Divisão Serviço da Hora Laboratório Primário de Tempo e Frequência 2010 SIM TFWG Workshop and Planning Meeting March 9 – 12 Lima, Peru. Time Scales Virtual

Divisão Serviço da HoraDivisão Serviço da HoraLaboratório Primário de Tempo e FrequênciaLaboratório Primário de Tempo e Frequência

2010 SIM TFWG Workshop and Planning MeetingMarch 9 – 12 Lima, Peru.

Time Scales

Virtual Clocks and Algorithms

Ricardo José de CarvalhoNational Observatory Time Service Division

March 11, 2010

Ricardo José de CarvalhoNational Observatory Time Service Division

March 11, 2010


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 named TA(k) 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 TA(k) from ensemble of atomic clocks?

How can we evaluated the stability of TA(k)?

How can we generated UTC(k) from TA(k)?


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.


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.


Time Scale System


Time Scale System

In practice a time scale system can be divided into four parts:

The first part is a clock ensemble that will be used as a reference for the generation of a local time scale TA(k);

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 time scale TA(k);

The fourth is the equipment to generate UTC(k) from TA(k) and how to do that.


The Clock Ensemble

An atomic clock delivers a series of physical electric pulses separeted from one another by a duration of 1 second, often designated as “series of 1pps”. Each pulse is an event that can be associated a number. This number is the reading of the clock for that particular event: for example, it may read as 2010 March 11 13h45min00s, summarizing is the time of the clock.

The time of an individual clock can not be measured, what we can do is to measure the time differences between clocks.

The 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.

Clocks are independent, but some correlations between clock frequencies may be detected. These correspond mainly to responses to changes in the environmental conditions in that the clocks are.


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.

The measurement results are affected by intrinsic noise measurement that is considered negligible in relation to the clock noise.

From an ideal point of view, the measurement system have frequent and precise mesurements, changing as little as possible the clock intercomparison data.


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.


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


Time Scale Formulation To Simple Mean of Two Clocks the time scale is defined by:


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

)3()()(2

1)( 211 ththtTA

)4(0)()(2

11

ii thtTA

1h

2/1h


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


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


Time Scale Formulation To Weighted Mean of n Clocks the time scale is defined by:


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

)7()(1

)(1

3

n

iii thw

ntTA

)8(0)()(1

3

n

iii thtTAw


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.


Time Scale Formulation A general model of clock i behavior may be write as:

is the time deviation of the clock i at time t + Ƭ;

is the synchronization error of the clock i at time t;

is the frequency offset (syntonization error) of the clock i at time t,which produces a linear ramp in the time deviations;

is the frequency drift term, which produces a quadratic time deviation;

is the all random fluctuations. It is this term which is typically characterized by one or more of the five power law processes.

)()()()( 221 ttDtytxtx iieieieie

)( txie

)(txie

)(tyie

ieD

)(ti


Time Scale FormulationA 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:

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

)()()(ˆ tytxtx ieieie

n

iijieije txtxwtx

1

)]()(ˆ)[()(

)()(

)(ˆtxtx

tY iij


Time Scale FormulationAn exponentially filtered estimate of the current average frequency of

clock i that will be used in the next prediction interval is given by:

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:

)()(ˆ1

1)( tYmtYm

tY iiii

i

2/1

2

2min

3

4

3

11

2

1

i

im

)(

1

)(

1

12

12

in

i i

iw


Time Scale Formulation The prediction error estimate is given by:

because ensemble time is a weighted average of each clock times, the prediction error estimate is biased, because each clock is a member of the ensemble, so it is necessary to correct this biasing by he condition imposed on the time scale is:

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:

iiii Ktxtx 22 )(ˆ)(ˆ

)1(

1

ii wK

)()(1

1)( 222 tNtN

t iii

)()( 222 yi t


A Case Study with Real Data

The art of building a time scale from an ensemble of clocks requires various choices. We will show some of them by the following example. We used in this example time differences of six commercial cesium clocks for the period of August 08, 2008 to September 30, 2009 with 1 hour of interval between measures.

The time difference measurement data are coming from an automated acquisition system that measures the time differences between the clocks with comercial eletronic counter.


A Case Study with Real Data


Figure 1. Time Difference between clocks.


Instability of the Clocks

To evaluate the individual instability of the clocks we use the N-cornered hat technique by:

m

jiji Bs

m 1

22

2

1

m

jiijsm

B 2

1

1


Instability of the Clocks

Figure 2. Comparison of Allan deviations of the clocks.


Initializing the Algorithm To initialize the algorithm the UTC(BIPM) was used as reference to calculate the following parameters

that is

T130 – UTC(BIPM) = -14.738960µs at 54704 00h00minT129 – UTC(BIPM) = -13.138160µs “T103 – UTC(BIPM) = -10.633442µs “T123 – UTC(BIPM) = -60.150347µs “T125 – UTC(BIPM) = +5.664442µs “T102 – UTC(BIPM) = -31.095ns “

)()()( tUTCtxtx BIPMiie


Initializing the Algorithm to

we calculated using Excel a linear regression of the time differences [clock - UTC(BIPM)], to all clocks we have

yT130 – UTC(BIPM) = -9.532ns/day (R2= 0.990)




yT125 – UTC(BIPM) = +4.283ns/day(R2= 0.999)

yT102 – UTC(BIPM) = +5.945ns/day (R2= 0.976)

period 54704 00h00min to 54734 00h00min

)()(

)(

txtx

ty ieieie


Initializing the Algorithm

Figure 3. Linear Fit of T130 – UTC(BIPM)

















Initializing the Algorithm To initialize

the prediction error estimate in the algorithm we used the individual instability of the clocks estimated by the N-cornered hat technique, reminding that Ƭ = 1 hour we have:

(εT130)

2= (3600)2x(1.978x10)-26

(εT129)2= (3600)2x(1.222x10)-26

(εT103)2= (3600)2x(1.243x10)-26

(εT123)2= (3600)2x(3.340x10)-26

(εT130)2= (3600)2x(3.781x10)-26

(εT130)2= (3600)2x(8.260x10)-26

)()( 222 yi t


Initializing the Algorithm To initialize

Ƭmini

is the value of Ƭ at minimum σy(Ƭ)

on Allan variance curve for

clock i. It was used the specification of the 5071A supplied by the

manual the value is Ƭmini

= 5 days so:

mi= 68.78263370667524

2/1

2

2min

3

4

3

11

2

1

i

im


Results

Figure 9. The time deviation of the clocks and scale (normalized).


Results

Figure 10. The time deviation of the clocks and scale (normalized).


Results

Figure 11. The time deviation of the scale (normalized).


Results

Ƭ(days) T130 T129 T103 T123 T125 T102 Scale

51.61E-14

1.31E-14

1.59E-14

1.71E-14

2.27E-14

1.37E-14

1.21E-14

101.20E-14

9.73E-15

9.50E-15

1.16E-14

2.51E-14

9.45E-15

7.87E-15

151.09E-14

7.44E-15

6.93E-15

9.27E-15

2.90E-14

8.37E-15

6.37E-15

201.18E-14

5.55E-15

5.76E-15

7.61E-15

3.35E-14

8.14E-15

5.17E-15

Table 1. Allan Deviation with UTC(BIPM) as reference


Results

Figure 12. The Change of Clock’s Weights with Time.


Results



Results



GENERATING UTC(k) FROM TA(k)

To With the equipment generate UTC(k) from TA(k) we have:

First, after each publication of the circular T we calculated the rate of TA(k) with reference UTC(BIPM), that is, we obtain the following value:

{TA(k) – UTC(BIPM)}ns/day (One sees a month)

Second, we chose one of the clocks as reference and we calculated de rate of the clock with TA(k) as reference, that is,

{Clock – TA(k)}ns/day (every hour)

finally we will have for every hour

{TA(k) – UTC(BIPM)}ns/day (One sees a month) +

{Clock – TA(k)}ns/day (every hour)____________________________________________________

{Clock – UTC(BIPM)}ns/day (every hour)


GENERATING UTC(k) FROM TA(k)

With an equipment denominated micro phase stepper that have as input the frequency of the chosen clock, we applied the correction:

{Clock – UTC(BIPM)}ns/day (every hour)

== == UTC(k)

clock clock frequencyfrequency

Micro Phase Micro Phase StepperStepper

-0.27ns/day-0.27ns/day


THE ENDTHE END

Ricardo José de Carvalho – [email protected]ório Nacional / Divisão Serviço da Hora

ON/DSHO – BRASILPhone: +55 21 2580-7781

Fax: +55 21 2580-6071http://www.horalegalbrasil.mct.on.br/

THANKS BY YOUR ATTENTION

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Divisão Serviço da Hora Laboratório Primário de Tempo e Frequência 2010 SIM TFWG Workshop and Planning Meeting March 9 – 12 Lima, Peru. Time Scales Virtual