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

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