Instrumentation & Measurement Magazine 23-2 - 58

where hi(tk) is the time (phase) of clock i (Clk i) at time instant tk. Then at the start of the TS calculations, the first 30
days, the rapid version of UTC (UTCr) at time tk (UTCr(tk))
was used instead of TA(tk) for computing the initial values
of xi(tk) for the rate calculation of Clk i (yi(tk)) given by (8)
[10], because TA(tk) was not yet determined.
	

yi ( t k ) =

xi ( t k ) − xi ( t k − T )
T

	(8)

where T is the interval over which the rate of the clock is
predicted. This interval must be chosen very carefully in
TS calculations to predict the clock rate correctly, which
reduces the effect of clock anomalies on the resultant time
scale as much as possible. For the 5071 A (high performance) Cs clocks, the best value of T is 30 days, because it
reaches its noise floor at that interval [10].
◗◗ The measured phase difference between Clki and Clks at
time tk (Xis(tk)) can be calculated according to (9) [10]:
	

Xis ( t k ) = hi ( t k ) − hs ( tk ) = xi ( tk ) − xs ( tk )	(9)

where hs(tk) is the time of the master clock (Clks) at time
tk and xs(tk) is the phase difference of Clks with respect to
TA(tk) at any time tk. Then, the national time scale of OP
(UTC(OP)) at time tk was used as the master clock (hs(tk))
in the computation of Xis(tk), because of its excellent stability specifications over all averaging times. Then, (9) can be
approximated as shown in (10):
	

Xis ( t k ) = hi ( tk ) −  UTC ( OP )( t k )	(10)

In that case, Xis(tk) values were obtained directly from the
comparison results of the OP clocks that were downloaded
from the time server at the BIPM without any additional
calculations.
With these two assumptions and the average TS algorithm
mentioned in [10], [16], different tests have been made to study
the effect of adding de-noised Cs clocks using EMD on the stability of the resultant TA(K).
In the first test, a TS was built using the simple average
TS algorithm of [10] using only the 4 real Cs clocks of OP before de-noising. Fig. 8 shows both the frequency stability of

Fig. 8. Frequency stability of Cs1, Cs2, Cs3, and Cs4 using UTC(OP) as a
reference, and the frequency stability of TA1 using UTCr as a reference.
58	

Fig. 9. Frequency stability of TA1 and TA2 using UTCr as a reference.

the resultant average time scale (TA1) using UTCr as a reference and the frequency stability of the 4 Cs clocks of OP using
UTC(OP) as a reference. This was done for averaging times
from 1 day (86400 s) to 81 days (6998400 s). From Fig. 8, we
note that the frequency stability of the resultant TA1 is better
than the stability of all the contributing clocks in the ensemble, as expected.
In the second test, the de-noised Cs1, Cs2, Cs3 and Cs4
clocks, as mentioned above, were used for building an average time scale (TA2) instead of the original clocks. The
frequency stability of TA2 was compared with that TA1 for
the same averaging times, as shown in Fig. 9. Obtained results show that using EMD for de-noising the Cs clocks of the
ensemble improves the frequency stability of the clocks, and
hence the stability of TA2 as compared toTA1 for most of the
averaging times up to nearly 40 days (3456000 s). The stability
improvement factor of TA2 with respect to TA1 is about 47%
and 42% for averaging times of 1 day and 10 days, respectively. Using the EMD method for Cs clocks signal de-noising
improves the frequency stability of TA(K) for averaging times
up to 38 days. Then, TA(K) can be used as a highly stable reference for steering UTC(K) time scales during the window
periods of UTCr which is 10 days and for longer periods up
38 days.

Comparison between the Effect of
Using EMD and Kalman Filter (KF) on
TA(K)
The KF as a signal processing technique is still used at some
time laboratories for the prediction step of the TS algorithm
and for de-noising atomic clock signals to improve the frequency stability of TA(K) and hence UTC(K) TS [3], [7], [15].
In this section, the KF is used for de-noising the WFM noise of
Cs1, Cs2, Cs3, and Cs4 of OP. Then, the de-noised Cs clocks are
used for building an average time scale (TA5). The frequency
stability of TA5 is compared with that of TA3 built using the
four Cs clocks before de-noising and that of TA4 build using the same Cs clocks de-noised using the EMD method, as
shown in Fig. 10. An even weight of 0.25 was given to each of
the Cs clocks in the three cases, so that the sum of all relative
weights equals one in the TS algorithm.

IEEE Instrumentation & Measurement Magazine	

April 2020
Instrumentation & Measurement Magazine 23-2

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