1

Time Scales

Virtual Clocks and Algorithms

Ricardo José de CarvalhoNational Observatory Time Service Division

February 06, 2008

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Introduction

• Let us assume that we have an ensemble of atomic clocks that are running at the same time and we want to build an independent atomic time scale from an ensemble of clocks. That scale will be used as an internal reference for the generation of a local UTC(k) time scale.

• What are the possibilities and the difficulties we are facing?

• What are the ingredients necessary to build UTC(k) from ensemble of atomic clocks?

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Time Scale from one Clock

Simple solution

• We can select one of the clocks and we can decide that our atomic time scale is identified with the output of this clock.

• This solution is valid if the clock exhibits qualities superior to the others.

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Time Scale System

Better Solution

• We can build a time scale system that can enable a time laboratory to keep time with stability, accuracy, and reliability beyond the performance level of the best physical clock in the laboratory by combining all the clocks resulting in a Virtual Clock.

5

Time Scale System

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Time Scale System

In practice a time scale system can be

divided into three parts: The first part is a clock ensemble that will

be used as a reference for the generation of a local UTC time scale;

The second, an automated data acquisition system that measures the time differences between the clocks;

The third is a time scale algorithm that computes the paper time scale.  

 

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The Clock Ensemble

• Atomic clock are very stable and accurate but they may be sensitive to the environmental conditions, so they are to be maintained in the most stable environment to avoid injecting instabilities.

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Automated Data Acquisition

• Because the time of an individual clock can not be measured, one must measure time differences between clocks. Inside the laboratory the time differences are obtained with a high degree of accuracy using electronic counter.

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Time Scale Algorithm

• By samplig an ensemble of clocks, an ideal time scale algorithm would generate time and frequency with more reliability, stability and frequency accuracy than one of the individual clocks in the ensemble.

• In general a time scale algorithm takes the time difference measurements between clocks and combines them mathematically to produce an average time scale.

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Time Scale Formulation

• The Simplest Time Scale of all is just the reading of a single clock, that is:

• The condition imposed on the time scale is:

)1()()( 10 thtTA

)2(0)()( 10 thtTA

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Time Scale Formulation

• To Simple Mean of Two Clocks the time scale is defined by:

• The condition imposed on the time scale is:

)3()()(2

1)( 211 ththtTA

)4(0)()(2

11

ii thtTA

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Time Scale Formulation

• The time scale defined by equation (3) will be affected by any anomalus behavior that one of the clocks present. For example, suppose that the clock 1 has jumped by an amount

• Then the time scale is affected by an amount

1h

2/1h

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Time Scale Formulation

• To Simple Mean of n Clocks, n > 2 the time scale is defined by:

•

• The condition imposed on the time scale is:

• Anomalous behavior can thus be detected with confidence incresing as n increases.

)5()(1

)(1

2

n

ii thn

tTA

)6(0)()(1

2

n

ii thtTA

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Time Scale Formulation

• To Weighted Mean of n Clocks the time scale is defined by:

•

• The condition imposed on the time scale is:

• Anomalous behavior can thus be detected with confidence increasing as n increases.

)7()(1

)(1

3

n

iii thw

ntTA

)8(0)()(1

3

n

iii thtTAw

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Time Scale Formulation

• The time scales considered previously possess a disadvantage that, if one the clocks stops or a new clock is introduced, a step and a rate change will generally occur in the time scale.

• To solve this problem the definition of the time scale should be modified, in such a way that, the considered average is on the time offset of clock with the time scale as reference.

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Time Scale System

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Brazilian Atomic Time Scale

• In the choice of the algorithm to calculated TA(ONRJ) we have consider that the optimum algorithm with a small group of clocks is not obvious and very difficult to find.

• The algorithm that generates TA(ONRJ) follows the same steps of the main ensemble algorithm used successfully in the NIST and is outlined here shortly.

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Brazilian Atomic Time Scale

• A first prediction of the time offset for each clock against the ensemble is given by

• The best estimate of the time offset of each clock at time given the measurements is

•

)()()(ˆ tytxtx ieieie

n

iijieije txtxwtx

1

)]()(ˆ)[()(

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• Once the are known the average frequency of each clock over the last interval can be estimate by

• An exponentially filtered estimate of the current average frequency of clock i that will be used in the next prediction interval is given by

)()(

)(ˆtxtx

tY iij

)()(ˆ1

1)( tYmtYm

tY iiii

i

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• where mi is an exponential time constant determined from the relative levels of white noise and random walk FM, that is

• The clock weights wi are calculated from

2/1

2

2min

3

4

3

11

2

1

i

im

)(

1

)(

1

12

12

in

i i

iw

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• The prediction error estimate is given by

because ensemble time is a weighted average of each clock times, the prediction error estimate (7) is biased, because each clock is a member of the ensemble, so it is necessary to correct this biasing by

•  

iiii Ktxtx 22 )(ˆ)(ˆ

)1(

1

ii wK

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• Since the noise characteristics of a cesium clock may not be stationary, the current prediction error of each clock is exponentially filter where the past prediction error are deweighted in the process, that is

• the time constant for the filter is typically chosen to be 20 days and the initial value of is estimated as:

)()( 222 yi t

)()(1

1)( 222 tNtN

t iii

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Brazilian National Time Scale System.

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CONCLUSIONS

• We showed that it is possible to generate an atomic time scale with a small group of clocks.

• The next step will consist in making the system almost fully automated, by implementing the automatic detection of clock anomalies and to allow for the maintenance, to add or remove clock from the system in easily fashion.

• The new UTC(ONRJ) is now inside of the limits recommended by the CCIR and by the CCDS. ±100ns of UTC.