Research Article Time Synchronization and Performance of...

Hindawi Publishing CorporationInternational Journal of Navigation and ObservationVolume 2013 Article ID 371450 5 pageshttpdxdoiorg1011552013371450

Research ArticleTime Synchronization and Performance ofBeiDou Satellite Clocks in Orbit

Han Chunhao Cai Zhiwu Lin Yuting Liu Li Xiao ShenghongZhu Lingfeng and Wang Xianglei

Beijing Satellite Navigation Center Beijing 100094 China

Correspondence should be addressed to Lin Yuting lyt1108163com

Received 24 March 2013 Revised 3 July 2013 Accepted 31 July 2013

Academic Editor Sandro Radicella

Copyright copy 2013 Han Chunhao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The time model of Beidou satellite clocks is analyzed The general relations of satellite clocks with the system time are studiedThe error sources of two-way radio time transfer between satellites and uplink stations are analyzed The uncertainty of type A isabout 03 ns in Beidou system All the satellite clocks in orbit of Beidou satellite navigation system are evaluated by the clock offsetsobserved by the two-way radio time transferThe frequency stabilities at a sample time of 10000 s and 1 day for all the satellite clocksare better than 10 times 10

minus13 It means that the performance of Beidou satellite clocks in orbit is consistent with the ground test andthe results in orbit are a little better than those in ground vacuum

1 Introduction

Beidou satellite navigation system began to provide regionalservice since December 2012 The constellation of Beidousystem is constituted of 14 satellites in orbit 5GEO satellites5IGSO satellites and 4MEO satellites Table 1 shows the basicinformation of the Beidou satellites Service area now coverslatitude 55

∘S sim 55∘N and longitude 55

∘E sim 180∘E Practical

operational accuracy of Beidou system is better than 10m(95) in horizontal and 15m (95) in vertical [1]

As we all know that time synchronization of satelliteclock plays a significant role in satellite navigations accurateand reliable satellite clock offset parameters are the base ofPNT service Time synchronization error of satellite clock ismainly caused by the time transfer from the master stationand its offset prediction The clock prediction error dependson its frequency instability Then the measurement predic-tion and evaluation of satellite clocks are very important fora satellite navigation system

GPS operates a worldwide monitoring stations networkand includes sixUSAF stations elevenNGA stations and twoIGS stations Geodetic receivers are equipped in these stationsto monitor the performance of satellite clocks On 28 May2010 the first Block IIF satellite designated SVN62PRN25was launched containing three atomic frequency standards

one DCBFS serial number 1010 (Cs 1010) and two RFSserial numbers 27 and 14 (Rb 27 and Rb 14) The frequencystabilities of SVN62 Cs 1010 and Rb 27 are respectively 5 times

10minus14 and 7times10

minus15 at 1 day in orbit And the frequency stabilityof GPS Block IIR Rb is about 9 times 10

minus15 at 1 day for October2010 (all using NGA data) [2ndash4] while the frequency stabilityat 1 day for Galileo satellite clock is about 5times10

minus14 for GIOVEA Rb clocks and 8 times 10

minus15 for GIOVE B PHM [5 6]How about the performance of Beidou clocks in orbits

It is a very concerned question for many GNSS users Thesystem signals and observations of Beidou regional systemare analyzed by Deutsches Zentrum fur Luft-und Raumfahrt(DLR) using a local monitoring network in March 2012The short-term stability and middle-term stability of Beidousatellite clocks are analyzed and compared to other systemsFrequency stability of Beidou RAFS is about 7 times 10

minus12

sim

1 times 10minus11 at 1 second Frequency stability of the best Beidou

satellite clock is about 1 times 10minus13 at 1000 seconds and GPS

Block IIF is not worse than 1times10minus13 [7] In the following sec-

tions the timemodel of satellite clocks used by Beidou systemis described then the error source and uncertainty of the two-way radio time transfer (TWTT) are analyzed which is usedtomeasure clock differences between satellites and the uplinkstations Finally the results and conclusions are detailed

2 International Journal of Navigation and Observation

Table 1 Basic information of Beidou satellite clocks

Num Type (Num) Date03 GEO-1 201011704 GEO-3 20106205 IGSO-1 20108106 GEO-4 201011107 IGSO-2 2010121808 IGSO-3 201141009 IGSO-4 201172710 IGSO-5 201112211 GEO-5 201222512 MEO-3 201243013 MEO-4 201243014 MEO-5 201291915 MEO-6 201291916 GEO-6 20121025

2 Time Model of Satellite Clocks

Considering the large-scale spacetime involved (about 1 times

105 km in space and several days even several months or

years in time) and the precision requirements (1m even 1 cm1mm level) the GNSS data process must be dealt with underthe framework of relativity and quantum theory Two kindsof conceptually different time scales are concerned in GNSSproper times and coordinate times Essentially for any twoevents the observed space interval and time interval betweenthem are dependent on the observer The time readingsgiven directly by ideal clocks located in satellites stations orobservers are proper times They are related to the observeror to the spacetime environments of the clocks This meansthat different observer has different clock due to its relativevelocity and position in the gravitation field In order to havea common time reference for all observers we must choose aspecial observer and construct a reference systemA referencesystem contains a 3-dimensional space reference frame anda time reference The former determines the spatial position(3 space coordinates) of an event and the latter gives thehappening time which is called coordinate time For Earthsatellites a nonrotating geocentric reference system is used todescribe their orbits The reference time is usually TCG (thegeocentric coordinate time) or TT (the terrestrial time) [8 9]

The relationship between the proper time 120591 of satellite andthe coordinate time TT (here noted by 119905) can be modeled as[8]

119905 = [1 minus(1198820

minus (32) (120583119886))

1198882

] (120591 minus 1205910

)

+2

1198882

radic120583119886 sdot 119890 (sin 119864 minus sin 1198640

)

(1)

where 120583 = GM119864

is the geocentric gravitation constant 1198820

the gravity potential of the geoid 119886 the orbit main axis 119890

the eccentricity 119864 the real eccentric anomaly and 119888 the speedof light respectively Then

120591 minus 119905 = [1198820

minus ((32) (GM119864

119886))

1198882

] (119905 minus 1199050

)

minus2

1198882

radic119886GM119864

sdot 119890 sin119864

(2)

If the satellite clock 119879(119905) is modeled as

119879 (119905) = 120591 (119905) + 1198860

+ 1198861

(119905 minus 1199050

) + 1198862

(119905 minus 1199050

)2

+ 120585 (119905) (3)

in which 1198860

1198861

and 1198862

are clock offset parameters and 120585(119905) isthe clock phase noise the offset of satellite clock reffered toBDT can be written as

119909 (119905) equiv 119879 (119905) minus BDT (119905)

= 1198860

+ 1198861

(119905 minus 1199050

) + 1198862

(119905 minus 1199050

)2

+ Δ119905119901

grav + 120585 (119905)

(4)

HereΔ119905119901

grav is the periodic term of relativistic effect as follows

Δ119905119901

grav = minus2

1198882

radic120583119886 119890 sin 119864 = minus2119878

sdot 119878

1198882

(5)

The relativistic effect must be taken into account for the eval-uation of clock performance If not the stability of frequencywill be influenced The quasi-half-day periodical terms inAllan deviations of GPS clocks and Galileo clocks [2 5] weguess may be caused by this term

3 Two-Way Satellite TimeTransfer and Error Analysis

In Beidou system TWTT between satellites and uplink sta-tions is used for the satellitetime synchronization The basicprinciple of TWTT is as follows The satellites and stationsgenerate and transmit pseudo-range signals controlled bytheir local clocks then the uplink pseudo-range 120588

119906

anddownlink pseudo-range 120588

119889

are measured by the satellitesand the stations respectively The uplink pseudo-range anddownlink pseudo-range can be written as

120588119906

(119879119903

119878

) =1003816100381610038161003816119878 (119905119903

119878

) minus 119877

(119905119890

119877

)1003816100381610038161003816 sdot

1

119888minus Δ119879119877

(119905119890

119877

) + Δ119879119878

(119905119903

119878

)

+ 120591119890

119877

+ 120591119903

119878

+ 120591tro + 120591ion (119891119906) + 120591grav

120588119889

(119879119903

119877

) =1003816100381610038161003816119877 (119905119903

119877

) minus 119878

(119905119890

119878

)1003816100381610038161003816 sdot

1

119888+ Δ119879119877

(119905119903

119877

) minus Δ119879119878

(119905119890

119878

)

+ 120591119903

119877

+ 120591119890

119878

+ 120591tro + 120591ion (119891119889) + 120591grav

(6)

where 119905119903

119878

and 119905119890

119878

are time of reception and emission of thesatellite signal 119905119903

119877

and 119905119890

119877

are time of reception and emission ofthe station signalΔ119879

119878

andΔ119879119877

are satellite and stationrsquos clockoffset 120591119903

119877

and 120591119890

119877

are time delay of reception and emission ofthe station equipment 120591119890

119878

and 120591119903

119878

are time delay of receptionand emission of the satellite equipment 120591tro and 120591ion are

International Journal of Navigation and Observation 3

troposphere delay and ionosphere delay119891119906

and119891119889

are uplinkfrequency and downlink frequency 120591grav relativistic timedelay caused by Earth gravitation

The clock differences between satellites and stations arecomputed in the master station by using the uplink pseudo-ranges and the downlink pseudo-ranges The satellite clockoffset can be given by the observed uplink pseudo-range anddownlink pseudo-range as follows

Δ119879119878

(119905119894

) = Δ119879119877

(119905119894

) +1

2

times [120588119906

(119879119894

119878

) minus 120588119889

(119879119894

119877

)]

minus1

119888( 119878

minus 119877

) sdot 119899119877119878

(Δ119879119878

minus Δ119879119877

minus 120591119877119878

)

+ Δ120591119877

minus Δ120591119878

minus Δ120591ion + sdot sdot sdot

(7)

where

Δ120591119877

equiv 120591119903

119877

minus 120591119890

119877

Δ120591119878

equiv 120591119903

119878

minus 120591119890

119878

Δ120591ion equiv 120591ion (119891119906) minus 120591ion (119891119889)

119899119877119878

equiv(119878

minus 119877

)

1003816100381610038161003816119878 minus 119877

1003816100381610038161003816

(8)

The random error of satellite clock difference includes thenoise of pseudo-range observable and the satellite clock phasenoise In short term (le1000 s) the influence of the frequencydrift and phase noise of satellite clock to clock offset can beneglected So the uncertainty of typeAof satellite-board clockoffset measurement can be calculated by the fluctuation ofclock difference Analysis shows that the uncertainty of typeA is less than 03 ns [10] In middle or long term (ge10000 s)the influence of the pseudo-range noise can be neglected andthe results of the Allan variance of satellite clocks are reliable

4 Performance Evaluation ofBeidou Satellite Clocks in Orbit

Satellites that include GEO satellites of serial number 03 0406 and 11 IGSO satellites of serial number 07 08 09 and 10andMEOsatellites of serial number 13 and 14 are evaluated Inorder to ensure the reliability of the evaluation result the timeinterval of satellite clock data is no less than 15 daysThe timescale reference for analysis is the high performance hydrogenclock in ground

Figures 1 2 and 3 show the linear residuals and second-order polynomial residuals of the observed satellite clock off-sets

The green curves are plots of the linear residuals of satel-lite clocks All of the linear residual of GEO-3 GEO-4 andIGSO-2 are smooth which mean that the rubidium clockshave significant frequency drifts The blue curves are thesecond-order polynomial residuals of satellite clocks which

0

100

200

0

2000

0 10 20 30 40 50 60minus200

minus100

Seco

nd-o

rder

pol

ynom

ial r

esid

uals

(ns)

Time (days)

Second-order polynomial

minus4000

minus2000 Line

ar re

sidua

ls (n

s)

Residual errors of GEO-3 satellite fromJanuary 1 2012 to February 21 2012

minus90644e minus 010t2 + 0016545t minus 9246854626

Figure 1 Residual of GEO-3 satellite clock

2205 2210 2215 2220 2225 2230 2235 2240

0

500

1000

Time (days)

0

10

20

minus1500

minus1000

minus500

minus30

minus20

minus10

1058t2 + 012814t + 4745457275


Seco

nd-o

rder

pol

ynom

ial r

esid

uals

(ns)


Line

ar re

sidua

ls (n

s)


0 10 20 30 40 50 60 70 80

0

100

200

300

400

Time (days)

0

1000

2000

minus200

minus100

Seco

nd-o

rder

pol

ynom

ial r

esid

uals

(ns)


Line

ar re

sidua

ls (n

s)Residual errors of IGSO-2 satellite from

January 13 2012 to March 31 2012

minus1000

minus2000

minus3000

minus4000


Figure 3 Residual of IGSO-2 satellite clock


Table 2 Frequency stability of Beidou system satellite clocks

GEO-1 GEO-3 GEO-4 GEO-5 IGSO-1 IGSO-2 IGSO-3 IGSO-4 IGSO-5Stability(10000 s) 731 times 10minus14 552 times 10minus14 758 times 10minus14 917 times 10minus14 813 times 10minus14 595 times 10minus14 794 times 10minus14 853 times 10minus14 898 times 10minus14

Stability(1 day) 671 times 10minus14 290 times 10minus14 383 times 10minus14 566 times 10minus14 938 times 10minus14 307 times 10minus14 253 times 10minus14 391 times 10minus14 445 times 10minus14

100 101 102 103 104 105 106

Averaging time (s)

Alla

n fre

quen

cy st

abili

ty120590y(120591)

10minus15

10minus14

10minus13

10minus12

10minus11

10minus10

GEO-1GEO-3GEO-4GEO-5IGSO-1IGSO-2

IGSO-3IGSO-4IGSO-5MEO-3MEO-4

Figure 4 Frequency stability of Beidou system satellite clocks

demonstrate that the frequency drafts are changing slowlyand the rubidium clocks in orbit have high-level noise char-acteristic such as flick and random walk

The frequency stability of Beidou satellite clocks is evalu-ated by use of the overlappingAllan deviation Figure 4 showsplots of the frequency stability of Beidou system satelliteclocks Table 2 shows the frequency stability at a sample timeof 10000 seconds and 1 day

The frequency stability of Beidou satellite clocks is of thelevel of 10minus14 at a sample time of 10000 seconds and 1 dayThefrequency stability at a sample time of 10000 seconds is about595 sim 917times10

minus14 and that at a sample time of 1 day is about253 sim 938 times 10

minus14Figure 5 gives the comparison of the clock performances

in orbit and in the ground vacuumThe results show that theperformances in orbits are conformable with those in groundAs a whole the results in orbit are a little better than those inground

5 Conclusion

The long-term evaluation for Beidou satellite clocks has beendone using TWTT between satellites and stationsThe results

1 2 3 4 5 6 7 8 9

Allan stability

In orbitIn ground vacuum

Alla

n fre

quen

cy st

abili

ty120590y(86400)

10minus13

Figure 5 Clock day stabilities in orbit and in ground vacuum pots

show that the performance of satellite clock is steady and ingood condition The frequency stabilities at a sample time of10000 s and 1 day for all the satellite clocks are better than 10times

10minus13 Itmeans that the performance of Beidou satellite clocks

in orbit is consistent with the ground test and the results inorbit are a little better than those in ground vacuum

Acknowledgments

Theauthors wish to thank the Editor SandroM Radicella theEditorial AssistantMs Joanna and the anonymous reviewerswhose comments helped improve this paper enormously

References

[1] H Qiaohua ldquoDevelopment of Beidou navigation satellite sys-temrdquo in Proceedings of the 5th Meeting of International Commit-tee on GNSS (ICG-5 rsquo12) Beijing China 2012

[2] F Vannicola R Beard J White and K Senior ldquoGPS Block IIFatomic frequency standard analysisrdquo in Proceedings of the 42thAnnual Precise Time and Time Interval (PTTI) Meeting pp 181ndash196 2010

[3] J Oaks J A Buisson andMM Largay ldquoA summary of theGPSconstellation clock performancerdquo in Proceedings of the 39thAnnual Precise Time and Time Interval (PTTI) Meeting pp 119ndash130 2007

[4] D M Manning and C P Petersen ldquoAFNGA GPS monitorstation high-performance cesium frequency standard stability


20072008 from NGA kalman filter clock estimatesrdquo in Pro-ceedings of the 40th Annual Precise Time and Time Interval(PTTI) Meeting pp 335ndash348 2008

[5] P Waller F Gonzalez S Binda et al ldquoThe in-orbit performan-ces of GIOVE clocksrdquo IEEE Transactions on Ultrasonics Ferro-electrics and Frequency Control vol 57 no 3 pp 738ndash745 2010

[6] P Waller F Gonzalez and S Binda ldquoLong-term performanceanalysis of giove clocksrdquo in Proceedings of the 42th Annual Pre-cise Time and Time Interval (PTTI) Meeting pp 171ndash180 2010

[7] O Montenbruck A Hauschild P Steigenberger U Hugento-bler P Teunissen and S Nakamura ldquoInitial assessment of thecompassbeidou-2 regional navigation satellite systemrdquo GPSSolutions vol 17 no 2 pp 211ndash222 2013

[8] H Chunhao ldquoTime measurement within the frame of relativ-ityrdquo Progress in Astronomy vol 20 no 2 pp 107ndash113 2002

[9] H Chunhao C Zhiwu L Yuting L Li et al ldquoTime synchroni-zation and performance evaluation of beidou satellite clocksrdquoin Proceedings of the 3rd China Satellite Navigation Conference2012

[10] L Liu L-F Zhu C-H Han X-P Liu and C Li ldquoThe modelof radio two-way time comparison between satellite and stationand experimental analysisrdquo Chinese Astronomy and Astrophys-ics vol 33 no 4 pp 431ndash439 2009

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Hindawi Publishing Corporation httpwwwhindawicom

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Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


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DistributedSensor Networks



Table 1 Basic information of Beidou satellite clocks

Num Type (Num) Date03 GEO-1 201011704 GEO-3 20106205 IGSO-1 20108106 GEO-4 201011107 IGSO-2 2010121808 IGSO-3 201141009 IGSO-4 201172710 IGSO-5 201112211 GEO-5 201222512 MEO-3 201243013 MEO-4 201243014 MEO-5 201291915 MEO-6 201291916 GEO-6 20121025

2 Time Model of Satellite Clocks

Considering the large-scale spacetime involved (about 1 times

105 km in space and several days even several months or

years in time) and the precision requirements (1m even 1 cm1mm level) the GNSS data process must be dealt with underthe framework of relativity and quantum theory Two kindsof conceptually different time scales are concerned in GNSSproper times and coordinate times Essentially for any twoevents the observed space interval and time interval betweenthem are dependent on the observer The time readingsgiven directly by ideal clocks located in satellites stations orobservers are proper times They are related to the observeror to the spacetime environments of the clocks This meansthat different observer has different clock due to its relativevelocity and position in the gravitation field In order to havea common time reference for all observers we must choose aspecial observer and construct a reference systemA referencesystem contains a 3-dimensional space reference frame anda time reference The former determines the spatial position(3 space coordinates) of an event and the latter gives thehappening time which is called coordinate time For Earthsatellites a nonrotating geocentric reference system is used todescribe their orbits The reference time is usually TCG (thegeocentric coordinate time) or TT (the terrestrial time) [8 9]

The relationship between the proper time 120591 of satellite andthe coordinate time TT (here noted by 119905) can be modeled as[8]

119905 = [1 minus(1198820

minus (32) (120583119886))

1198882

] (120591 minus 1205910

)

+2

1198882

radic120583119886 sdot 119890 (sin 119864 minus sin 1198640

)

(1)

where 120583 = GM119864

is the geocentric gravitation constant 1198820

the gravity potential of the geoid 119886 the orbit main axis 119890

the eccentricity 119864 the real eccentric anomaly and 119888 the speedof light respectively Then

120591 minus 119905 = [1198820

minus ((32) (GM119864

119886))

1198882

] (119905 minus 1199050

)

minus2

1198882

radic119886GM119864

sdot 119890 sin119864

(2)

If the satellite clock 119879(119905) is modeled as

119879 (119905) = 120591 (119905) + 1198860

+ 1198861

(119905 minus 1199050

) + 1198862

(119905 minus 1199050

)2

+ 120585 (119905) (3)

in which 1198860

1198861

and 1198862

are clock offset parameters and 120585(119905) isthe clock phase noise the offset of satellite clock reffered toBDT can be written as

119909 (119905) equiv 119879 (119905) minus BDT (119905)

= 1198860

+ 1198861

(119905 minus 1199050

) + 1198862

(119905 minus 1199050

)2

+ Δ119905119901

grav + 120585 (119905)

(4)

HereΔ119905119901

grav is the periodic term of relativistic effect as follows

Δ119905119901

grav = minus2

1198882

radic120583119886 119890 sin 119864 = minus2119878

sdot 119878

1198882

(5)

The relativistic effect must be taken into account for the eval-uation of clock performance If not the stability of frequencywill be influenced The quasi-half-day periodical terms inAllan deviations of GPS clocks and Galileo clocks [2 5] weguess may be caused by this term

3 Two-Way Satellite TimeTransfer and Error Analysis

In Beidou system TWTT between satellites and uplink sta-tions is used for the satellitetime synchronization The basicprinciple of TWTT is as follows The satellites and stationsgenerate and transmit pseudo-range signals controlled bytheir local clocks then the uplink pseudo-range 120588

119906

anddownlink pseudo-range 120588

119889

are measured by the satellitesand the stations respectively The uplink pseudo-range anddownlink pseudo-range can be written as

120588119906

(119879119903

119878

) =1003816100381610038161003816119878 (119905119903

119878

) minus 119877

(119905119890

119877

)1003816100381610038161003816 sdot

1

119888minus Δ119879119877

(119905119890

119877

) + Δ119879119878

(119905119903

119878

)

+ 120591119890

119877

+ 120591119903

119878

+ 120591tro + 120591ion (119891119906) + 120591grav

120588119889

(119879119903

119877

) =1003816100381610038161003816119877 (119905119903

119877

) minus 119878

(119905119890

119878

)1003816100381610038161003816 sdot

1

119888+ Δ119879119877

(119905119903

119877

) minus Δ119879119878

(119905119890

119878

)

+ 120591119903

119877

+ 120591119890

119878

+ 120591tro + 120591ion (119891119889) + 120591grav

(6)

where 119905119903

119878

and 119905119890

119878

are time of reception and emission of thesatellite signal 119905119903

119877

and 119905119890

119877

are time of reception and emission ofthe station signalΔ119879

119878

andΔ119879119877

are satellite and stationrsquos clockoffset 120591119903

119877

and 120591119890

119877

are time delay of reception and emission ofthe station equipment 120591119890

119878

and 120591119903

119878

are time delay of receptionand emission of the satellite equipment 120591tro and 120591ion are



and119891119889



Δ119879119878

(119905119894

) = Δ119879119877

(119905119894

) +1

2

times [120588119906

(119879119894

119878

) minus 120588119889

(119879119894

119877

)]

minus1

119888( 119878

minus 119877

) sdot 119899119877119878

(Δ119879119878

minus Δ119879119877

minus 120591119877119878

)

+ Δ120591119877

minus Δ120591119878


(7)

where

Δ120591119877

equiv 120591119903

119877

minus 120591119890

119877

Δ120591119878

equiv 120591119903

119878

minus 120591119890

119878


119899119877119878

equiv(119878

minus 119877

)

1003816100381610038161003816119878 minus 119877

1003816100381610038161003816

(8)






0

100

200

0

2000

0 10 20 30 40 50 60minus200

minus100

Seco

nd-o

rder

pol

ynom

ial r

esid

uals

(ns)

Time (days)


minus4000

minus2000 Line

ar re

sidua

ls (n

s)




2205 2210 2215 2220 2225 2230 2235 2240

0

500

1000

Time (days)

0

10

20

minus1500

minus1000

minus500

minus30

minus20

minus10

1058t2 + 012814t + 4745457275


Seco

nd-o

rder

pol

ynom

ial r

esid

uals

(ns)


Line

ar re

sidua

ls (n

s)


0 10 20 30 40 50 60 70 80

0

100

200

300

400

Time (days)

0

1000

2000

minus200

minus100

Seco

nd-o

rder

pol

ynom

ial r

esid

uals

(ns)


Line

ar re

sidua

ls (n



minus1000

minus2000

minus3000

minus4000







100 101 102 103 104 105 106

Averaging time (s)

Alla

n fre

quen

cy st

abili

ty120590y(120591)

10minus15

10minus14

10minus13

10minus12

10minus11

10minus10










5 Conclusion


1 2 3 4 5 6 7 8 9

Allan stability


Alla

n fre

quen

cy st

abili

ty120590y(86400)

10minus13





Acknowledgments


References















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











and119891119889



Δ119879119878

(119905119894

) = Δ119879119877

(119905119894

) +1

2

times [120588119906

(119879119894

119878

) minus 120588119889

(119879119894

119877

)]

minus1

119888( 119878

minus 119877

) sdot 119899119877119878

(Δ119879119878

minus Δ119879119877

minus 120591119877119878

)

+ Δ120591119877

minus Δ120591119878


(7)

where

Δ120591119877

equiv 120591119903

119877

minus 120591119890

119877

Δ120591119878

equiv 120591119903

119878

minus 120591119890

119878


119899119877119878

equiv(119878

minus 119877

)

1003816100381610038161003816119878 minus 119877

1003816100381610038161003816

(8)






0

100

200

0

2000

0 10 20 30 40 50 60minus200

minus100

Seco

nd-o

rder

pol

ynom

ial r

esid

uals

(ns)

Time (days)


minus4000

minus2000 Line

ar re

sidua

ls (n

s)




2205 2210 2215 2220 2225 2230 2235 2240

0

500

1000

Time (days)

0

10

20

minus1500

minus1000

minus500

minus30

minus20

minus10

1058t2 + 012814t + 4745457275


Seco

nd-o

rder

pol

ynom

ial r

esid

uals

(ns)


Line

ar re

sidua

ls (n

s)


0 10 20 30 40 50 60 70 80

0

100

200

300

400

Time (days)

0

1000

2000

minus200

minus100

Seco

nd-o

rder

pol

ynom

ial r

esid

uals

(ns)


Line

ar re

sidua

ls (n



minus1000

minus2000

minus3000

minus4000







100 101 102 103 104 105 106

Averaging time (s)

Alla

n fre

quen

cy st

abili

ty120590y(120591)

10minus15

10minus14

10minus13

10minus12

10minus11

10minus10










5 Conclusion


1 2 3 4 5 6 7 8 9

Allan stability


Alla

n fre

quen

cy st

abili

ty120590y(86400)

10minus13





Acknowledgments


References















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













100 101 102 103 104 105 106

Averaging time (s)

Alla

n fre

quen

cy st

abili

ty120590y(120591)

10minus15

10minus14

10minus13

10minus12

10minus11

10minus10










5 Conclusion


1 2 3 4 5 6 7 8 9

Allan stability


Alla

n fre

quen

cy st

abili

ty120590y(86400)

10minus13





Acknowledgments


References















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Time Synchronization and Performance of...

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