KALMAN FILTER CHARACTERIZATION OF CESIUM CLOCKS AND HYDROGEN MASERS

Transcription

1 34 h Annual Precie Time and Time Inerval PTTI Meeing KALMAN FILTER CHARACTERIZATION OF CESIUM CLOCKS AND HYDROGEN MASERS Lee A. Breakiron U.S. Naval Obervaory Wahingon, DC 39-54, USA Abrac Our previou PTTI paper demonraed ha a wo-ae Kalman filer involving he parameer frequency and frequency drif, when properly implemened, produce frequencie for hydrogen maer ha generae a able mean imecale, a lea for cae involving imulaed and low-noie real daa. The curren paper exend he inveigaion o he noiier daa acually employed in he USNO operaional imecale. Reul for abou 5 day of poproceed daa for maer and 5 ceium clock indicae comparable frequency abiliie o hoe obained wih he currenly operaional imecale algorihm. INTRODUCTION Mean imecale baed on Kalman filer modelling of individual clock have been inveigaed by oher in order o provide an enemble ime ha i more uniform han he ime kep by any one of he coniuen clock and whoe eimaed ae are opimal in he minimum quared error ene. Tryon and Jone [], Barne and Allan [], Sein [3], and Greenhall [4] employed a ae parameer he phae, frequency, and frequency drif of an enemble of ceium clock relaive o a maer clock. A problem wih hi ype of filer i ha, ince ime i unobervable, elemen of he covariance marix grow wihou bound [5]. Wei and Weier [5] uilized a one-ae filer o deermine he frequency of ceium clock ued in heir AT imecale. Only Tryon and Jone and Wei and Weier eed heir algorihm on real daa. Barne and Allan [] propoed a Kalman filer ha uilized only frequency and drif, poining ou ha only hee wo parameer are phyically meaningful for a frequency andard, and indeed howed by imulaion ha uch a wo-parameer filer performed beer han he hree-parameer one. Since he phae ime of a clock i inegraed from he frequency, i add no informaion, and oluion for a phae parameer i unneceary. Conequenly, a reaonable model for boh our ceium-beam frequency andard and our hydrogen maer which generally have inheren frequency drif i: f + f + d + ε + d + d + + 5

2 34 h Annual Precie Time and Time Inerval PTTI Meeing where f i a clock' frequency a ime,,...; d i i frequency drif; ε and are independen random variable wih zero mean and normal diribuion uncorrelaed in ime zero auocorrelaion, i.e. whie noie procee; and Δ i he ime ep. The laer i hour in our cae, which i in he whie FM noie regime of ceium and in he flicker FM regime of maer. For he maer, meauremen yem noie dominae clock noie for ampling ime up o a few hour and i characerized by whie phae noie. A Kalman filer ha aume he preence of only whie FM and random-walk FM noie canno exacly model -hour daa from a maer. However, in our previou paper [6], i wa hown ha, for an enemble of hydrogen maer conneced o a low-noie Timing Soluion Corporaion or TSC meauremen yem, uch a filer wih he parameer of frequency and frequency drif i quie capable of producing a mean imecale wih a frequency abiliy ha i no only viable, bu a ampling ime up o everal day, acually uperior o ha produced by a convenional lea-quare algorihm uch a he currenly operaional a USNO [7]. Among oher iue, hi paper addree wheher a viable imecale can be imilarly generaed for he noiier daa from our currenly operaional Daa Acquiiion Syem. In order o minimize proce noie, one mu chooe a reference ha i able a poible. The Mean imecale, a lea a derived from he clock under conideraion, canno o erve becaue i i no ye available a he fir ep of a recurion. One could refer all he frequencie and drif o one of he clock or o a Mean prediced from previou daa. However, i would be eaier and more error- and correlaionfree o refer one ype of clock, ay he maer, o he Mean of he oher ype of clock ceium; hi i our objecive. In USNO operaional imecale, he maer rae and drif are calibraed again he ceium Mean anyway, which i alway available. THE KALMAN FILTER The ae raniion equaion are: f d f d f d f d + ε + + ε + + or: X + Φ X + W + where he ubcrip denoe clock and Φ i he ae raniion marix. Le z i be he phae of clock i a ime relaive o he ame reference. The obervaion equaion are: 5

3 34 h Annual Precie Time and Time Inerval PTTI Meeing 53 + / / v v d f d f z z z z or:, V X H Z + where Z,Δ i he vecor of meauremen, H i he obervaion marix ime-dependen in ha he number of weighed clock wih available daa may vary, and V i he vecor of meauremen error ν i. For uncorrelaed parameer, he covariance marix Q Δ of he proce error W, for ufficienly mall Δ, i uch ha: ε ε d dq where i ε i he noie pecral deniy omeime called variance rae of random-walk FM and i i he noie pecral deniy of random-run FM random walk of frequency drif. The effec of hee error on he yem for any inerval Δ i: Φ Φ d d dq Q T + + / / 3 / / / 3 / 3 3 ε ε where Φ T i he ranpoe of marix Φ.

4 34 h Annual Precie Time and Time Inerval PTTI Meeing If he meauremen error are uncorrelaed, he meauremen error vecor V + Δ ha he covariance marix: v R v where v i he noie pecral deniy of random walk of phae, or whie FM. The aumpion of zero auocorrelaion doe require prefilering for uch oulier a erroneou phae meauremen hared by conecuive fir difference. The real daa in hi and he previou paper were ubjeced o robu prefilering. Raher han noie pecral deniie, le u inead olve for he variance σ ν ν Δ, σ ε ε Δ, and σ Δ a our noie parameer. The Hadamard variance, σ H, when he noie ource coni only of whie noie FM, random-walk FM, and random-run FM, can be expreed a [8]: σ H σ v τ + ε τ + 3 / 6 σ / σ τ where τ i he ampling ime. Our noie parameer may be olved by lea quare from hi equaion of condiion. Hadamard variance ha he beneficial propery of being ineniive o frequency drif. If he reference were a mean imecale, improved value could be obained from fi of he individual clock and recompued occaionally a ime when he performance of a clock appear o have changed ignificanly. The fir ep in he verion of he Kalman recurion in which no knowledge of he proce noie i required i o calculae he invere of he raniion covariance: P [ P ] + H T R H where he raniion covariance marix prediced for he nex ep Pˉ whoe upercrip denoe prediced can be aumed iniially o conain huge covariance. Second, we compue he Kalman gain K: K P H T R where P i he raniion covariance marix of he curren ep. Nex, we updae he parameer eimae huly where ^ denoe eimaed : X X + K [ Z H X ] Then we make he following predicion for he nex ep 54

5 34 h Annual Precie Time and Time Inerval PTTI Meeing X + Φ X T P + Φ P Φ + Q and o on. Each applicaion of he recurion yield an eimae of he yem ae ha i a funcion of he elaped ime ince he la filer updae, which can occur any ime, i.e. Δ i no necearily conan, and he daa need no be equally paced. Noe ha here are wo marix inverion inead of one a in he more common formulaion of he Kalman filer. We aw in our previou paper ha hi i of no conequence [6]. If prefiler analyi of he daa indicae he preence of a ignifican frequency ep, he covariance in P could be ree o huge value in order o reiniialize he filer. A formulaed here, he Kalman filer regain he ae informaion wihin a few ep, and he daa are appropriaely downweighed unil ha ha occurred. Thi reeing of he covariance affec only he clock involved, no he enire imecale, which i generaed from averaging he Kalman frequencie acro he enire enemble, a explained in he nex ecion. 3 THE KALMAN MEAN The frequencie and drif produced by hi wo-ae Kalman filer could be averaged and inegraed o generae a Kalman-filer-baed mean imecale. The relaive ize of he clock covariance Pˉ would be inverely relaed o he individual clock weigh. Robune would require he impoiion of an upper limi on any one clock' relaive weigh. If he filer and model operae correcly, he clock variance hould lowly converge o eady-ae value. While here i ome degradaion of he frequency abiliy of he imecale a a reul of he capping of each clock relaive weigh, hi degradaion i rarely ignifican in pracice unle he enemble i mall or very inhomogeneou, neiher of which i anicipaed o be he cae upon implemenaion. In any cae, robune mu be purchaed a ome price, and hi price will have o be decided upon during implemenaion. If, in forming he mean imecale, each clock frequency i weighed according o he invere of i variance, he imecale would be biaed by he clock-enemble effec. The predicion error i alway oo mall becaue he frequency of a clock i correlaed wih he frequency of he enemble, ince he enemble include a conribuion from each clock. The weigh, which are proporional o he invere of he variance, are herefore yemaically oo large, cauing a poiive feedback ha increaingly biae he imecale oward he enemble be clock [9,]. Thi can be avoided if he weigh of each clock i baed on i abiliy relaive o a Mean of which i i no a par, eiher by: uing anoher Mean enirely, ay one baed on ceium clock raher han maer; or compuing a Mean coniing of he re of he clock in he enemble and referring he clock in queion o ha Mean. can be accomplihed by uing he relaion: σ u σ / w where σ i he uncorreced variance, σ u i he correced unbiaed variance, and weigh i he relaive weigh, aumed o be conan wih ime [7]. In pracice, an upper limi would alo have o be placed on any one clock weigh. 55

6 34 h Annual Precie Time and Time Inerval PTTI Meeing Parameer and error eimae would be available from he filer in near real ime, bu pracical implemenaion would require robu deecion and rejecion of oulier meauremen whoe error are unlikely o have originaed in he clock, a well a promp recogniion of ime and frequency ep. Kalman filer provide boh a foreca of he nex daa poin and an eimae of i uncerainy, o i i poible o implemen robu oulier deecion, hough one mu allow for he poibiliy ha dicrepan poin are indicaion of a ep in ime or frequency. Sein propoed an alered gain funcion o moohly deweigh oulier and an adapive filer o recognize ime and frequency ep [3]. Daa ep deecion will be addreed in Secion 6. Iniially he filer' knowledge of he parameer will be crude, bu in poproceing one can make ue of fuure daa by running forward- and backward-moving filer hrough he daa and averaging he correponding parameer weighed by he number of ime ep conribuing o each uing weigh baed on filer variance moohing []. In order o avoid uing he ame daum wice, he ae a any one ime from one of he filer ha o be averaged wih he prediced ae a ha ime from he oher filer. Thi kind of proceing of ime daa ha only been done, o our knowledge, by NIST a poeriori for heir AT imecale []. Thi could be done in near real ime every ime ep for ome inerval of he lae daa, hough reroacive recompuaion would be involved. 4 THE MASERS The wo-ae Kalman filer decribed above, including he forward-and-backward averaging and a correcion for he clock-enemble effec, wa applied o 539 day of hourly daa eleced from hoe of Sigma Tau/Daum/Symmericom maer a USNO. The daa were produced by our recenly renovaed Daa Acquiiion Syem DAS, a meauremen yem involving univeral ime-inerval couner and muliplexed wiche. The dominan noie of hi yem i a phae noie of ~35 p. The daa were referenced o he Mean of he ame maer a compued by he curren USNO imecale algorihm [7], which removed conan frequencie and drif from each eady-ae egmen of every maer by modelling i again he ceium Mean compued by he ame algorihm. In pracice, he inceuou naure of he enemble and i reference can be avoided by referencing he maer o he Mean prediced by he filer and modelling he maer again a Kalman ceium Mean. Bu in hi paper, an independen reference i no neceary, ince our objecive are o: e he abiliy and lack of yemaic error of he Kalman Mean relaive o he operaional Mean for DAS daa of boh maer and ceium, raher han he lower-noie maer daa inveigaed in our previou paper [6]; deermine he ime conan of he filer in modelling imulaed hock o he clock, which will involve only comparion beween hocked and unhocked daa; and 3 earch for an opimal clock weighing cheme, whoe relaive value hould no change ignificanly beween wo rongly correlaed reference i.e. an independen ceium Mean and a ceium-calibraed Mean. The Kalman filer yielded very aifacory reul for he DAS daa of each maer. For example, Figure compare he Kalman frequencie wih he hourly frequencie of phae for maer NAV. The 537 day of daa hown would have o have been lea-quare fi by more han five linear egmen in order o be derended for frequency and drif wih our curren USNO algorihm [7]. Deciding on he placemen of uch egmen require labor and ome ubjecive judgmen ha he Kalman filer permi u o avoid. The Kalman frequencie were no very eniive o choice of proce noie parameer uning of he filer, i.e. he fi of each clock enire Hadamard variance curve by Equaion gave ufficienly accurae noie parameer in pie of he occaional mall change in he rae and drif parameer encounered here. The noie parameer for he maer had he following range: 56

7 34 h Annual Precie Time and Time Inerval PTTI Meeing σ σ σ v ε The phae error of he DAS meauremen yem i ~35 p, o for an hourly fir difference, he obervaional noie would be 35 p/hr THE KALMAN MASER MEAN Generaion of a mean imecale by he imple inegraion of he average inananeou Kalman frequency eimae for he clock enemble yield a preernaurally able reul reflecing only he evoluion of one clock model. Connecion wih realiy mu be mainained hrough ue of he baic imecale equaion [7,3]: z z Σ w i { x i + i x i + f i d i } where z i he phae of he reference minu he Mean of he enemble; i x i i he phae of he reference minu clock i; f i i he frequency and i w i he weigh of clock i; d i he frequency drif of clock i, repecively, relaive o he Mean, all a ime ; and i he ime ep. In our cae, hour. Boh he Kalman Mean and i reference, he curren USNO Mean, were calculaed wih hi equaion; he only difference i ha f i and d i are he ae parameer of he filer and he produc of lea-quare fi o fir difference, repecively. Unle noed oherwie, he individual clock weigh are uniy unle a daa poin i rejeced in he prefilering. One of he compuaional advanage o averaging clock frequencie a above inead of clock phae a in imecale algorihm ha ue anoher form of he baic equaion i he obviaion of he ime ep in he mean imecale when weighed clock are added or removed from he enemble. In he cae of he curren USNO imecale algorihm, he clock rae relaive o he Mean i deermined prior o i addiion o he enemble in order ha hi rae can be removed by he baic equaion, preerving he aumed conan frequency of he enemble. In he cae of he Kalman filer under e, however, he clock frequency i deermined relaive o he curren USNO maer Mean, raher han he Kalman Mean being compued, o he rae o be removed i ha relaive o he USNO maer Mean, o which he Kalman Mean will hen be referenced. Thi will no be he cae in an acual implemenaion, however. In ha cae, he ceium will be referenced o he Kalman ceium Mean being compued, and he maer will be calibraed by and referenced o he Kalman ceium Mean. Averaging he maer ogeher over he ime pan MJD hough ome of he maer came in a lae a MJD 5347, we obained he reuling Kalman Mean, relaive o he USNO maer Mean, i hown in Figure. The wo Mean do no differ more han.4 n from one anoher. Since boh Mean have been generaed from he ame raw daa and he clock in each Mean were ulimaely 57

8 34 h Annual Precie Time and Time Inerval PTTI Meeing calibraed again he USNO ceium Mean, hey are of very imilar long-erm abiliy, and heir phae hould random-walk away from one anoher, which indeed appear o be he cae. In order o deermine he abolue frequency abiliie of he wo e of maer Mean curren and Kalman, we divided he maer ino hree ub-enemble of four maer each and performed a hreecornered-ha analyi on each e uing he mehod of Torcao e al. [4], which correc for ub-enemble inercorrelaion. The reul are given in Figure 3. The Kalman imecale abiliie are a lea a grea a hoe of he curren imecale. Apparenly he ignifican meauremen phae noie a he hore ampling ime and he flicker FM noie of he maer are no impedimen o he ue of hi filer on maer, hough he eparaion beween he wo e of abiliy curve a he mid-erm ampling ime in Figure 3 indicae a differen apporioning of he proce and meauring error by he wo imecale algorihm. In order o ue daa efficienly and combine clock properly in a Kalman mean, we hould e poible clock weighing cheme. The crierion for uch a cheme would be maximum abiliy in he ampling ime region of mo inere for he eering of our Maer Clock, which range from day he ime conan of our frequency eering o 5 day he ime conan of our phae eering [5]. I migh eem logical o ue a clock weigh he invere frequency variance eimaed by he Kalman filer. However, i i no poible o inveigae weighing cheme uing hee daa, becaue hey were calibraed wih a Mean baed on he ame equally weighed clock, o any abiliy difference beween he Kalman and curren Mean due o he weighing would only indicae ha heir repecive weighing cheme were differen. A we will ee in Secion 7 for he ceium, weigh baed on frequency abiliie for ampling ime horer han he ime conan of he filer found o be day in Secion 6 yield beer reul han hoe baed on longer ampling ime. Bu he cae for he maer i complicaed by he fac ha maer are already beyond he whie FM noie regime a our hore ampling ime hour. Indeed, whiene e wih he MATLAB ofware package indicae ha: he frequency reidual of he individual maer are ignificanly auocorrelaed a ampling ime horer han 6 hour due o each clock prevailing flicker noie; and he frequency reidual from he Kalman maer Mean are ignificanly auocorrelaed a ampling ime horer han 6 day, probably due o he fac ha he maer are being calibraed again an inceuou Mean one compued from he ame daa proceed by anoher imecale algorihm and o he -day ime conan of he filer. 6 KALMAN FILTER TIME CONSTANT Before moving on o he noiier daa of he ceium, le u ue he maer daa o deermine he ime conan of hi Kalman filer when operaing on DAS daa. Thi ime conan i imporan becaue of he effec of proce noie, uch a ha due o environmenal hock o he clock. Significan hock on longer imecale han hi ime conan can be properly handled by he filer in i normal operaion, bu uch hock ha ake place on horer imecale would require pecial handling, e.g. daa rejecion if he effec i ranien, or clock downweighing and covariance reiniializaion if he effec appear o be permanen, a lea unil he filer can again aifacorily model he clock new parameer. Failure o ake appropriae acion could reul in he deabilizaion of he Maer Clock if i were eered o he Kalman-filer-baed Mean. In view of hi evenualiy, before hi filer can be implemened in real ime, a e will need o be deigned ha will recognize frequency excurion in he daa ha are oo rapid o be followed cloely by he filer. Perhap he frequency innovaion produced by a Kalman filer uned for whie phae and whie FM noie and, hence, having a much horer ime conan han ha of our propoed filer, wih any ignifican frequency divergence aken a indicaing a recen frequency ep by he clock. Anoher 58

9 34 h Annual Precie Time and Time Inerval PTTI Meeing poibiliy, a lea for he maer, i o monior heir frequencie on a yem wih a lower meauring noie han he DAS, e.g. he TSC yem ued in our previou udy [6]; ceium are no profiably udied by uch a yem becaue of heir greaer proce noie exceed he meauring noie of even he DAS. Prior o developing uch a e, we mu deermine he ime conan of he curren filer. Thi can be done by inering an arificial ep in he fir-difference daa and oberving how long he filer frequencie require o reurn o eady ae. Ramp hock of variou ize and over inerval from hour o day were inered in he daa for each of he maer. A ample of he reul i hown in Figure 4 for four of he maer. The average recovery ime wa found o be day, which we will ake a he ime conan of hi filer. Conequenly, he filer require a rae-change deecor wih a ime conan much horer han day. 7 THE CESIUMS AND THE KALMAN CESIUM MEAN Having eed our filer on he abler daa of he maer, le u now proceed o he noiier daa of he ceium clock, which mu ulimaely provide he yemaic calibraion of all he clock. A aed earlier, however, in hi e we will calibrae he frequencie of he ceium again hoe of he maer, whoe frequencie were in urn calibraed wih hoe of hee very ceium, o here will be ince no preen in he enviioned implemenaion. Doing o for 498 day of daa for 5 ceium clock, we obained he following range for he proce noie parameer: σ σ v ε σ The noie model aumed by our filer i valid for he ceium becaue he DAS meauremen phae noie, even had i been a problem for he maer, i wamped by he whie FM proce noie of he ceium. Auming equal clock weigh for he momen, we compued a Kalman ceium Mean whoe difference from he curren USNO ceium mean are given in Figure 5 and whoe frequency abiliy relaive o he USNO maer Mean i depiced in Figure 6. Figure 5 how no deparure of he wo Mean greaer han 8 n, abou wha he mean would be expeced o depar from one anoher a he reul of random walk FM [6]. In Figure 6, we ee ha he wo Mean are of comparable abiliy on he long erm. The mall decreae in abiliy on he hor erm i likely due o he more realiic a oppoed o opimiic modelling of he whie noie of he individual ceium compared o he lea-quare fi of he curren USNO algorihm. In Secion 5 we alo noed effec from he differen noie modelling for he maer. Unlike he maer, he ceium ill diplay whie FM noie for ampling ime a long a our -hour meauremen rae, o i migh be poible o increae he abiliy of he Kalman ceium Mean by weighing he clock unequally. We have no been able o accomplih hi wih he curren USNO algorihm, parly becaue he clock are o imilar in frequency abiliy and parly becaue any clock diplaying a change in i parameer i immediaely deweighed unil i can be remodelled, a procedure 59

10 34 h Annual Precie Time and Time Inerval PTTI Meeing ha end o make he performance of all weighed clock appear imilar. Sill, perhap he more realiic reamen of he proce noie by he Kalman filer may make differeniaion beween he clock poible. The Kalman frequencie poe he dominan clock noie, o i i logical o ry o weigh he clock according o he invere frequency variance eimaed every ime ep by he filer. Weighing he clock huly, however, only worened he abiliy ee Figure 7, probably becaue he frequency variance are moly a meaure of he noie prevailing around ampling ime imilar o he ime conan of he filer day, while he relaive abiliie of he individual clock evidenly change more rapidly. Perhap, hen, i migh be poible o increae he abiliy of he Kalman ceium Mean by weighing he clock according o he invere Allan variance for ampling ime much horer han day. Conequenly, he ceium daa were weighed uing Allan variance for ampling ime of hour, 6 hour, hour, and day. Indeed, he horer he ampling ime, he beer he abiliy became ee Figure 7, bu he abiliy did no quie aain ha of our equally weighed reul. Conequenly, i i no likely ha one can improve he abiliy of hi Kalman-filer-baed Mean wih Allan-variance weigh. CONCLUSION We have hown ha: he DAS meauremen daa for he maer, he ame ued in he operaional USNO maer mean imecale, can alo be uccefully uilized by our Kalman filer o generae a Mean ha i boh a able a he operaional maer Mean and a ha produced from he le noiy TSC daa udied previouly [6], a lea over ampling ime including a few week, which i neceary if uch a imecale were o replace he curren one for purpoe of eering a Maer Clock. DAS meauremen daa for he ceium clock, he ame ued in he operaional USNO ceium mean imecale, can alo be uccefully uilized by our Kalman filer o generae a ceium Mean abou a able a he operaional ceium Mean, wihin he limiaion of he inceuou frequency calibraion. Thi capabiliy i neceary if uch a imecale were ued o replace he curren one for purpoe of calibraing he frequencie and drif of he maer in our enemble. The ime conan of our filer/daa combinaion i day, meaning ha ignifican hock o he clock occurring over inerval horer han day mu be iolaed and heir effec eliminaed from he Mean uing a mehod wih a more alacriou ep repone han ha of our filer. No beer weighing cheme wa found for he ceium han ha of equaliy among each. Before our final objecive, real-ime implemenaion of hi mean imecale algorihm, can be aemped, we mu: e and implemen a mehod capable of idenifying and allowing for ae change oo rapid for hi propoed filer o handle; e wheher a free-running Mean prediced by hi algorihm can erve a a ufficienly able reference, raher han he operaional USNO maer Mean ued in hi paper; 5

11 34 h Annual Precie Time and Time Inerval PTTI Meeing decide wheher averaging he ae parameer from he forward- and backward-moving filer i preferable o uing hoe of he forward-moving filer, given ha he moohing by he former i purchaed a he co of phae ep in he real-ime Mean. The advanage of replacing our curren imecale algorihm wih a Kalman-filer-baed one would be a reducion in labor and an increae in he mahemaical rigor of he Mean compuaion. ACKNOWLEDGMENTS The auhor would like o hank Paul Koppang and Jame Skinner for ueful dicuion and he laer for help uilizing MATLAB. REFERENCES [] P. V. Tryon and R. H. Jone, 983, Eimaion of parameer in model for ceium beam aomic clock, Journal of Reearch of he Naional Bureau of Sandard, 88, 3-6 and R. H. Jone and P. V. Tryon, 983, Eimaing ime from aomic clock, ibid., 7-4. [] J. A. Barne and D. W. Allan, 985, Time cale abiliie baed on ime and frequency Kalman filer, in Proceeding of he 39 h Annual Sympoium on Frequency Conrol, 9-3 May 985, Philadelphia, Pennylvania, USA NTIS AD-A744, pp. 7-. [3] S. R. Sein and R. L. Filler, 988, Kalman filer analyi for real ime applicaion of clock and ocillaor, in Proceeding of he 4 nd Annual Sympoium on Frequency Conrol, -3 June 988, Balimore, Maryland, USA IEEE Publicaion 88CH588-, pp ; S. R. Sein, 989, Kalman filer analyi of preciion clock wih real-ime parameer eimaion, in Proceeding of he 43 rd Annual Sympoium on Frequency Conrol, 3 May- June 989, Denver, Colorado, USA IEEE Publicaion 89CH69-6, pp. 3-36; S. R. Sein, 99, Inveigaion of he predicabiliy of clock and he udy of i applicaion o a digial DCS, Ball Communicaion Syem Diviion Conrac Repor DCA--88-C-7. [4] C. A. Greenhall,, Kalman plu weigh: a ime cale algorihm, in Proceeding of he 33 rd Annual Precie Time and Time Inerval PTTI Syem and Applicaion Meeing, 7-9 November, Long Beach California, USA U.S. Naval Obervaory, Wahingon, D.C., pp [5] M. Wei and T. Weier, 99, A new ime cale algorihm: AT plu frequency variance, in Proceeding of he Annual Precie Time and Time Inerval PTTI Applicaion and Planning Meeing, 8-3 November 989, Redondo Beach, California, USA U.S. Naval Obervaory, Wahingon, D.C., pp [6] L. A. Breakiron,, A Kalman filer imecale for aomic clock and imecale, in Proceeding of he 33 rd Annual Precie Time and Time Inerval PTTI Syem and Applicaion Meeing, 7-9 November, Long Beach, California, USA U.S. Naval Obervaory, Wahingon, D.C., pp [7] L. A. Breakiron, 99, Timecale algorihm combining ceium clock and hydrogen maer, in Proceeding of he 3 rd Annual Precie Time and Time Inerval PTTI Applicaion and Planning 5

12 34 h Annual Precie Time and Time Inerval PTTI Meeing Meeing, 3-5 December 99, Paadena, California, USA NASA Conference Publicaion 359, pp [8] S. T. Huell, 996, Relaing he Hadamard variance o MCS Kalman filer clock eimaion, in Proceeding of he 7 h Precie Time and Time Inerval PTTI Applicaion and Planning Meeing, 9 November- December 995, San Diego, California, USA NASA Conference Publicaion 3334, pp [9] K. Yohimura, 98, Calculaion of unbiaed clock-variance in uncalibraed aomic ime cale algorihm, Merologia, 6, [] J. Levine, 996, Incorporaing daa from a primary frequency andard ino a ime cale, in Proceeding of he 996 IEEE Inernaional Frequency Conrol Sympoium, 5-7 June 996, Honolulu, Hawaii, USA IEEE Publicaion 96CH35935, pp [] R. G. Brown and P. Y. C. Hwang, 99, Inroducion o Random Signal and Applied Kalman Filering John Wiley & Son, New York, pp [] M. A. Wei and T. P. Weiar, 994, Sifing hrough Nine Year of NIST Clock Daa wih TA, Merologia, 3, 9-9. [3] D. B. Percival, 978, The U.S. Naval Obervaory Clock Time Scale, IEEE Tranacion on Inrumenaion and Meauremen, IM-7, [4] F. Torcao, C. R. Ekrom, E. A. Bur, and D. N. Maaki, 999, Eimaing frequency abiliy and cro-correlaion, in Proceeding of he 3 h Annual Precie Time and Time Inerval PTTI Syem and Applicaion Meeing, -3 December 998, Reon, Virginia, USA U.S. Naval Obervaory, Wahingon, D.C., pp [5] L. A. Breakiron and P. Koppang, 996, Frequency eering of hydrogen maer, in Proceeding of he 996 IEEE Inernaional Frequency Conrol Sympoium, 5-7 June 996, Honolulu, Hawaii, USA IEEE Publicaion 96CH35935, pp. 3-. [6] D. W. Allan, 987, Time and frequency ime-domain characerizaion, eimaion, and predicion of preciion clock and ocillaor, IEEE Tranacion on Ulraonic, Ferroelecric, and Frequency Conrol, UFFC-34, [7] C. R. Ekrom and P. A. Koppang,, Degree of freedom and hree-cornered ha, in Proceeding of he 33 rd Annual Precie Time and Time Inerval PTTI Syem and Applicaion Meeing, 7-9 November, Long Beach, California, USA U.S. Naval Obervaory, Wahingon, D.C., pp

13 34 h Annual Precie Time and Time Inerval PTTI Meeing Par in MEASURED FREQUENCIES KALMAN FREQUENCIES MJD Figure. A comparion beween hourly frequencie and Kalman frequencie for maer NAV relaive o he curren USNO maer Mean. nanoecond MJD Figure. The phae difference beween he Kalman maer Mean and he USNO maer Mean. The clock are equally weighed in boh. 53

14 34 h Annual Precie Time and Time Inerval PTTI Meeing LOG ALLAN DEVIATION LOG TAU ec CURRENT ALGORITHM ENSEMBLE # CURRENT ALGORITHM ENSEMBLE # CURRENT ALGORITHM ENSEMBLE #3 KALMAN ALGORITHM ENSEMBLE # KALMAN ALGORITHM ENSEMBLE # KALMAN ALGORITHM ENSEMBLE #3 Figure 3. Frequency abiliy of maer Mean for hree ub-enemble a generaed by he Kalman filer and he curren USNO imecale algorihm. The error bar were compued uing he mehod of Ekrom and Koppang [7]. 54

15 34 h Annual Precie Time and Time Inerval PTTI Meeing Figure 4. Kalman filer frequencie before and afer inerion of a imulaed n/day ep a MJD 53. for maer NAV red, NAV4 green, NAV8 blue, and NAV orange. nanoecond MJD Figure 5. The phae difference beween he Kalman ceium Mean and he USNO maer Mean. The clock are equally weighed in boh. 55

16 34 h Annual Precie Time and Time Inerval PTTI Meeing LOG ALLAN DEVIATION LOG TAU ec USNO KALMAN Figure 6. The frequency abiliy v. ampling ime τ for he Kalman and USNO ceium Mean relaive o he USNO maer Mean. LOG ALLAN DEVIATION LOG TAU ec EQUAL WEIGHTS -HOUR WEIGHTS 6-HOUR WEIGHTS -HOUR WEIGHTS -DAY WEIGHTS Figure 7. The frequency abiliy v. ampling ime τ for Kalman ceium Mean, relaive o he USNO maer Mean, compued for equal clock weigh and clock weigh baed on Allan variance for ampling ime of hour, 6 hour, hour, and day. 56