Instrumentation & Measurement Magazine 23-2 - 57

and use them to verify the ability of EMD to enhance their
frequency stability. The 7-digit codes given by BIPM to highperformance 5071A Cs clocks maintained by OP are 1352804,
1350770, 1350774 and 1350396, which are designated from now
on as Cs1, Cs2, Cs3 and Cs4. The phase measurements of these
clock versus UTC(OP) for 365 days (from MJD 57692 to 58056)
were downloaded from the BIPM time department server and
de-noised using EMD. MATLAB program was designed for
this purpose and the main steps of this program are as follows:
◗◗ The raw phase measurement data of Cs1, as shown in Fig.
6, is decomposed into eight IMF components and a residue using sifting process of EMD, as explained above.
◗◗ By analyzing the obtained eight IMFs, it is found that
the high-frequency noise is concentrated into the first
four IMFs from IMF1 to IMF4, so these components are
cancelled to reduce the noise.
◗◗ The useful signal existed in the last IMFs from IMF5 to
IMF8, so these components are reconstructed with the
residue to give the de-noised signal, as shown in Fig. 6.
◗◗ The decomposition process of the Cs clock signal into
eight IMFs is explained in detail in [8], and the final result
of the de-noising process is shown in Fig. 6. The number
of IMFs cancelled from the Cs signal to achieve the best
accuracy of the de-noising process is chosen by analyzing
the obtained eight IMFs and the results of Fig. 6 to obtain
a de-noised signal that keep the main features of the raw
signal, as shown in Fig. 6.
◗◗ The same procedure is repeated for raw phase measurements data of Cs2, Cs3 and Cs4 to obtain its de-noised
signals.
The trend (relative frequency offset) of the phase data of
Cs1 was removed before applying EMD. From Fig. 6, we note
that the EMD method is used for CS1 de-noising while keeping
the main features of the signal in time domain.
To check the effectiveness of EMD method for Cs1 de-noising, the frequency stabilities (expressed by Overlapping Allan
Deviation (OADEV)) [2] of Cs1 and Cs1 de-noised were computed and plotted using UTC(OP) as a reference for averaging
time from 1 day (=86400 s) to 89 days (=7689600 s), as shown
in Fig. 7. From Fig. 7, it is noticed that the frequency stability of
Cs1 is greatly improved due to using EMD for signal de-noising for averaging times from 1 day (86400 s) to almost 34 days

Fig. 6. De-noising Cs1 raw phase measurements using EMD.
April 2020	

Fig. 7. Frequency stability of Cs1 and Cs1 de-noised using EMD.

(3456000 s). These results show that EMD is an effective and efficient simple method for 5071A Cs clock signal de-noising as
compared to the complicated KF that can be used for the same
purpose. The same conclusion was obtained for Cs2, Cs3, and
Cs4.
In the next section, the de-noised Cs1, Cs2, Cs3, and Cs4
were used to build an average TS algorithm, to show the effect
of using EMD method for Cs clock signal de-noising on the frequency stability of the resultant average time scale (TA).

Effect of EMD on the Frequency
Stability of UTC(K) through Simulations
The TS algorithms used in time keeping laboratories for generating UTC(K) time scales can be divided into two main
categories. The first one is the simple average method that
combines the weighted clock errors to generate the average
time scale (TA(K)) which is more stable than any clock in the
ensemble [4], [5], [10]-[13]. Then, TA(K) is used for steering
UTC(K) more frequent to follow UTC, as mentioned in [14].
The other category depends on using signal processing techniques such as KF together with the average TS algorithm for
prediction and signal de-noising to enhance the performance
of the resultant TA(K) and hence UTC(K) TS [3], [7], [15].
In this section the simple average TS algorithm described in
[10], [16] was applied for building an average TS TA(K) using
an ensemble of the four Cs clocks of OP before and after denoising to show the effect of de-noising on the stability of the
resultant TA(K). The daily phase comparison results of Cs1,
Cs2, Cs3, and Cs4 for a period of 365 days (MJD 57692 to 58056)
before and after de-noising, as mentioned previously, are used
in the average TS algorithm. Also, the rapid TS difference
(UTCr-UTC(OP)) during the same period is used in the algorithm. The clocks maintained at OP and UTC(OP) were chosen
because it is one of the best TSs around the world.
During the TS computations, the following assumptions
were made:
◗◗ Since the phase difference of clock i(xi(tk)) with respect to
TA(tk) at any time tk can be determined from (7) as follows
[10],:
	

xi ( tk ) =  hi ( t k ) −   TA ( t k )	(7)

IEEE Instrumentation & Measurement Magazine	57



Instrumentation & Measurement Magazine 23-2

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