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Stability analysis of a time scale algorithm
CHINESE ASTRONOMY AND ASTROPHYSICS PERGAMON
Chinese
Astronomy
and Astrophysics
26 (2002)
245-253
Stability Analysis of a Time Scale Algorithmt * KE Xi-zheng1y2 ’Shaanxi
LI Xiao-hui’
Observatory,
2 The Second
Artillery
Chinese
LIU Zhi-ying’ Academy
Engineering
of Sciences,
College
DENG Fang-lin2 L&tong
710600
of PLA, Xi ‘an 710025
Abstract The principles of ALGOS atomic time algorithm of the Time and Frequency Division of International Bureau of Metrology are introduced, and a new time scale algorithm based on wavelet decomposition is proposed. With the new algorithm it is possible to extract the different stabilities of the atomic clock in various frequency ranges. As shown by experimental results, the atomic time given by this wavelet-based algorithm is more stable than those given by the existing Key
algorithms.
words:
atomic
time-wavelet
decomposition-frequency
stability
1. INTRODUCTION In the history of time measurements, people have always tried to find more uniform and more accurate standards of time. There are many time scales which have played important functions in mankind’s production, mode of living and scientific research. One of these by the earth’s rotation. The is Universal Time (UT), which is a time scale determined discovery of non-uniformity in the earth’s rotation prompted the introduction of Ephemeris Time. While it is more uniform than Universal Time, the Ephemeris Time cannot be easily obtained from measurements, and because it requires rather long time of observation it cannot be conveniently used on various occasions. Moreover, the attainable accuracy of the measurements is rather limited and cannot satisfy the increasing demand for the greatest accuracy in time and frequency measurements. The invention of atomic clock has altered the traditional situation of mankind’s dependence on the time determined by astronomical observations. t Supported
by Foundation of Chinese Academy of Sciences for Excellent Received 2000-08-24; revised version 2001-04-02 * A translation of Acta Astron. Sin. Vol. 42, No. 4, pp. 420-427, 2001
0275-1062/02/$-see PII:SO275-
front
1062(02)00063-2
matter
@ 2002 Elsevier
Science
B. V. All rights
Young
reserved.
Scholars
246
KE Xi-theng
et al. /Chinese
Astronomy
and Astrophysics
26 (2002)
245-253
The seconu in Atomic Time is defined by the the cesium atom. Atomic Time started on 1955 January 1, Oh Om UT, and it has ever since been counted continuously. It may be acquired by computation with many atomic clocks. The International Atomic Time (TAI) is evaluated via computation using the algorithm ALGOS on more than 200 atomic clocks located all over the world. The atomic clocks of the various laboratories are connected to TAI through the Global Positioning System, and also by two-way comparison of satellites and other means. The frequency accuracy of TAI in 1988 is f5 x 10-14, and the frequency stability is ‘T~(~,T) = 2 x lo- l4 , for r E (2 months, several years). For convenience of application in geodesy and navigation, some form of international coordination is necessary. The Coordinated Universal Time (UTC) adopts the rate of TAI and uses leap seconds so that its time can be made as close to UT as possible. The UTC is a product of compromise between TAI and UT. It is expected that no more than one leap second is needed for the UTC in each year, so that the difference between UTC and UT is less than 0.9second. The essence of time scale algorithm is the coordination of interrelations of the atomic clocks, which constitute the TAI, as well as the weighting of the various laboratories according to the historical characteristics of their clocks and the adjustment of the relations among the clocks via their weights. For the establishment of a uniform and stationary time scale, we need an advanced algorithm and a reasonable method of weighting. Even now, the International Telecommunication Union still regards time scale algorithm as a research topic that countries of the world are encouraged to engage in. In order to keep the accuracy and continuity of the time scale and to make the time unit to be as close as possible to the second of the international unit system, all the time keeping laboratories are equipped with own cesium clocks. Each atomic clock is used to keep a time scale, but it is possible that no physical apparatus is ever free from malfunctioning. Therefore, a laboratory engaged in the work of time-keeping usually maintains more than one caesium clock. Thus it is required that the times of the various atomic clocks are synthesized to a standard time, and it is for this purpose that an algorithm of atomic time is needed. For an international standard of time like TAI, a good accuracy and a high long-term stability are required. In order to attain this goal, one has to compute the TAI with the data coming from a large number of atomic clocks all over the world and to get these clocks connected with the ALGOS algorithm 11121.In some countries and regions there are similar algorithms in use. All such algorithms belong to the classical method of weighting.
2. ALGOS
(BIPM)
The first step of establishing the TAI is is obtained with the weighted means of ALGOS. TAI results after the frequency of the EAL frequency with the standard 2.1
The Definition of TAI The EAL is updated at the following t=to+nT/6
ALGORITHM
to set up the EAL (echelle atomique libre), which several hundred atomic clocks in the world, using calibration of EAL is made through a comparison frequency.
times: n = O,l, 2...6.
KE Xi-zheng
et al. /Chinese
Astronomy
and Astrophysics
26 (2002)
245-253
247
Here T = 30d and to is the time of the last reduction in the preceding month: thus the EAL is updated every five days and calculated every 30 days. At time t, the definition of EAL is
i=l
where hi(t) represents the reading of Clock i at t, and its value cannot be directly obtained through experiment; N is the number of clocks; pi is the weight of Clock i; h: is the time correction of Clock i at moment t, and the correction is for ensuring the continuity of time scale in case of variation in the weights or in the number of clocks. Let us denote the clock difference xi(t) between Clock i and the EAL by
xi(t) = EAL(t) At moment
t the difference
xi,j(t)
between
- hi(t).
(2)
Clock i and Clock j is
X:i,j(t) = hj(t) - hi(t).
(3)
The value of zi,j (t) may be directly obtained by experiment. In one and the same laboratory, it can be gained by direct comparison, and between different laboratories by some technique of time transmission. xi(t) is the difference between a clock and the average time scale, and it cannot be directly obtained by experiment. It merely denotes the result of computing EAL. From this the users may get the time scale. According to the above definition, we have the following system of equations:
i~lPiXi(t) =i$Pihi(t) Y
{ xi,j(t)
= xi(t)
- xj(t)
(4)
.
For N atomic clocks there are N such equations, and from them we may get xi(t). For the time span of computation, the difference xi(to + nT/6) (n = 0,1,2...6) of each Clock i relative to the EAL can be obtained, and this is its computed time scale. 2.2 Selection of Time The time correction
Correction
term
Term
h:(t) is the sum of two terms h:(t) = ai
where ai
is the time correction ai
+ Bip(t)(t
for Clock i relative
131:
- to). to EAL at moment
= EAL(to) - hi(to) = xi(to) .
Bip(tc) is the predicted frequency of Nowadays the predicted frequency keeps a is computed from the frequency in the last variance of {xi(to - T + nT/6), n = 0,1...6}
(5)
to. We have
(6)
Clock i relative to EAL in the interval [to, t]. fixed value within one month of [to,t + T]. It period of reduction. In other words, the least decreases, and this is one-step linear prediction.
KE Xi-zheng
248
et al. /Chinese
Astronomy
and Astrophysics
26 (2002)
245-253
This hypothesis has the following prerequisite: the atomic clock behaves similarly in the last month and in this month. If the time of measurement is suitably chosen, this hypothesis will be valid. In practice, when T = 30d, the dominant noise of the clock is expressed as a random walk noise, and the best estimate of B+,(ts) is the value of the last reduction. 2.3 Selection
of Statistical
Weights
The selection of weight is to identify the value that leads to optimum long-term stability of the time scale. The weight is determined by the behavior of the clock in the last month. The concrete procedure is as follows: (1) The solution of the system of equations (4) in the time span [to, t + T]is determined by the weight in the last reduction For the next reduction, the weight determined in the present reduction is used. (2) The estimated value of frequency Bip(to + T) is calculated with the xi(t) solved from the system of equations (4). (3) The variance $(S, t) is computed with Bi,(to + T) as well as the Bip'sin the preceding five reductions. (4) For the weight of Clock i, the temporary value computed with the following formula is used: 1000 p: = $(6,t)
.
(5) Except the following two cases, the new weight pi(t) is equal to p:(t): a. The limit of maximum weight is 1000; b. If the clock exhibits abnormal behavior, its weight is set to zero. From the once-every-5-days computation, ALGOS (BIPM) produces a delayed time scale TAI. The weights of the ALGOS algorithm belong to the classical method of weighting. Other places and laboratories use similar algorithms, such as the MOWA algorithm of JATC 131.
3. WAVELET
DECOMPOSITION
ALGORITHM
OF TIME SCALE
This is an atomic time algorithm developed by us. It is established on the basis of wavelet decomposition, so\we call it Wavelet Decomposition Algorithm (WDA). With this algorithm we set up the Wavelet Decomposition Atomic Time (WDAT) . The classical weight algorithm has some inherent limitations. It assigns to each clock one single weight and doe not synthetically consider the frequency stabilities of all the clocks. The average time scale can only minimize one particular noise, and cannot minimize all the noises. In the ALGOS algorithm, an atomic clock may be weighted according to its longterm stability, so ‘it is possible that due to bad long-term stability an atomic clock with good short-term stability can get very small weight. Then this leads to a waste of resource. In view of this many improved algorithms have been proposed, but they all possess certain limitations. Now wavelet transformation can extract the components of the atomic clock in different frequency ranges, and the frequency stability is different in different frequency ranges, so we have proposed the wavelet decomposition algorithm: the signal from the
KE Xi-zheng
et al. /Chinese
Astronomy
and Astrophysics
26 (2002)
245-253
249
clock is decomposed in the wavelet domain, the components in different frequency ranges are extracted, weighted mean is applied in the wavelet domain, then, after inversion, the synthetic time scale is obtained. Because in this method the different degrees of stability of the atomic clocks in different frequency ranges have been taken into account, all the noises of the synthetic time scale are minimized. It is a weighting method with multiple resolution. 3.1 Setup of WDAT Let the total number of atomic clocks participating in the reduction be C and let the A(c) and frequency reading of Clock c at a certain time be be T(c), the phase correction correction B(c) of the clock having been applied. In the time interval of reduction I the corrected reading TM(c, t) may be written as: TM(c,t)
The weight of Clock c is pt(c) (the ALGOS then the WDAT may be written as:
weighting
method
EPt(c)TM(c>t) cgl PWvk WDAT(t)
c = 1,2,3 ,... C.
= T(c, t) + A(c) + B(c)[t - to],
is used in the time domain),
t) + 44 + B(c)(t- h)l
=
= ‘=’ 5
(7)
(8)
pt(c)
c=l The
difference
between
Clock c and WDAT
is defined
DC(t) = WDAT(t) In practice, e, i.e.,
what
we measure
as
- T(c, t) .
is the difference
between
(9) Clock c and some other
Dc,e(t)= T(e,t) - T(c, t) .
Clock
(10)
The atomic clock with the best performance is taken to be the Master Clock. Let it be Clock C. D~c(t) represents the difference of Master Clock relative to WDAT, and it cannot be obtained by experiment. It expresses the computed time scale and can provide users with the correct value of WDAT. According to the above definition, we have the following system of equations:
5
c=l
{
Pt(c)R(t)
= c&Pt(c)[A(c)
=
Ve,
t)
-
T(c,
Dc,Mc(t).
t> .
In order to get DMc(t),
5 pt(c)[A(c) + B(c)(t =
- to)], (11)
Dc,e(t)
By experiment we can obtain are solved and we have
Dj&t)
+ B(c)(t
‘=’
to)] +
the above equations
5 Pt(c)Dc,Mc(t)
c=l
(12)
KE Xi-zheng
250
et al. /Chinese
Astronomy
and Astrophysics
26 (2002)
245-253
The above expression gives the difference between Master Clock and WDAT, so from Master Clock we may obtain WDAT. Eq.(12) consists of two terms. The first is the weighted mean of the time correction of C atomic clocks, i.e.,
m(t)
=
c=l
(13) 5 pt(c)
.
c=l
Here A(c) is a constant adopted to keep the continuity of time scale. In general it is taken B(c) is the predicted frequency to be the frequency correction at the end of reduction. correction of the current reduction. If the number of clocks does not change and the weight Otherwise, in order to keep of every clock remains the same, then A and B are unchanged. the continuity of the time scale, the values of A and B have to be altered. In the general case, for one and the same atomic clock these values do not, and should not change too much. The method of weighting in the ALGOS algorithm is effective for these two terms, so it is adopted by us. The second term of Eq.(12)
is the mean value of clock difference: C-l
C-l
z-D(t) =
TD(t)
c Pt(c) c Pt(c)DcW(t) c=l
c=l
c-l
c-l
c_lPt(c)Dc,MC(t)c =
c-l
5 PN4 c=l
=
is the mean difference
cgPto . c=lc pt(c>
CL1
pt(c> l
RF(t)
.
(14)
5 pt(c)
c=l
between
the Master Clock and the C-l other clocks. It is random process completely undetermined. For each D c,~~(t), this expresses a non-steady and it is quite different for different times of reduction. When the weight is taken according to the historical characteristics of the clock, the result is extremely bad. slightly deformed it and picked up a term RF(t) for individual analysis.
For this, we have
3.2 Weighted Mean With Multiple Resolution We proposed the method of weighted mean with multiple resolution 141. The wavelet decomposition is carried out for Dc,MC(t). The various frequency components are picked up. The wavelet variance is used to characterize the frequency stability. The weighting is made for different scales, and the signals are reconstructed. Because the wavelet variance characterizes the stability of the clock in various frequency ranges an because by means of wavelet decomposition the different frequency components of the signals of the clock are separated, so the method can bring out the best characteristics of the individual clocks. 3.2.1 Wavelet decomposition The clock difference signal between Clock c and Master Clock Dc,MC(t) may be written as
Dc,dt)
=
2
k=-co
&,,k%,,k
(t) +
5 2
j=-co
@;,&j,k(t).
(15)
k=-m
where Pjo,k = (0, pjo,k) is the coefficient of the rough term of scale function vj,, and ‘Yj,k = (D, 4j,k) is the coefficient of the fine term of wavelet function 4j. Therefore, from
KE Xi-zheng
et al. /Chinese
Astronomy
and Astrophysics
26 (2002)
245-253
251
the viewpoint of wavelet scale the signal is divided into two levels. The level above jr, is extracted for the fundamental characteristic of Dc,~c(t), and that below je is the fine detail approximation of Dc,~c(t). As the scale increases, the separated frequency becomes lower and lower. 3.2.2 Weighted multiple resolution The weight in a certain local frequency range may be written as :
where aj expresses the weighted multiple resolution below the wavelet scale j. This method of weighting takes into account that the stabilities of the clock in various frequency ranges are different, and it is clearly an improvement on the classical method of weighting. 3.2.3 Reconstruction of signals According to the relation of wavelet transformation and its reconstruction, the weighted signal can be written as:
where 0: expresses
the weighted
multiple
resolution
4. EXPERIMENTAL
of Dc,~c(t)
in wavelet
scale j.
STUDY
For the measured data of a certain laboratory, we made a comparison between WDAT and the Synthetic Atomic Time (TA(K)) of the laboratory. Because the first term of Eq.(12), i.e., TC(t), is the same, so only the second term, i.e., TD(t), is written out. Fig.1 shows the difference between the four cesium clocks of the laboratory and the TAI in a lOOd-period from 1984 November to 1985 January. Fig.2 illustrates the comparison between the atomic time TA(K) of the laboratory and the TD(t) term of WDAT. For the TA(K) the weight is taken according to the historical properties of the clocks: for CLKl the weight is rather large and it can only suppress noise of one kind. Therefore, the fluctuation in the atomic time provided by it is far larger than that of WDAT. A comparison of their Allan variances is shown in Table 1. We see that for both short-term and long-term stabilities the WDAT is better. Table
1
Comparison between the Allan variances of TA(K) and WDAT
Allan variance TA”Kc WDAT
1 day duration 0.1697x lo-‘” 0.0441x 10-12
10 days duration 0.6481x10-‘” 0.1168x10-12
30 days duration 0.2160x IO-‘” O.O389x1O-‘2
252
KE Xi-zheng
1.5
-
r B ,_ .& P cp'o.5-
,’ ..’ ,
et al. /Chinese
---. ---
*,_’ :
,_f I
’ ‘L,.
and Astrophysics
26 (2002)
1
CLKl-TAI CLKZ-TAI
-
245-253
TA(K)-TAI
- - CLK3-TAI
CLK4-TAI
‘,.*
/” I
Astronomy
\ \
k. ' 8' '.\ 8.; '. ,,-.,.______,-.-____-..-___ '\ -'-.--______._------._____________,-.. -..__. 46020
46040
46060
46080
46100
46020
46040
MJD Fig. 1 Time difference between the 4 cesium clocks of a certain laboratory and the TAI in 1984 Nov - 1985 Jan
Table
2
Weights
Fig. 2
Comparison
a certain laboratory
of four atomic
clocks in various
46060 MJD
46080
4610
of the atomic time TA(K)
of
and WDAT
scales
Weights of clocks Scale 0.90 1.30 1.70 3.30 4.90 5.70 7.30
CLKl 190.668 80.343 32.746 3.423 0.761 0.409 0.145
CLK2 278.000 278.000 233.049 68.939 27.419 21.308 17.843
CLK3 184.581 97.513 59.701 18.835 9.094 7.105 5.043
CLK4 278.000 142.960 62.784 10.120 5.190 4.295 3.022
Table 2 compares the weights of the four atomic clocks in different scales. We see that the weights are very different for the different scales. The fluctuation of CLKl is the largest and in it weight is usually the smallest. But for scale 0.9, its weight is larger than that of CLKS. This implies that it has one frequency component whose fluctuation is comparatively small. Similarly, CLK3 and CLK4 have large weights. The fluctuation of CLK2 is the smallest and for all scales its weights are always the largest, but the amount by which it is larger varies from scale to scale, and this just corresponds to the fact that different frequency components of the clocks have different intensities and the weights are different. Here is precisely the superiority of WDAT. It takes into account the differences in stability of the clocks in various frequency ranges, and makes use of the best individual characteristics of the clocks. As shown in Table 2, the weight of CLK2 is the largest, so WDAT should be closer to CLK2. The comparison between CLK2 and WDAT is illustrated in Fig.3. As shown by this figure, although CLK2 is close to WDAT, the latter is far smoother, having been freed of many points of sudden changes.
5.
A new time algorithm. ALGOS, it influence of
CONCLUSIONS
scale algorithm is proposed in this work. It inherits the merits of the ALGOS Under the pre-condition that its long-term stability is not lower than that of leads to a great improvement of the short-term stability. It can suppress the sudden changes and make the time scale more steady and uniform.
KE Xi-zheng
et al. /Chinese
-‘.O*
Fig. 3
I
Astronomy
.
46020
and Astrophysics
46040
The time differences
46060 MJD
26 (2002)
I
46080
CLKZTAI
245-253
46100
and WDAT-TAI
References 1
Guinot B., Thomas
2
Patrizia
3
Pan
4
Ke Xi-zheng,
Tavella,
Xiao-pei,
C., BIPM
Claudine
Tu Lu-zheng, Wu Zhen-sen,
Annual
Thomas,
Report
Luo Ding-chang, Yang
of the Time
Metrologie,
Ting-gao,
Section,
1988, 1, Dl-D22
1991 28, 57 Science
Journal
in China,
1988, 8, 851
of Electronics,
1999, 27, 135
253
Chinese
Astronomy
and Astrophysics
26 (2002)
245-253
Stability Analysis of a Time Scale Algorithmt * KE Xi-zheng1y2 ’Shaanxi
LI Xiao-hui’
Observatory,
2 The Second
Artillery
Chinese
LIU Zhi-ying’ Academy
Engineering
of Sciences,
College
DENG Fang-lin2 L&tong
710600
of PLA, Xi ‘an 710025
Abstract The principles of ALGOS atomic time algorithm of the Time and Frequency Division of International Bureau of Metrology are introduced, and a new time scale algorithm based on wavelet decomposition is proposed. With the new algorithm it is possible to extract the different stabilities of the atomic clock in various frequency ranges. As shown by experimental results, the atomic time given by this wavelet-based algorithm is more stable than those given by the existing Key
algorithms.
words:
atomic
time-wavelet
decomposition-frequency
stability
1. INTRODUCTION In the history of time measurements, people have always tried to find more uniform and more accurate standards of time. There are many time scales which have played important functions in mankind’s production, mode of living and scientific research. One of these by the earth’s rotation. The is Universal Time (UT), which is a time scale determined discovery of non-uniformity in the earth’s rotation prompted the introduction of Ephemeris Time. While it is more uniform than Universal Time, the Ephemeris Time cannot be easily obtained from measurements, and because it requires rather long time of observation it cannot be conveniently used on various occasions. Moreover, the attainable accuracy of the measurements is rather limited and cannot satisfy the increasing demand for the greatest accuracy in time and frequency measurements. The invention of atomic clock has altered the traditional situation of mankind’s dependence on the time determined by astronomical observations. t Supported
by Foundation of Chinese Academy of Sciences for Excellent Received 2000-08-24; revised version 2001-04-02 * A translation of Acta Astron. Sin. Vol. 42, No. 4, pp. 420-427, 2001
0275-1062/02/$-see PII:SO275-
front
1062(02)00063-2
matter
@ 2002 Elsevier
Science
B. V. All rights
Young
reserved.
Scholars
246
KE Xi-theng
et al. /Chinese
Astronomy
and Astrophysics
26 (2002)
245-253
The seconu in Atomic Time is defined by the the cesium atom. Atomic Time started on 1955 January 1, Oh Om UT, and it has ever since been counted continuously. It may be acquired by computation with many atomic clocks. The International Atomic Time (TAI) is evaluated via computation using the algorithm ALGOS on more than 200 atomic clocks located all over the world. The atomic clocks of the various laboratories are connected to TAI through the Global Positioning System, and also by two-way comparison of satellites and other means. The frequency accuracy of TAI in 1988 is f5 x 10-14, and the frequency stability is ‘T~(~,T) = 2 x lo- l4 , for r E (2 months, several years). For convenience of application in geodesy and navigation, some form of international coordination is necessary. The Coordinated Universal Time (UTC) adopts the rate of TAI and uses leap seconds so that its time can be made as close to UT as possible. The UTC is a product of compromise between TAI and UT. It is expected that no more than one leap second is needed for the UTC in each year, so that the difference between UTC and UT is less than 0.9second. The essence of time scale algorithm is the coordination of interrelations of the atomic clocks, which constitute the TAI, as well as the weighting of the various laboratories according to the historical characteristics of their clocks and the adjustment of the relations among the clocks via their weights. For the establishment of a uniform and stationary time scale, we need an advanced algorithm and a reasonable method of weighting. Even now, the International Telecommunication Union still regards time scale algorithm as a research topic that countries of the world are encouraged to engage in. In order to keep the accuracy and continuity of the time scale and to make the time unit to be as close as possible to the second of the international unit system, all the time keeping laboratories are equipped with own cesium clocks. Each atomic clock is used to keep a time scale, but it is possible that no physical apparatus is ever free from malfunctioning. Therefore, a laboratory engaged in the work of time-keeping usually maintains more than one caesium clock. Thus it is required that the times of the various atomic clocks are synthesized to a standard time, and it is for this purpose that an algorithm of atomic time is needed. For an international standard of time like TAI, a good accuracy and a high long-term stability are required. In order to attain this goal, one has to compute the TAI with the data coming from a large number of atomic clocks all over the world and to get these clocks connected with the ALGOS algorithm 11121.In some countries and regions there are similar algorithms in use. All such algorithms belong to the classical method of weighting.
2. ALGOS
(BIPM)
The first step of establishing the TAI is is obtained with the weighted means of ALGOS. TAI results after the frequency of the EAL frequency with the standard 2.1
The Definition of TAI The EAL is updated at the following t=to+nT/6
ALGORITHM
to set up the EAL (echelle atomique libre), which several hundred atomic clocks in the world, using calibration of EAL is made through a comparison frequency.
times: n = O,l, 2...6.
KE Xi-zheng
et al. /Chinese
Astronomy
and Astrophysics
26 (2002)
245-253
247
Here T = 30d and to is the time of the last reduction in the preceding month: thus the EAL is updated every five days and calculated every 30 days. At time t, the definition of EAL is
i=l
where hi(t) represents the reading of Clock i at t, and its value cannot be directly obtained through experiment; N is the number of clocks; pi is the weight of Clock i; h: is the time correction of Clock i at moment t, and the correction is for ensuring the continuity of time scale in case of variation in the weights or in the number of clocks. Let us denote the clock difference xi(t) between Clock i and the EAL by
xi(t) = EAL(t) At moment
t the difference
xi,j(t)
between
- hi(t).
(2)
Clock i and Clock j is
X:i,j(t) = hj(t) - hi(t).
(3)
The value of zi,j (t) may be directly obtained by experiment. In one and the same laboratory, it can be gained by direct comparison, and between different laboratories by some technique of time transmission. xi(t) is the difference between a clock and the average time scale, and it cannot be directly obtained by experiment. It merely denotes the result of computing EAL. From this the users may get the time scale. According to the above definition, we have the following system of equations:
i~lPiXi(t) =i$Pihi(t) Y
{ xi,j(t)
= xi(t)
- xj(t)
(4)
.
For N atomic clocks there are N such equations, and from them we may get xi(t). For the time span of computation, the difference xi(to + nT/6) (n = 0,1,2...6) of each Clock i relative to the EAL can be obtained, and this is its computed time scale. 2.2 Selection of Time The time correction
Correction
term
Term
h:(t) is the sum of two terms h:(t) = ai
where ai
is the time correction ai
+ Bip(t)(t
for Clock i relative
131:
- to). to EAL at moment
= EAL(to) - hi(to) = xi(to) .
Bip(tc) is the predicted frequency of Nowadays the predicted frequency keeps a is computed from the frequency in the last variance of {xi(to - T + nT/6), n = 0,1...6}
(5)
to. We have
(6)
Clock i relative to EAL in the interval [to, t]. fixed value within one month of [to,t + T]. It period of reduction. In other words, the least decreases, and this is one-step linear prediction.
KE Xi-zheng
248
et al. /Chinese
Astronomy
and Astrophysics
26 (2002)
245-253
This hypothesis has the following prerequisite: the atomic clock behaves similarly in the last month and in this month. If the time of measurement is suitably chosen, this hypothesis will be valid. In practice, when T = 30d, the dominant noise of the clock is expressed as a random walk noise, and the best estimate of B+,(ts) is the value of the last reduction. 2.3 Selection
of Statistical
Weights
The selection of weight is to identify the value that leads to optimum long-term stability of the time scale. The weight is determined by the behavior of the clock in the last month. The concrete procedure is as follows: (1) The solution of the system of equations (4) in the time span [to, t + T]is determined by the weight in the last reduction For the next reduction, the weight determined in the present reduction is used. (2) The estimated value of frequency Bip(to + T) is calculated with the xi(t) solved from the system of equations (4). (3) The variance $(S, t) is computed with Bi,(to + T) as well as the Bip'sin the preceding five reductions. (4) For the weight of Clock i, the temporary value computed with the following formula is used: 1000 p: = $(6,t)
.
(5) Except the following two cases, the new weight pi(t) is equal to p:(t): a. The limit of maximum weight is 1000; b. If the clock exhibits abnormal behavior, its weight is set to zero. From the once-every-5-days computation, ALGOS (BIPM) produces a delayed time scale TAI. The weights of the ALGOS algorithm belong to the classical method of weighting. Other places and laboratories use similar algorithms, such as the MOWA algorithm of JATC 131.
3. WAVELET
DECOMPOSITION
ALGORITHM
OF TIME SCALE
This is an atomic time algorithm developed by us. It is established on the basis of wavelet decomposition, so\we call it Wavelet Decomposition Algorithm (WDA). With this algorithm we set up the Wavelet Decomposition Atomic Time (WDAT) . The classical weight algorithm has some inherent limitations. It assigns to each clock one single weight and doe not synthetically consider the frequency stabilities of all the clocks. The average time scale can only minimize one particular noise, and cannot minimize all the noises. In the ALGOS algorithm, an atomic clock may be weighted according to its longterm stability, so ‘it is possible that due to bad long-term stability an atomic clock with good short-term stability can get very small weight. Then this leads to a waste of resource. In view of this many improved algorithms have been proposed, but they all possess certain limitations. Now wavelet transformation can extract the components of the atomic clock in different frequency ranges, and the frequency stability is different in different frequency ranges, so we have proposed the wavelet decomposition algorithm: the signal from the
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clock is decomposed in the wavelet domain, the components in different frequency ranges are extracted, weighted mean is applied in the wavelet domain, then, after inversion, the synthetic time scale is obtained. Because in this method the different degrees of stability of the atomic clocks in different frequency ranges have been taken into account, all the noises of the synthetic time scale are minimized. It is a weighting method with multiple resolution. 3.1 Setup of WDAT Let the total number of atomic clocks participating in the reduction be C and let the A(c) and frequency reading of Clock c at a certain time be be T(c), the phase correction correction B(c) of the clock having been applied. In the time interval of reduction I the corrected reading TM(c, t) may be written as: TM(c,t)
The weight of Clock c is pt(c) (the ALGOS then the WDAT may be written as:
weighting
method
EPt(c)TM(c>t) cgl PWvk WDAT(t)
c = 1,2,3 ,... C.
= T(c, t) + A(c) + B(c)[t - to],
is used in the time domain),
t) + 44 + B(c)(t- h)l
=
= ‘=’ 5
(7)
(8)
pt(c)
c=l The
difference
between
Clock c and WDAT
is defined
DC(t) = WDAT(t) In practice, e, i.e.,
what
we measure
as
- T(c, t) .
is the difference
between
(9) Clock c and some other
Dc,e(t)= T(e,t) - T(c, t) .
Clock
(10)
The atomic clock with the best performance is taken to be the Master Clock. Let it be Clock C. D~c(t) represents the difference of Master Clock relative to WDAT, and it cannot be obtained by experiment. It expresses the computed time scale and can provide users with the correct value of WDAT. According to the above definition, we have the following system of equations:
5
c=l
{
Pt(c)R(t)
= c&Pt(c)[A(c)
=
Ve,
t)
-
T(c,
Dc,Mc(t).
t> .
In order to get DMc(t),
5 pt(c)[A(c) + B(c)(t =
- to)], (11)
Dc,e(t)
By experiment we can obtain are solved and we have
Dj&t)
+ B(c)(t
‘=’
to)] +
the above equations
5 Pt(c)Dc,Mc(t)
c=l
(12)
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The above expression gives the difference between Master Clock and WDAT, so from Master Clock we may obtain WDAT. Eq.(12) consists of two terms. The first is the weighted mean of the time correction of C atomic clocks, i.e.,
m(t)
=
c=l
(13) 5 pt(c)
.
c=l
Here A(c) is a constant adopted to keep the continuity of time scale. In general it is taken B(c) is the predicted frequency to be the frequency correction at the end of reduction. correction of the current reduction. If the number of clocks does not change and the weight Otherwise, in order to keep of every clock remains the same, then A and B are unchanged. the continuity of the time scale, the values of A and B have to be altered. In the general case, for one and the same atomic clock these values do not, and should not change too much. The method of weighting in the ALGOS algorithm is effective for these two terms, so it is adopted by us. The second term of Eq.(12)
is the mean value of clock difference: C-l
C-l
z-D(t) =
TD(t)
c Pt(c) c Pt(c)DcW(t) c=l
c=l
c-l
c-l
c_lPt(c)Dc,MC(t)c =
c-l
5 PN4 c=l
=
is the mean difference
cgPto . c=lc pt(c>
CL1
pt(c> l
RF(t)
.
(14)
5 pt(c)
c=l
between
the Master Clock and the C-l other clocks. It is random process completely undetermined. For each D c,~~(t), this expresses a non-steady and it is quite different for different times of reduction. When the weight is taken according to the historical characteristics of the clock, the result is extremely bad. slightly deformed it and picked up a term RF(t) for individual analysis.
For this, we have
3.2 Weighted Mean With Multiple Resolution We proposed the method of weighted mean with multiple resolution 141. The wavelet decomposition is carried out for Dc,MC(t). The various frequency components are picked up. The wavelet variance is used to characterize the frequency stability. The weighting is made for different scales, and the signals are reconstructed. Because the wavelet variance characterizes the stability of the clock in various frequency ranges an because by means of wavelet decomposition the different frequency components of the signals of the clock are separated, so the method can bring out the best characteristics of the individual clocks. 3.2.1 Wavelet decomposition The clock difference signal between Clock c and Master Clock Dc,MC(t) may be written as
Dc,dt)
=
2
k=-co
&,,k%,,k
(t) +
5 2
j=-co
@;,&j,k(t).
(15)
k=-m
where Pjo,k = (0, pjo,k) is the coefficient of the rough term of scale function vj,, and ‘Yj,k = (D, 4j,k) is the coefficient of the fine term of wavelet function 4j. Therefore, from
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the viewpoint of wavelet scale the signal is divided into two levels. The level above jr, is extracted for the fundamental characteristic of Dc,~c(t), and that below je is the fine detail approximation of Dc,~c(t). As the scale increases, the separated frequency becomes lower and lower. 3.2.2 Weighted multiple resolution The weight in a certain local frequency range may be written as :
where aj expresses the weighted multiple resolution below the wavelet scale j. This method of weighting takes into account that the stabilities of the clock in various frequency ranges are different, and it is clearly an improvement on the classical method of weighting. 3.2.3 Reconstruction of signals According to the relation of wavelet transformation and its reconstruction, the weighted signal can be written as:
where 0: expresses
the weighted
multiple
resolution
4. EXPERIMENTAL
of Dc,~c(t)
in wavelet
scale j.
STUDY
For the measured data of a certain laboratory, we made a comparison between WDAT and the Synthetic Atomic Time (TA(K)) of the laboratory. Because the first term of Eq.(12), i.e., TC(t), is the same, so only the second term, i.e., TD(t), is written out. Fig.1 shows the difference between the four cesium clocks of the laboratory and the TAI in a lOOd-period from 1984 November to 1985 January. Fig.2 illustrates the comparison between the atomic time TA(K) of the laboratory and the TD(t) term of WDAT. For the TA(K) the weight is taken according to the historical properties of the clocks: for CLKl the weight is rather large and it can only suppress noise of one kind. Therefore, the fluctuation in the atomic time provided by it is far larger than that of WDAT. A comparison of their Allan variances is shown in Table 1. We see that for both short-term and long-term stabilities the WDAT is better. Table
1
Comparison between the Allan variances of TA(K) and WDAT
Allan variance TA”Kc WDAT
1 day duration 0.1697x lo-‘” 0.0441x 10-12
10 days duration 0.6481x10-‘” 0.1168x10-12
30 days duration 0.2160x IO-‘” O.O389x1O-‘2
252
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---. ---
*,_’ :
,_f I
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1
CLKl-TAI CLKZ-TAI
-
245-253
TA(K)-TAI
- - CLK3-TAI
CLK4-TAI
‘,.*
/” I
Astronomy
\ \
k. ' 8' '.\ 8.; '. ,,-.,.______,-.-____-..-___ '\ -'-.--______._------._____________,-.. -..__. 46020
46040
46060
46080
46100
46020
46040
MJD Fig. 1 Time difference between the 4 cesium clocks of a certain laboratory and the TAI in 1984 Nov - 1985 Jan
Table
2
Weights
Fig. 2
Comparison
a certain laboratory
of four atomic
clocks in various
46060 MJD
46080
4610
of the atomic time TA(K)
of
and WDAT
scales
Weights of clocks Scale 0.90 1.30 1.70 3.30 4.90 5.70 7.30
CLKl 190.668 80.343 32.746 3.423 0.761 0.409 0.145
CLK2 278.000 278.000 233.049 68.939 27.419 21.308 17.843
CLK3 184.581 97.513 59.701 18.835 9.094 7.105 5.043
CLK4 278.000 142.960 62.784 10.120 5.190 4.295 3.022
Table 2 compares the weights of the four atomic clocks in different scales. We see that the weights are very different for the different scales. The fluctuation of CLKl is the largest and in it weight is usually the smallest. But for scale 0.9, its weight is larger than that of CLKS. This implies that it has one frequency component whose fluctuation is comparatively small. Similarly, CLK3 and CLK4 have large weights. The fluctuation of CLK2 is the smallest and for all scales its weights are always the largest, but the amount by which it is larger varies from scale to scale, and this just corresponds to the fact that different frequency components of the clocks have different intensities and the weights are different. Here is precisely the superiority of WDAT. It takes into account the differences in stability of the clocks in various frequency ranges, and makes use of the best individual characteristics of the clocks. As shown in Table 2, the weight of CLK2 is the largest, so WDAT should be closer to CLK2. The comparison between CLK2 and WDAT is illustrated in Fig.3. As shown by this figure, although CLK2 is close to WDAT, the latter is far smoother, having been freed of many points of sudden changes.
5.
A new time algorithm. ALGOS, it influence of
CONCLUSIONS
scale algorithm is proposed in this work. It inherits the merits of the ALGOS Under the pre-condition that its long-term stability is not lower than that of leads to a great improvement of the short-term stability. It can suppress the sudden changes and make the time scale more steady and uniform.
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-‘.O*
Fig. 3
I
Astronomy
.
46020
and Astrophysics
46040
The time differences
46060 MJD
26 (2002)
I
46080
CLKZTAI
245-253
46100
and WDAT-TAI
References 1
Guinot B., Thomas
2
Patrizia
3
Pan
4
Ke Xi-zheng,
Tavella,
Xiao-pei,
C., BIPM
Claudine
Tu Lu-zheng, Wu Zhen-sen,
Annual
Thomas,
Report
Luo Ding-chang, Yang
of the Time
Metrologie,
Ting-gao,
Section,
1988, 1, Dl-D22
1991 28, 57 Science
Journal
in China,
1988, 8, 851
of Electronics,
1999, 27, 135
253