciprocal—there is likely to besmall lack of reciprocity even in the atmosphere40 and thereceiver and transmitter at each of the end points are unlito be perfectly balanced.41 Even if this balance between incoming and outgoing delays at each station couldachieved, the delays through the different componentsunlikely to have identical temperature coefficients so thatbalance cannot be maintained in normal operations withrepeated re-calibrations.42

Two-way time transfer can be further characterizedhow transmissions in the two directions are related. A hduplex system uses a single transmission path that is ‘‘turaround’’ to transmit in the reverse direction. A full-duplesystem, on the other hand, transmits in both directions simtaneously using either two nominally identical unidirectionchannels or some form of multiplexing on a single bidiretional path. All of these arrangements have advantagesdrawbacks. The half-duplex system is more likely to beciprocal since the same path is used in both directions,that advantage is partially offset by the fact that fluctuatioin the delay may degrade the time delay measuremeespecially if these fluctuations have periods on the ordethe turn-around time. A full-duplex system is not degradas much in this way since the measurements in the tworections are more nearly simultaneous, but incoherent fltuations in the delays of the two unidirectional channels~orthe cross talk between the two counter-propagating signathe case of a single, bidirectional channel! may be an equallyserious limitation.

Two-way methods can be used in many different haware environments. Kihara and Imaoka43 realized frequencysynchronization of better than 10213 and time synchronization of better than 100 ns using the two-way method in2000 km synchronous digital hierarchy based on opticabers. Even higher accuracy has been realized using comnications satellites to transfer time between timilaboratories44 and using dedicated optical-fiber circuits.45

The same two-way method is used in the NIST aumated computer time service~ACTS!46 to transmit time in-formation using standard dial-up telephone lines with an


certainty of a few milliseconds. Although the method is thesame as that used in the previous cases, the timing unc


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tainty is much greater because of variations in the reciproof the path—that is, in the lack of equality of the delays ovthe inbound and outbound circuits. Note that jitter in toverall delay itself does not produce a problem if it doesaffect the reciprocity, but it may be difficult to exploit thifact in a real-time system, which often uses the measudelay of a previous round-trip measurement to correctcurrent one. While this strategy is simple in principle, it rsults in a system that is sensitive both to jitter in the ovedelay itself and to its asymmetry.

C. Common-view time transfer

The third commonly used method is called commview.47 Two receivers, which are approximately equidistafrom a timing source, each measure the arrival time ofsame time signal using their local clocks. Since the sigarrives at the two stations at very nearly the same time,difference between the two time tags is approximatelytime difference between the two clocks. The two stationsbe synchronized with an uncertainty that is proportionalthe difference between the two paths back to the transmiCommon-mode fluctuations in the two paths and fluctuatiin the timing source itself cancel out~at least in first order!.There need not be any explicit correction for the path delathe two paths are very nearly the same, but it is hardarrange this in practice, and some correction for the residpath difference is usually required. The receivers must kntheir locations relative to the transmitter to compute tcorrection—a requirement that has no analogue in the tway scheme. If the transmitter and the receivers are mov~as in the global positioning system! then computing thesedistances may involve complicated transformations.

Common-view techniques lend themselves naturallybroadcast services, since many receivers can be synchronsimultaneously using signals from a single transmitter. Thhave been an important technique in time transfer for myears. Early systems used common-view observationstelevision signals, Loran stations, and similar transmissioThere have even been proposals to use local power distrtion systems for common-view time transfer within a singdistribution region. In many of these cases the common-vtransmitter is designed for some other purpose, and it dnot even need to ‘‘know’’ that it is being used for time ditribution. All of the receivers must know about each othhowever, and must agree on a common observation scheand processing algorithm. As we will show below, thknowledge can be acquired after the fact in some cases

Two-way systems and common-view systems are setive to fluctuations in the path delay in different ways. Twway systems depend on the reciprocity of a single path—the fact that it has the same delay in both directions.limitation arises from fluctuations in the symmetry of thpath and not on its overall delay. Common view, on the othand, depends on the equality of the delays in two sepapaths transmitting signals in the same direction, and i

Judah Levine

er-limited by effects that are not common to both paths. Whenthe paths are mostly through the atmosphere, both methods


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XI. GLOBAL POSITIONING SYSTEM

The signals from the global positioning system satelliare widely used for high-accuracy time transfer using bone-way and common-view methods. A similar system,erated by Russia, is called GLONASS. We will first descrthe satellites and the characteristics of the signals they brcast, and we will then discuss how these messages caused for time and frequency distribution.

The GPS system uses 24 satellites in nearly circularbits whose radius is about 26 600 km. The orbital periodthese satellites is very close to 12 h, and the entire conlation returns to the same point in the sky~relative to anobserver on the earth! every sidereal day~very nearly 23 h56 m!.

The satellite transmissions are derived from a singlecillator operating at a nominal frequency of 10.23 MHzmeasured by an observer on the earth. In traveling fromsatellite to an earth-based observer, the signal frequencmodified by two effects—a redshift due to the second-orDoppler effect and a blueshift due to the difference in gratational potential between the satellite and the obserThese two effects produce a net fractional blueshift of ab4.4310210 ~38 ms/day!, and the proper frequencies of thoscillators on all of the satellites are adjusted to compenfor this effect, which is a property of the orbit and is therfore common to all of them. In addition to these commoffsets, there are two other effects—the first-order Doppshift and a frequency offset due to the eccentricity oforbit, which vary with time and from satellite to satellite.

The primary oscillator is multiplied by 154 to generathe L1 carrier at 1575.42 MHz and by 120 to generateL2 carrier at 1227.6 MHz. These two carriers are modulaby three signals: the precision ‘‘P’’ code—a pseudo-randomcode with a chipping rate of 10.23 MHz and a repetitiperiod of 1 week, the coarse acquisition~‘‘C/A’’ ! code witha chipping rate of 1.023 MHz and a repetition rate of 1 mand a navigation message transmitted at 50 bits/s. Undermal operating conditions, the C/A is present only on theL1carrier.

The satellites currently in orbit also support ‘‘antspoofing.’’ ~AS! When the AS mode is activated~which isalmost always at the present time!, the P code is encryptedand the result is called theY code. TheP andY codes havethe same chipping rate, and this code is therefore oftenferred to asP(Y).

Each satellite transmits at the same nominal frequenbut uses a unique pair of codes which are constructed sohave very small cross correlation at any lag and a very smauto-correlation at any nonzero lag~CDMA—code divisionmultiple access!. The receiver identifies the source of thsignal and the time of its transmission by constructing locopies of the codes and by looking for peaks in the crcorrelation between the local codes and the received


sions. Since there are only 1023 C/A code chips, it is feasibto find the peak in the cross correlation between the local an


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received copies using an exhaustive brute-force methWhen this procedure succeeds, it locks the local clock totime broadcast by the satellite modulo 1 ms~with an offsetdue to the transmission delay!, and allows the receiver tobegin searching for the 50 b/s navigation data. Once this dstream has been found, the receiver can read the sattime from the navigation message, use it to construct a locopy of theP code, and begin looking for a cross-correlatiopeak using this faster code. This cross-correlation procfails when a nonauthorized receiver processes theY code,and these receivers usually drop back automatically to usthe C/A code only in this case.

The pseudo-random codes are Gold codes,48 which areimplemented using shift registers that are driven at the chping frequency of the code. The details of the design ofcode generators, the correlation properties of the codes,the relationship between the codes and the satellite timediscussed in the GPS literature. A particularly clear expotion is in Ref. 49.

The navigation message contains an estimate of theand frequency offsets of each satellite clock with respecthe composite GPS time, which is computed usingweighted average of the clocks in the satellites and intracking stations. This composite clock is in turn steeredUTC~USNO!, which is in turn steered to UTC as computeby the BIPM. The time difference between GPS time aUTC~USNO! is guaranteed to be less than 100 ns, andestimate of this offset, which is transmitted as part of tnavigation message, is guaranteed to be accurate to 25 npractice, the performance of the system has almost alwsubstantially exceeded its design requirements.

GPS time does not incorporate additional leap secobeyond the 19 that were defined at its inception; the tidiffers from UTC by an integral number of additional leaseconds as a result. This difference is currently 13 s, andincrease as additional leap seconds are added to UTC.accuracy statements in the previous paragraph should thfore be understood as being modulo 1 s. The number of lseconds between GPS time and UTC is transmitted as pathe navigation message, but is not used in the definitionGPS time itself. Advance notice of a future leap secondUTC is also transmitted in the navigation message.

The GLONASS system is similar to GPS in general cocept. In its fully operational mode, there will be 24 satellitorbiting in 3 planes 120° apart. At the present time~October,1998! there are 13 operational satellites. The GLONASSPcode has a chipping frequency of 5.11 MHz and a repetitperiod of 1 s; the C/A code chipping frequency is 511 kHand repeats every millisecond. All satellites transmit tsame codes, but the carrier frequencies are differ~FDMA—frequency division multiple access!. These fre-quencies are defined by

f 15160210.5625k MHz ~24!

and

f 25124610.4375k MHz, ~25!

2583Judah Levine

ledwherek is an integer that identifies the satellite. These ratherwidely spaced frequency allocations complicate the design of


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the front end of the receiver—especially the temperaturebility of the filters that are usually needed to reject stroout-of-band interference. On the other hand, cross talktween signals from the different satellites is much less oproblem; interference from a single-frequency source is aless of a problem with GLONASS, since the interfering snal is less likely to affect the signals from the entire conslation.

In the original design,k was a unique positive integer foeach satellite. The resulting frequencies interfered with raastronomy measurements, and a set of lower frequenwith negative values ofk will be phased in over the next fewyears. In addition, satellites on opposite sides of the emay be assigned the same value fork. This would notpresent a problem for ground-based observers, sincecould not see both of them simultaneously, but satellbased systems may require additional processing~similar tothe tracking loops in GPS receivers! to distinguish betweenthe two signals.

As with GPS, the satellites transmit the offset betwethe satellite clock and GLONASS system time, but the retionship to UTC is not as well defined. Signals from tGLONASS satellites are therefore usually used in commview, since the time of the satellite clock cancels in thprocedure.

The signals from the GPS constellation are usuallyfected by anti-spoofing~AS!, which encrypts theP code, andby selective availability~SA!, which dithers the primary oscillator and degrades the frequency stability of the sign~The implementation of SA may also dither parameters inephemeris message which produce a dither in the compposition of the satellite. This aspect of SA is usuaswitched off.! From the point of view of a nonauthorizeuser, the primary effect of AS is to deny access to thePcode, since it can only be processed by a keyed recewhen it is transmitted in its encryptedY-code form. An au-thorized user would see this encryption as authenticatingtransmission, since only a real satellite can produce vencrypted messages. Hence the name anti-spoofing. Thscription of the GLONASSP code has not been publisheand is officially reserved for military use, but the GLONASC/A code is not degraded.

The operation of the receiver is basically the sameeither system. The receiver generates a local copy ofpseudo-random code transmitted by each satellite in viand finds the peak in the cross correlation between eacthese local copies and the received signal. The local cgenerator is driven from a local reference oscillator, andtime offset of the local clock needed to maximize the crocorrelation is thepseudo range—the raw time difference between the local and satellite clocks. Using the contents ofnavigation message, the receiver corrects the pseudo rfor the travel time through the atmosphere, for the offsetthe satellite clock from satellite system time, etc.~If the re-ceiver can process both theL1 andL2 frequencies, then thereceiver can also estimate the delay through the ionospby measuring the difference in the pseudo ranges obse


using theL1 andL2 frequencies. The details of this calcu-lation are presented below. If the receiver can process on


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the L1 signal then it usually corrects for the ionosphedelay using a parameter transmitted in the navigation msage.! The result is an estimate of the time difference btween the local clock and GPS or GLONASS system tim

The time difference between the local clock and satelsystem time can be used directly to discipline a local oslator or for other purposes. However, these signals haveterministic fluctuations due to imperfections in the broadcephemeris which is used by the receiver to correct the psdorange. In addition, signals from the GPS constellationusually also degraded by the frequency dithering of selecavailability. These effects are attenuated or removed in comon view; if the stations are not too far apart, the commoview method also attenuates fluctuations in the delay duthe troposphere and the ionosphere since the paths to thestations often have similar delays and the fluctuationsthese delays often have significant correlation as well.

To facilitate time transfer among the various nationlaboratories and timing centers, schedules for observingsignals from the GPS and GLONASS satellites are publisby the International Bureau of Weights and Measu~BIPM!,50 and all timing laboratories adhere to them asminimum. ~Not all timing laboratories can receive GLONASS signals at the present time.! Other users often observthis schedule as well so as to facilitate the evaluation of thlocal clocks in terms of national time scales. The compariswith national time scales may be required for legal traceaity purposes, and using GPS time directly may not satithese legal requirements.

The need for common-view schedules arose initially bcause the first GPS receivers could only observe one sateat a time. Newer receivers can observer many satellitesmultaneously~as many as 12 for the newest units!, and thismakes it possible to eliminate formal schedules in principEach station records the difference between its local cland GPS or GLONASS system time using all of the satelliin view, and the time difference between any two stationscomputed after the fact by searching the data sets from bstations for all possible common-view measurements. Indition to eliminating the need for formal tracking schedulethis method can be used to construct common views betwtwo stations that are so far apart that they never have aellite in common view. A receiver in Boulder, Colorado, foexample, might have a common view link to Europe usione satellite and a simultaneous second link to Asia usinsecond one. These data could be combined to produccommon-view link between Europe and Asia that was inpendent of the intermediate clocks in Boulder and onsatellites.

XII. MULTIPLE FREQUENCY DISPERSION

In practice, it is difficult to arrange things so that the twreceivers in a common-view arrangement are exactly equdistant from the transmitter, and common-view data mustcorrected for this path difference. If the transmission paththrough the atmosphere or some other refractive medi

Judah Levine

lythen knowing the difference in the physical path lengths isnot enough—it is also necessary to know the refractivity of


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the medium~the difference between the actual index of rfraction and unity!. If the refractivity is dispersive~that is, ifit depends on the frequency of the carrier used to transmitinformation!, and if the dispersion is separable into a pthat depends on the carrier frequency and a part thatfunction of the physical parameters of the path, then itoften possible to compute the refractivity by measuringdispersion, even when both of these parameters vary fpoint to point along the path. If the length of the physicpath isL then the transit times measured using two differcarrier frequenciesf 1 and f 2 will be

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The first term on the right-hand side is the time delay duethe transit time of the signal along the path of physical lenL, and the second term is the additional delay due torefractivity of the medium. Note that this second term dpends only on the measured time difference between thenals at the two frequencies and on the known frequencypendence of the refractivity. It does not depend onphysical lengthL itself or on the details of the spatial depedence ofG, and we would have obtained exactly the saresult if only a small piece of the pathL8,L was actuallydispersive. This is an important conclusion, because it methat a receiver can estimate the delay through the ionospby measuring the difference in the arrival times of identimessages sent using theL1 andL2 frequencies. Note thathis method fails if the medium is refractive without beindispersive—that is, if the index of refraction differs frounity but is independent of the carrier frequency. This issituation with the effect of the troposphere on microwa


frequencies near 1 GHz, and other means must be foundmodel the refractive component of its delay.~There is no


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easy way to estimate the correction due to the troposphand its effect is usually ignored at the present time aresult. This is not adequate when the phase of the carrieused as in geodetic measurements, and more sophisticestimates of the troposphere are required in thapplications.51!

Although the time difference arising from the dispersimay be small, it is multiplied by the refractivity divided bthe dispersion, a ratio that is larger than unity.~The value isabout 1.5 for theL1/L2 frequencies used by GPS.! The re-sult is that the ionospheric refractivity makes a significacontribution to the time of flight of a signal from a satellitignoring this contribution is only adequate for common-vieobservations over short base lines where the effect cancethe subtraction. Furthermore, the noise in the dispersmeasurement (t12t2) is amplified by this same ratio, so thathe price for removing the refractivity of the path is an icrease in the noise in the measurements by roughly a faof 3 compared to a single-frequency measurement usingL1.

XIII. EXAMPLES OF TIME TRANSFER USING GPS

In this section we illustrate the principles of time transfusing signals from the GPS satellites.~The same principleswould apply to time transfer using GLONASS signals.52!Since the entire GPS constellation usually has AS enabtheP code is not available to most time-transfer users.~Evenwhen AS is turned on, theP code can still be used for dispersion measurements. A cross correlation of theP-code sig-nals onL1 andL2 can be performed without knowing thactualP code—the fact that it is the same on both carriersenough.! All of the time-difference results in this section abased on measurements using the C/A code or the caitself. In addition, the code-based data were acquired wsingle-frequency receivers, which cannot measure theL1-L2dispersion and must estimate or measure the contributiothe ionosphere using some other method. This is generdone using the parameters in the navigation message trmitted by the satellite, although other techniques exist.53

The C/A code cross-correlation process results in aries of ticks with a period of 1 ms, and decoding the epheeris message allows the receiver to identify the ticks in tstream that are associated with integer GPS seconds.54 Whenthe arrival times of these ticks are corrected for the offsethe satellite clock from GPS time, for the path delay andthe other effects described above, the result is a stringHz pulses that are synchronized to GPS time.~In order toapply these corrections, the receiver must parse thestream transmitted by the satellite.! The time differences between these pulses and the corresponding ticks from a lclock are then measured using a standard time intecounter. Single-channel receivers can only perform this msurement on one satellite at a time, but newer ‘‘all in viewreceivers can measure the time difference between the ticthe local clock and GPS time using up to 8 or 12 satellisimultaneously. In the simplest version of this scheme,

2585Judah Levine

toreceiver measures the time difference between the localclock and a composite 1 Hz tick. This tick is positioned


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using data from all of the satellites that are being tracked,the better receivers can provide independent time differenbetween the local clock and GPS time using each sateThis is done by providing a single 1 Hz tick to the timinterval counter together with a data stream that specifiesoffset between the time of the physical tick and GPS timecomputed using data from each satellite being tracked.less the two stations are very close together, this latter cbility is necessary for common-view measurements, aswill show below.

Figure 1 shows an example of these time-differenmeasurements. The plot shows the time difference betwelocal cesium standard and GPS time measured usingfrom a single satellite—SV 9. The time of the local clockoffset from GPS time by about 500 ns, and this explainsmean value of the data that are displayed. The time dission in the figure is totally due to the effects of SA on tGPS signal—the contributions due to receiver noise andthe frequency dispersion of the clock are negligible on tscale. These data arenot white phase noise at periods shortthan a few minutes, and the mean time difference over aminutes is not stationary. Averages over longer periods beto approximate white phase noise, but the contribution ofto the variance does not become comparable to the noisthe clock itself until the data are averaged for several da~Unfortunately, averaging white phase noise only decreathe standard deviation of the mean at a rate proportionathe square root of the averaging time, and this improvemdoes not even begin until we enter the domain wherevariance is dominated by white phase noise.!

As we pointed out above, the common-view procedcancels the contribution of the satellite clock to the varianof the time differences, and we would therefore expect tthis procedure would cancel SA. This expectation is cfirmed by the data in Fig. 2. This figure shows the timdifferences between two identical receivers whose anten

FIG. 1. The time difference between a local cesium clock and GPS tmeasured using data from satellite 9 only. The observed variation is duthe effects of SA on the GPS signal; the contributions of the clockreceiver to the variance are negligible on this scale.


were located about 1 m apart. The two receivers used thesame local clock as a reference; the mean time differen


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shown in the figure is due to the difference in the lengthsthe two cables between the receivers and the common clThese data have an amplitude of 1.7 ns rms, and theymuch closer to white phase noise—at least at short perio

The data shown in Fig. 2 represent a best-case resultreference clock is the same for both units and the antenare very close together so that almost all propagation effare the same for both of them and therefore cancel incommon-view difference. Likewise, the fluctuations in tclock itself are also common to both channels. However,advantages of common view are still significant even olonger base lines and different clocks. Figure 3 shows dfrom a common-view experiment between NIST in BouldColorado, and the United States Naval Observatory~USNO!in Washington, DC. Even though these stations are ab

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FIG. 2. The time differences measured in a short-base line common-vexperiment between two multichannel GPS receivers. The two receiwere connected to the same clock and the antennas were about 1 m apart.The average time difference between the two receivers was computedall satellites in view.

FIG. 3. The time differences between a receiver at NIST in Boulder, Corado, and a second receiver at USNO in Washington, DC, about 2400away. As in the previous figure, the average time difference between thestations was computed using all satellites in common view at each time.

Judah Levine

cemembers of the common-view ensemble change slowly during the observa-tion period.


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2400 km apart, the common-view time difference data sha short-term noise of about 3 ns rms—not as good asshort base line results, but still much better than the one-data shown in the first figure.

The common-view differences shown in both of thefigures use data from all satellites that are in common viewany time~at least four and usually five or six!. It is importantto compute the common-view differences on a satellitesatellite basis and then average these differences~rather thansubtract the average time difference computed using allellites in view at each station!. The reason for this is thastations that are 500 km or more apart are likely to be traing a different set of satellites with only some of themcommon. The fluctuations due to SA are different for easatellite, so that the two stations will have a different averaSA degradation that will not be cancelled in the commview procedure. It is important that multichannel receiveprovide the time-difference data by satellite for this reasSingle-channel receivers can do almost as well at cancethe SA, although the noise is somewhat higher because tare fewer independent measurements to average.55

Since the GPS carrier is derived from the same oscillathat produces the code, several groups,56–59 have begun ex-perimenting with time transfer~or frequency calibration60!using the phase of the carrier rather than the code.61 Thereare a number of conceptual difficulties with this approaThe most serious are probably the difficulty in calibrating teffective delay through the receiver and the ambiguitiesestimating the integer number of cycles in the time of fligbetween the satellite and the receiver. These experimentalmost always done using common-view methods, sincedifficult to estimate the various systematic corrections wsufficient accuracy in a one-way experiment. In addition,increased resolution that is available using the phase ofcarrier requires a corresponding increase in the accuracthe satellite ephemerides, and most carrier-phase anaused postprocessed ephemerides for this reason.

A number of newer code-based receivers use the catracking loop to aid the code-based correlator. These teniques include delta pseudo range and integrated Doppl62

The ionospheric effect on the code and the carrier haveposite signs, so that it is also possible in principle to usecode-carrier dispersion to estimate the ionospheric refracity.

One of the primary problems with GPS data is the effof multipath on the propagation of the signals throughatmosphere. This effect arises from the fact that the sigmay travel along many different paths in going from tsatellite to the receiver. Reflectors that are near the receiantenna are most troublesome, but even objects that arefar away can cause trouble. If the local clock is sufficienstable, it is possible to average the multipath effects by traing each satellite for as long as possible and averagingresults. This procedure is most effective in averagingcontributions due to reflectors that are many wavelengaway from the antenna, since the multipath contributtends to fluctuate rapidly in this case.63


The magnitude of the multipath effect varies from satellite to satellite and from second to second as the satellite


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move through the sky and the geometry of the path andreflectors changes. However, as viewed by an observethe earth, the satellites return to the same point in theevery sidereal day~approximately every 23 h 56 min!, andthe contribution should repeat with this period. Figuredemonstrates this. These data show the common-viewdifferences~computed using 1 s measurements with no averaging! between two receivers whose antennas were abom apart~the same setup as was used for the results showFig. 2!. Time differences for consecutive days have be‘‘stacked’’—that is, they have been offset by2240 s in timeand displaced vertically on the plot for clarity. The poinwith the sameX coordinate on successive plots were thefore recorded at the same sidereal time on consecutive dNote that although the pattern of the time differences is vcomplex, it shows very high correlation from one trace to tnext. This is exactly what we would expect from multipatThe geometry and the details of the reflection change wtime in a complicated way, but the entire pattern repeatsthe next sidereal day when the satellite comes back tosame position relative to the ground-based observer andreflectors that are near the antenna. Also note that this figshows thedifferentialmultipath effects on two antennas thare only 1 m apart. We do not know the full multipath effeon either signal, although we would expect that it wouldnot smaller than the magnitude shown in this figure.

It is tempting to try to correct for the multipath effect bmeasuring it once as in Fig. 4 and by then correctingmeasurements on subsequent days using these data asplate. This does not work for three reasons. In the first plathe plot shows the differential multipath effect, and it is nclear what the actual effect on each receiver is. In the secplace, the pattern is not absolutely stationary in time becathe satellites do not return toprecisely the same positionevery sidereal day. This drift is small over the few dashown in the figure, but it becomes significant after a we

FIG. 4. The results of a short-base line common-view experiment usingfrom SV 9 only. The figure shows 1 s measurements with no averaginTime differences from successive days are offset by2240 s and are dis-placed vertically by an arbitrary amount for clarity.

2587Judah Levine

-sor so, so that the template would not be useful for very long.Finally, it is often difficult to separate multipath effects from


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The tracking schedules published by the BIPM incluan advance of 4 min every day so that tracks on consecudays are observed in approximately the same multipathvironment. This method does not remove or correct foreffect of multipath—it simply converts it from a fluctuatinsignal to a nearly static offset which is different for easatellite. This is not necessarily a desirable result, since tare already satellite-to-satellite offsets due to errors inephemerides.

XIV. DELAY MEASUREMENTS IN VIRTUAL CIRCUITS

We now turn to a discussion of time distribution usincommunications networks. Everything that we have saidfar about measuring the transmission delay assumed imitly that the path between the transmitter and receiver hadwell-defined characteristics that we intuitively associate wa physical circuit or a well-defined path through the atmsphere. These ideas are still useful in characterizing mpaths through the atmosphere, but transmission circuitscable and fiber networks are increasingly packet switcrather than circuit switched. This means that there is no eto-end physical connection between the transmitter andreceiver. Packets are sent from the transmitter to the receover a common high-bandwidth physical channel, whichshared among many simultaneous connections using tdomain multiplexing~or some other equivalent strategy!.Each packet passes through a series of routers which readestination address and direct the packet accordingly. Dappear to enter at one end and emerge at the other, butneed not be a physical connection between the two. Inextreme case, consecutive packets may travel over veryferent paths because of congestion or a hardware failThis variability complicates any attempt to measure thefective delay of the circuit.

These problems are likely to become more importanmultiplexed digital circuits become more widely used.number of strategies have been proposed for dealingthese problems. A shorter path is likely to have smaller vability as well, since it would probably have fewer routeand gateways, and this suggests that the absolute delay mbe used as an estimate of its variability. It is temptingassert that the absolute magnitudes of these problems wsmaller on higher-speed channels, but this is by no measure thing.

XV. ENSEMBLES OF CLOCKS AND TIME SCALES

You will probably want to have more than one clockyou are serious about time keeping. Although a second ccan provide a backup when the first one definitively fathree clocks provide the additional ability to detect part


failures or occasional bad data points using a simplmajority-voting scheme. Unfortunately, such schemes a


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not as simple as they first appear to be for reasons tooutlined below, and the full advantage of using seveclocks can be realized only when they are combined imore complex time-scale procedure.

The problem with simple majority voting~or with com-paring the time of a ‘‘master’’ clock with a number of subsidiary ones! is that the stability of the frequency of almoany commercial clock~even a cesium-based device! is betterthan its accuracy by three orders of magnitude or moComparisons among clocks are almost always realized utime-difference measurements, and the large and stablequency differences between nominally identical devicessult in linearly increasing time differences as well.~Manyclocks have deterministic frequency aging as well, and tfurther complicates the issue by adding a quadratic depdence to the time-difference data.! It is no longer a simplematter to detect a hardware failure in this situation. It woube nice to think that the deterministic frequency offsets ofensemble of clocks would scatter uniformly about the truefrequency, but there is no reason to believe that this is trugeneral. In any case, it is much less likely to be true fosmall group of devices. Thus neither a simple average oftimes nor of the frequencies of the ensembles of devicesgood statistical properties. The machinery of a time scprovides a natural way of dealing with this problem.

A second advantage of a time scale is in providingframework for combining a heterogeneous group of deviso as to achieve a time scale that has the best propertieeach of them. For example, a time scale might be usedcombine the excellent short-term stability of a hydrogen mser with the better long-term performance of a cesium-badevice. Finally, a time scale can provide a flywheel fevaluating the performance of primary frequency standaor other devices such as prototype standards. Such evations can be done using only one clock, of course, but this no way of identifying the source of a glitch in that cofiguration.

As we will show below, the ensemble-average time afrequency have many of the properties of a physical clockparticular, the ensemble time and frequency must be settially using some external calibration. Conceptually, thisdone by measuring the time difference and frequency ofbetween each member clock and the external standardthough other methods in which a single member clockcalibrated and the other clocks are then compared to itoften used in practice. These initial time and frequency osets can be used to define the corresponding parametethe ensemble or an additional administrative offset canadded to them. This offset might be used to steer the scaa primary standard, for example. As we discussed aboTAI has been steered from time to time in this way usidata from primary frequency standards.

A. Clock models

As we discussed above, one of the problems withusing a formal time-scale algorithm but simply compariclocks using time differences, is the deterministic frequen

Judah Levine

eredifferences between nominally identical devices. A naturalway to deal with these deterministic effects is to include


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them in the definition of a model of how we expect the timdifferences to evolve, and to compare our observations wthe predictions of the model in order to detect problemSince the physical devices that make up an ensemble chas old devices fail and new ones are added as replacemit is customary to define the model parameters of each mber clock with respect to the ensemble average ratherwith respect to any particular master device. The goal ismake the characterization of each of the clocks in thesemble independent of the parameters of the others.

The deterministic performance of most clocks canvery well characterized using three parameters:x, the timedifference,y, the frequency difference, andd, the frequencyaging, where each of these parameters characterizessurements between the clock and the ensemble. Thesenitions imply that the ‘‘ensemble’’ is a clock too, althougwe have not yet specified how it can be read, set or chaterized. In fact, different algorithms address this problemdifferent ways.

Using these three parameters, we can construct a mof the deterministic performance of the clock. Assuming twe have the values of the three model parameters at stime t, the values at some later timet1t are given by

x~ t1t!5x~ t !1y~ t !t1 12d~ t !t2,

y~ t1t!5y~ t !1dt, ~30!

d~ t1t!5d~ t !.

It is possible to extend these equations to include highorder terms, but this is not necessary~or even useful! formost devices. These model equations assume that theredeterministicvariation in the frequency aging—an assumtion that is justified more by experience than by rigoroanalysis.

In addition to these deterministic parameters, our preous analysis of oscillator performance would lead us topect that our data would also have stochastic contributionthe variation of each of these parameters. For simplicity,could add terms for white phase noise, white frequennoise, and white frequency aging. These names are chosparallel the deterministic description in Eqs.~30!—we canalso use the alternative names we have defined above.~Aswe discussed above, the variance in each parameter hastributions both from its own white noise term and from allthe noise terms below it. The combined phase or frequenoises are very definitely not white.!

The next step would be to compare the predictions ofmodel equations with measurements and to test if the difences are consistent with the assumed model for the nOur measurements usually consist of pairwise time diffences between our physical clocks, and there is no obviuniversal prescription for using these data to evaluatmodel of the type specified in Eqs.~30!. There are two dif-ficulties: the first is that the time difference data are ambious to within an overall additive constant and the secon


that we have not specified a model for how the ensembclock itself is measured or how it evolves.


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B. AT1 algorithm

Several versions of the AT1 algorithm have been useNIST for many years. Although it works quite well with readata, I discuss it in detail not because it is theoretically otimum but because it is relatively simple to implement.addition, it illustrates the general ideas that underlie a nuber of other algorithms including ALGOS, which is used bthe BIPM to compute EAL and TAI.

The starting point for a discussion of AT1 is the obsevation that it is difficult to make robust estimates of thdeterministic frequency aging for most clocks. Even atlargest values oft ~where the effects of frequency aging agreatest!, these deterministic effects are usually swampedthe flicker and random-walk frequency noises in the deviAlmost the only exceptions are the very best hydrogen msers, whose frequency noise is so small that robust estimof the deterministic frequency aging are possible, and rapoor quartz oscillators~which are not normally used as members of ensembles anyway! whose frequency aging is slarge that it is readily observable even for small averagtimes. Thus the assumption thatd(t)50 for all t is consistentwith the data for almost all clocks, except for the best hydgen masers. For those clocks we would use a constant foaging that is determined outside of the ensemble algoritA typical value for such a maser would be of ord10221s21.

If we model the frequency aging as a noiseless consvalue then this is equivalent to assuming that the frequefluctuations are dominated by white noise processes. Tha reasonable assumption provided thatt is not too large. Themaximum value oft that still satisfies this requirement variewith the type of device; a typical value for a cesium osciltor would be less than a day or two. On the other handmust not be made too small either, because otherwisetime difference data will be dominated by the white phanoise of the measurement process.~This is not necessarily abad thing, but taking full advantage of the data in this carequires a change in strategy. This point is discussed bein connection with techniques used for transmitting timeing the Internet.! As specific examples, the NIST time scalwhich uses measurement hardware with a noise floor of seral ps usest52 h, while the BIPM computation of AL-GOS, which uses the much noisier GPS system to meathe time differences between laboratories, usest55d.

As we pointed out above, the initial values of the timand frequency of the ensemble must be knowna priori. Theepoch associated with these initial values ist0 . The algo-rithm then continues with periodic measurements of the tidifferences between each clock and one of them, whichdesignated as the reference clock or working standard. Thmeasurements are made at epochst1 ,t2 ,...,tn . These mea-surements are usually equally spaced in time, but this is mfor convenience than for any formal requirement of thegorithm.

If at some epochtk the time difference and frequencoffset between thej th clock and the ensemble werexj (tk)

2589Judah Levine

leand yj (tk), respectively, then AT1 models the time differ-ence at some later epochtk11 as being given by


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x j~ tk11!5xj~ tk!1yj~ tk!t, ~31!

where

t5tk112tk . ~32!

Note that the right side of Eq.~31! is an estimate based othe two model parameters of the clock using a simple linmodel. As we discussed above, this is appropriate for allthe best hydrogen masers. We would use the first of Eqswith a constant value ford for these devices. This constafrequency aging does not really have any effect at this poSince the algorithm normally operates with a fixed interbetween computations, the effect of adding frequency agis to add a constant offset to each of the predictions. Sucoffset can be absorbed into the starting values without losgenerality.

The mth clock in the ensemble is the working standaIt is characterized using the same model as any other [email protected]., Eq.~31! with j 5m#. The working standard is not necessarily the clock with the best statistical properties—sincmay be awkward to change this designation in a runnensemble, it is often the clock that is likely to be the moreliable or to live the longest.

At each epoch we measure the time differences betwthe working standard and each of the other clocks usintime interval counter or any of the other methods wescribed above. These measured time differences areXjm(t),wherej takes on all possible values except form. Combiningthe predicted time of each clock with respect to the ensemusing Eq.~31! with the corresponding measured time diffeence between that clock and the working standard, weestimate the time difference between clockm and the en-semble via clockj using

xmj ~ tk11!5 x j~ tk11!2Xjm~ tk11!, ~33!

where the superscriptj on the left side indicates that thestimate has been made using data from clockj. Note thatEq. ~33! could equally well have been thought of as a denition of the time of the ensemble with respect to physiclock m estimated using data from clockj. Also note that, asexpected, Eq.~33! reduces to Eq.~31! when j 5m sinceXmm(t)[0. Thus Eq.~33! applies to the working standarjust as it does to any other clock in the group.

If Eq. ~31! is an accurate model of the time of clockj,then Eq. ~33! is an unbiased estimate of the time of tworking standard with respect to the ensemble since theterministic time and rate offset of clockj have been properlyestimated. Note that clockm may have a deterministic timor frequency offset with respect to the ensemble—there isrequirement that its parameters be the same as those oensemble. Its statistical parameters are also not importanthis procedure to work.

If there areN clocks in the ensemble, there areN21independent time-difference measurementsXjm , and thesame number of equations identical to Eq.~33! with all pos-sible values ofj exceptj 5m. The previous state of clockmis used to predict its time with respect to the ensemble us


Eq. ~31!. ~Alternatively, we could use Eq.~33! for this pur-pose, settingXmm50. We adopt this latter strategy in what


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follows, since it simplifies the notation and emphasizesfact that clockm is not a special clock with respect to thensemble—only with respect to the time-difference haware.! If we assume that the fluctuations in the timdifference measurements are not correlated with each othe N estimates of the time of clockm with respect to theensemble are characterized by a well-defined mean, wwe can compute using

xm5(j

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(j

wj51. ~35!

If the clocks are all identical then a simple assumption wobe to set all of the weights equal to 1/N, so that Eq.~34!reduces to a simple average, but this is usually not the omum choice in a real ensemble.

Equation~34! can be equally well thought of as defininthe time of the ensemble with respect to the working stdard, clockm, computed by averaging the independent emates obtained from the time difference between clockmand each of the member clocks. The time of clockj withrespect to the ensemble can then be calculated by combiEq. ~34! with the measured time differenceXjm:

xj5Xjm1xm . ~36!

These values are then used as the starting point for themeasurement cycle.

The difference between Eq.~31!, which predicted thetime of clock j with respect to the ensemble based onprevious state, and Eq.~36!, which gives the value of clockjbased on the data from the current measurement cycle, isprediction error,e j . The average of this prediction error ismeasure of the quality of the clock in predicting the timethe ensemble. The average of this value can be used to oan estimate for the weight to be used for clockj in Eq. ~34!

wj5C

ê j2&

, ~37!

whereC is chosen so that the weights satisfy the normalition condition, Eq.~35!. Defined in this way, the weights arrelated to the TVAR statistic, which we defined above. Asour previous discussions, a good clock is a predictaclock—not necessarily one that has a small deterministicquency offset, because we can always remove that ofusing Eq.~31!.

The dispersion in the prediction error can also be usetest for a clock failure. That is, on each cycle we could eamine the statistic

e j22ê j

2&

ê j2&

~38!

and decide that clockj has a problem if this statistic is

Judah Levine

greater than some thresholdsmax. We might drop clockjfrom the ensemble calculation in this case and repeat the


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computation@Eq. ~34!# with its weight set to 0. If the statisticis close tosmax, we could choose to decrease the weightthe clock gradually so that there is no abrupt step in permance exactly atsmax. The AT1 time scale currently used aNIST setssmax54, and linearly decreases the weight ofclock starting when its average prediction error reaches 7of this value.

This procedure is a more formalized version of a simmajority-voting scheme. Its advantage is that the systemdifferences between the clocks have been removed bymodel equations and therefore do not affect the erdetection process. Furthermore, the averaging proceduapplied to quantities that have no first-order determinisvariation. The working standard enters into these calcutions exactly as any other member of the ensemble, andata can still be rejected by the prediction-error test justcussed even though it continues to play a central role intime-difference measurements.

If the clock frequencies were noiseless and constant tthat would be the end of the story. The time-difference msurements would be composed of deterministic frequeand time offsets which would be accounted for by the moequations, and the residual time difference data wouldcharacterized by pure white phase noise, which wouldoptimally attenuated by the averaging procedure. Unfonately, the frequency of the clock is characterized by whfrequency noise at intermediate times and by flicker arandom-walk fluctuations at longer times. As we showabove, these frequency fluctuations do not produce timeferences whose variation is characterized as pure white pnoise, and the average in Eq.~34! is no longer completelyjustified.

These frequency fluctuations can be accommodatedreplacing the static frequency parameteryj in Eq. ~31! with aparameter that is allowed to evolve with time. In ordercompute this dynamic frequency, the short-term frequeestimates

f j~ tk11!5xj~ tk11!2xj~ tk!

t~39!

are averaged using a simple exponential low-pass filteproduce the frequency estimate used in the model equa

yj~ tk11!5yj~ tk!1a f j~ tk11!

11a, ~40!

where the time constanta is set to the transition at the uppeend of the domain in which white frequency fluctuatiodominate the spectrum of the clock noise. The time consparameter is actually a dimensionless constant less thathe effective time constant ist/a. The value of the timeconstant is set to attenuate white frequency noise at interdiate periods; the gain of the filter approaches unity at vlong periods. As we discussed above, the primary reasonusing an exponential filter of this type is that it can be re


ized using the same kind of iteration as is used for the othparameters of the model. At each step in the process t


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model parameters computed on the previous step are cbined with the current measurements—older data needbe saved for this computation.

Several versions of this algorithm have been usedNIST for many years.64,65 The AT1 time scale provides optimum performance when the noise in the measurementtem and the clocks is no more divergent than white phnoise. Unfortunately, clocks exhibit flicker and random-wafrequency noise at longer term, and the scale itself tendbecome dominated by this type of noise at longer periods~onthe order of months!. The ALGOS algorithm used to compute EAL ~and from that TAI and UTC! tends to have thesame problem; as we discussed above, the solution incase is to steer TAI using data from the primary frequenstandards. This limitation has been addressed in the desigAT2.66

The AT1 algorithm has other limitations. It cannot dewith a device~such as a primary frequency standard! thatoperates only occasionally. We would like to be able to chacterize such a device using only a well-defined frequencits time with respect to the ensemble is not interestingimportant. In fact, since it operates only intermittently,time with respect to the ensemble may not be well definAs we mentioned above, the data from a primary frequestandard can be incorporated into the ensemble frequencadministrative adjustment as is done with TAI.

The AF167 procedure presents another solution to tproblem. Using an algorithm analogous to AT1, it estimathe frequency~rather than the time! of the working standardusing the measured frequency differences between it andother members of the ensemble. Data from a primaryquency standard can be incorporated into this scale in a nral way, and the fact that the standard operates only intertently is not a problem. Since it is quite similar to AT1 in ibasic design, it will also become dominated by flicker arandom-walk processes at longer periods—probably onorder of 1–2 yr.

In addition to ALGOS, there are a number of other agorithms that are similar to AT1. These algorithms habeen widely discussed in the literature.68 Other algorithmsare based on Kalman filters.69,70 This formalism has someadvantages in dealing with clocks that operate irregularycan also provide more robust definitions of the ensemtime and frequency than is possible using AT1 or its retives. On the other hand, Kalman filter algorithms often hastartup transients before the filter enters the steady-stategime. These transients can result in instabilities whenmembership in the ensemble changes frequently.

A fundamental problem with many time scales is tway the weights of the member clocks are calculated. TAT1 algorithm and its relatives assign a weight to each clobased on its prediction error—a good clock is one thapredictable over the time interval between measuremcycles.~We have already seen this principle in the discussof the Allan variance.! However, a clock whose weight iincreased because of this will tend to pull the ensemthrough Eq.~34!. A small increase in weight thus tends

2591Judah Levine

erheproduce positive feedback, and a clock that starts out a littlebit better than the other members of the ensemble can


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quickly take over the scale. The problem, of course, is tEqs.~31! and~36! are correlated, so that the prediction erris always systematically too small. The correlation cleagets worse as the weight of a clock increases.

This correlation can be addressed by administrativlimiting the maximum weight any clock can have, eitheran absolute maximum value~as in ALGOS! or to a maxi-mum percentage of the scale~as in the NIST implementationof AT1!. It is also possible to estimate the magnitude ofcorrelation and correct for it.71

These methods limit the maximum weight that any clocan have in the ensemble, even if that large weight wohave been deserved on statistical grounds. The reason isone of the motivations for using an ensemble is safetyrobustness in case of a failure. An ensemble whose timeffectively computed using only one clock will have a seous problem when that high-weight clock fails or has a dglitch. These problems can be more easily addressescales that are not computed in real time~such as TAI, whichis computed about one month after the fact!. It is much easierto recognize a problem retrospectively and to administively remove a clock from the ensemble when later dshows that it was starting to behave erratically.

Since the weights are normalized using Eq.~35!, anymethod that limits the maximum weight that a good clocan have implicitly gives poorer clocks more weight ththey deserve. The price for keeping good clocks from takover the scale is therefore a scale that is not as goodcould be because poorer clocks have more weight thandeserve based on statistical considerations.

A similar effect happens when a member clock halarge prediction error and is dropped from the ensembleresult. The contribution that this clock would have madethe ensemble is divided among the remaining members,the result that the ensemble time~and, indirectly, its fre-quency! may take a step when this happens. This canespecially serious in a small ensemble with one clock thasignificantly poorer than the others. If one of the good clocis dropped because it fails a test based on its prediction eits weight will be distributed among the remaining membeIf the weights of the other clocks are already limited by tadministrative constraints that we discussed above, the reis to transfer the weight of this good clock to the poorclock in the ensemble.

The methods used to calculate the frequency andweight of each clock are effectively trying to separateobserved variance in the time difference data into determistic and stochastic components. This separation, whicimplicitly based on a Fourier decomposition of the variancan never be perfect. These methods tend to have thedifficulty distinguishing between deterministic frequency aing and random-walk frequency noise because the two hsimilar ‘‘red’’ Fourier decompositions. This is the reasoAT1 and similar time scales often do not include a termdeterministic frequency aging into the model equationsthe time differences of the clocks—it is very difficult to etimate these parameters in the presence of the random-


noise in the data. This can be a serious problem in an esemble containing hydrogen masers, which tend to have s


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nificant long-term frequency aging combined with very smshort-term prediction errors. The very good short-term pformance means that these clocks will be given a very hweight in an AT1-type procedure, and this high weight wthen adversely affect the long-term stability of the scale dto the frequency aging of these devices. One solution isincorporate a constant frequency-aging term that is demined administratively using procedures outside of the sccomputations~using data from primary frequency standardfor example!. Another solution is to modify the algorithm toinclude explicitly different weights for short-term and longterm predictions.

The time and frequency of the scale itself can be maalmost totally insensitive to a change in the membershipthe ensemble, provided that the initial parameters of a nclock are accurately determined. These initial parametersusually determined by running a new clock in the scale wits weight administratively kept at 0.

One of the practical difficulties in running an ensembis that there is no physical clock that realizes its timefrequency. This deficiency must be remedied if the ensemtime is used to control a physical device. One solution issteer a physical clock so that its time and frequency trackcorresponding ensemble parameters. There are a numbdifferent ways of realizing this steering: the algorithm cemphasize time accuracy or frequency smoothness, butboth. An algorithm that emphasized time accuracy wouldlarge frequency adjustments and time steps to keepsteered clock as close as possible to ensemble time. Angorithm that emphasized smoothness would limit the magtude of frequency adjustments and never use time steps.choice between the two extremes~or the relative importanceof accuracy and smoothness in a mixed solution! cannot bespecifieda priori—the choice depends on how the data ato be used. Historically, steering algorithms at NIST hatended to place a higher value on frequency smoothnesson accuracy, with the inevitable result that the time dispsion of UTC~NIST! is larger than it would be if time accuracy were emphasized more strongly.

XVI. SYNCHRONIZING A CLOCK USING TIMESIGNALS

Finally, we turn to the question of how best to use a timsignal that we receive from a distant source of uncertaccuracy over a channel whose delay may be variable. Insimplest distribution system, the receiver is a simp‘‘dumb’’ device. Its internal state~whether time or fre-quency! is simply reset whenever a signal is received. Tsignal may be corrected for transmission delay or some osystematic correction might be used, but the important pois that the previous state of the receiver makes little orcontribution to the process. This may be the best strategthe uncertainty in the receiver state is so high as to makeffectively worthless, but it is important to recognize ththis is not universally true, and that the state of the receibefore the new information is received also contains inf

72

Judah Levine

n-ig-mation that may be useful.

The performance of the receiver clock can be character-


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ized in terms of deterministic and stochastic componeThe deterministic aspects of its performance can be usepredict the data that will be received from the transmittersome future time. This prediction is never perfect, of courbecause the deterministic components may not be knprecisely, because even if they are known at one instantare likely to evolve in time, and because the stochastic cponents of the model are always present and add noise tmeasurements. Although this prediction might not be ascurate as we would like, it nevertheless can often provideimportant constraint on the received data.

The most useful situation arises when the spectrumthe noise in the receiver clock has little or no overlap wthe spectrum of the noise in the transmission channel.73 Thishappy situation supports the concept of ‘‘separationvariance’’—the ability to distinguish between a ‘‘true’’ signal, which should be used to adjust the state of the receclock, and a ‘‘false’’ signal which is due to an un-modelefluctuation in the channel delay and which should therefbe ignored. In an extreme situation, this strategy couldused to detect a failure in the transmitter clock itself.

Decisions based on separation of variance are problistic ~rather than deterministic! in nature, and will only becorrect in an average sense at best. Whether or notimprove matters in a particular situation depends on thelidity of the assumption of separation of variance and onaccuracy of the various variance estimators. Since themators must be based on the data themselves, a ceamount of ambiguity is unavoidable.

This method obviously fails in those spectral regiowhere separation is impossible. There is no basis for deing whether the observed signal is ‘‘real’’ and should be usto adjust the local clock or represents transmission noiseshould be ignored. If the resulting ambiguity results in unceptably large fluctuations, then either the receiver orchannel must be improved. The crucial point is that neitthe receiver clock nor the transmission channel need be geverywhere in the frequency domain—it is perfectly aequate if they have complementary performance characttics, and this may in fact be the ‘‘optimum,’’ least-cost altenative for achieving a given accuracy in the distributiontime or frequency.

These considerations explain why it may not be opmum to use the signal received over a noisy channel ‘‘asIt may be possible to remove the channel noise if theceived data can be used to discipline an oscillator whinherent noise performance can be characterized by aance that separates cleanly from the noise added by the cnel in the sense we have discussed. These ideas can beto filter the intentional degradations imposed on the Gsatellites because of selective availability74 and to improvethe distribution of time on widearea networks such asInternet.75

These strategies can become quite complicated in debut all of them tend to share a number of general characistics. Individual measurements of the time differenceusually dominated either by white phase noise arising fr


the measurement process itself or by something like selecti


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availability, which is a pseudo-random process that loolike noise in the short term. In the usual state of affairs,frequency of the local clock is more stable than the timdifference measurements would indicate—in other wothese fluctuations in the measured time difference doimply a corresponding fluctuation in the frequency of tlocal clock. The optimum strategy is clearly to leave tlocal clock alone in the short term, since its free-runnistability is better than the time-difference data would incate. This implies some form of averaging of the measuments, with the details depending on the application.

When the variance in the time-difference data canmodeled as white phase noise, then the optimum strategclearly to make a large number of rapid-fire time-differenmeasurements and to average these data. This procescontinue for as long as the variance continues to look lwhite phase noise. For synchronization over packet netwosuch as the Internet, it is quite common to find periods ofto 15–30 s when this assumption is valid, and it is a simmatter to make 50~or more! time-difference measuremenduring this period.~The same idea can be used to improthe measurement noise in a time-scale system. In that cawell, the short-term variance is well modeled as white phnoise.! These data would be averaged to form a single powhich was then used to discipline the local clock usingmodel such as Eqs.~30! or ~31!. Although the average cosof the synchronization process depends only on the numof calibration cycles, clustering the measurements in tway may improve the performance of the synchronizatalgorithm by optimally averaging the white phase noisethe measurement process. Although the average cost osynchronization process would be the same if the measments were spread out uniformly in time, the performanwould not necessarily be as good because the increaseterval would put the measurements outside of the timemain where the fluctuations are dominated by white phnoise.76 While the cost of realizing this synchronization prcess increases linearly with the number of calibration cycthat are used in each group, the variance only improvesthe square root of this number, so that incremental improments in the performance of the algorithm become increingly expensive to realize.

At some longer time, the variance of the averaged tidifferences drops below the variance of the free-runnclock, and that is the point at which some form of correctito the time of the local clock is appropriate.~Note that thecrossover is not determined by the frequency offset oflocal clock but by the stochastic noise in this parameDeterministic offsets may be a nuisance in practice, but tcan always be removed in principle and should not plarole in a variance analysis. Furthermore, if the variancethe local clock can be characterized as white frequenoise, then averaging consecutive, closely spaced freqcies will yield a statistically robust frequency estimate. Tdomain in which white frequency noise dominates the vaance is often quite small, so that frequency averaging odoes not produce an appreciable improvement in the per

2593Judah Levine

vemance of the synchronization process.! This crossover point


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is never absolute, however. The local clock will always cotribute some information to the measurement process—even if this information is only used to detect a gross failuof the distant clock or of the calibration channel it shouprobably never be completely ignored.77

In the usual situation, the noise in the data from tdistant clock continues to decrease~relative to the localclock! at longer averaging times, so that the weight attacto the time-difference data in steering the local clockcreases by a corresponding factor. This is certainly truethe degradation imposed on the GPS satellites by selecavailability, and it is also a common feature of noise incommunications channel. Since both of these fluctuationsbounded in time, they make contributions to the Allan vaance that must decrease at least as fast as the averagingfor sufficiently long intervals.

These considerations tend to imply relatively slow crection loops—the time constant for steering a good comercial cesium standard to a GPS signal is usually a dalonger. The corresponding time constant for a compuclock synchronized using messages transmitted over theternet is usually a few hours. Although such loops maximthe separation of variance and uses both the local hardwand the calibration channel in a statistically optimum wathey have problems nevertheless. The very slow responsthe correction loop means that it takes a long time to sedown when it is first turned on and also after any transienstep of the local clock either in frequency or in time mpersist for quite some time before it is finally removed. Tloop may never reach steady-state operation in the extrcase. More frequent calibrations and shorter time constare usually used when worst-case~rather than long-term average! performance is the principal design metric.

Another situation which requires more frequent calibtions and shorter time constants occurs when the local olator has a relatively rapid variation in frequency that canbe characterized as a noise process. An example wouldlarge sensitivity to ambient temperature. Such an oscillawill tend to have large nearly diurnal frequency variationand much of the variation will often be concentrated nearstart and end of the working day. Although the time dispsion averaged over 24 h will tend to be small, the maximdispersion during the day may be unacceptably large,relatively frequent calibrations may be required to keeptime dispersion within specified limits.

The opposite problem happens when it is the chanthat has a large diurnal fluctuation in the delay~as withWWVB, for example!. The optimum strategy is clearly tincrease the interval between calibrations so as to reducesensitivity of the measurements to these spurious fluctions. This is not too hard to do in the case of WWVB, sinit is usually used with local oscillators that are stable enouto support the required increase in averaging time. Tanalogous problem in computer networks is more difficultdeal with, since the clock circuits in many computers are


stable enough to support the increase in averaging time th


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XVII. DISCUSSION

I have described some of the principles that are usedesign clocks and oscillators, and I have introduced thetistical machinery that is used to describe time and frequedata. This machinery can be used to characterize the osctors themselves and also to understand the degradationresults from transmitting these data over noisy channels wvarying delays.

Time and frequency measurements will continue to pan important role in science, engineering, and life in geneSeveral new requirements are likely to be most importanthe near term:

~a! The need for higher-accuracy frequency standardsapplications ranging from research in atomic physand astrophysics to synchronizing high-bandwidcommunications channels. Devices such as cesiumrubidium fountains and trapped-ion devices are alreain the prototype stage, and devices that can potentirealize the SI second with an uncertainty of signicantly less than 10215 are already being designedThere is no reason to believe that the trend towaimproving the accuracy and stability of frequency stadards will stop, or that applications for these improvstandards~both in science and in engineering! will notfollow their realization.

~b! Making use of these new devices will require improvments in distribution methods. Methods based onphase of the GPS carrier may be adequate—at linitially, but other methods will probably be neededwell. Digital transmission methods—especially metods using optical fibers—are likely to become moimportant for many of these applications.

~c! If these devices are to make a contribution to TA1, thincorporating them will also require a better undestanding of the transformation from the proper timrealized by the device to coordinate time—the exshape of the geoid, for example, especially at reworld laboratories, which are often located at plac~such as Boulder, Colorado! which have inhomoge-neous and inadequately documented crustal structu

~d! Applications such as electronic commerce, digital ntaries, and updating distributed databases need metfor simple, reliable, authenticated, and legally traceadistribution of time information with moderate accuracy on the order of milliseconds. The demand fthese services is growing exponentially, and the servmust be protected against attacks by a significant nuber of malicious users. These requirements for secuand reliability are not unique to digital time services,course, but addressing them while maintaining thecuracy of the time messages presents some un

Judah Levine

atproblems, which may not be completely solved by sim-ply scaling up the current designs.


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ACKNOWLEDGMENT

The work on the algorithms and statistics used for nwork synchronization was supported in part by Grant NNCR-941663 and NCR-911555 from the NSF to the Univsity of Colorado.

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of Clocks and Oscillators, NIST Technical Note No. 1337.~U.S. Gov-ernment Printing Office, Washington, DC, 1990!. This is a book ofreprints that contains some of the papers cited in the specific refersection. J. Vanier and C. Audoin,The Quantum Physics of Atomic Frequency Standards, in 2 volumes. Bristol:~Hilger and Institute of Phys-ics, London, 1989!.

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REVIEW ARTICLE Introduction to time and frequency … metrology, including a discussion of some of the types of measurement hardware in common use and the statistical machinery that