and non-linear signals in both time and frequency domains. The EMD method is used to decompose the signals or data series into a finite and a small number of Intrinsic Mode Functions (IMFs). Then, the HSA method is applied to the IMFs to generate instantaneous frequency data. Since the signal decomposition occurs in time domain and the length of the IMFs is the same as the original signal, the HHT preserves the varying frequency characteristics of the signal. The EMD method was first proposed for high performance 5071A atomic clock signal de-noising and frequency prediction in [8] but with no more details or applications on the TS algorithm to study its effect on the frequency stability of the resultant average TS. The work presented in this paper is an extension of [8]. In this paper, the EMD method is used for enhancing the frequency stability of national time scale UTC(K) for averaging times from 1 day to 40 days by de-noising the WFM noise of Cs atomic clocks. The EMD is first applied on denoising the Cs clocks comparison data of the Observatory of Paris (OP) published on the time department server of the Bureau International des Poids et Measures (BIPM). Then, these de-noised Cs clocks are used for building an average TS to study the effect of using EMD on its resultant frequency. To verify the effectiveness of the EMD method, it is compared with the KF as famous de-noising method used with TS in the literature. The KF is used for de-noising the same Cs atomic clocks of OP, and then the de-noised clocks are used for building an average TS. The frequency stability of the resultant TS is compared in both cases. Obtained results showed that the EMD method is as effective as the KF for improving the frequency stability of the resultant average TS for most of the averaging times from almost 1 day to 40 days. The EMD method is found to be adaptive, simple and efficient as compared to the complicated KF. The reminder of this paper is organized as follows: an overview of the EMD is introduced; the use of the EMD method for de-noising Cs clock signal is explained in detail; different TS tests are carried out with and without de-noising to study the effect of EMD on the frequency stability of the resultant average TS; a comparison between EMD and KF for enhancing the frequency stability of average time scales is introduced; and conclusions and remarks on the work of this paper are presented. The IMF can be defined as a simple oscillatory signal as the simple harmonic function, but it is more general and can have variable amplitude and frequency along the time axis. For any data set, IMF can be defined as any function that meets the following two requirements [6]: ◗◗ The number of maximum points and zero-crossing points must be either equal or vary by at most by one. ◗◗ The envelope of the maximum points and that of the minimum points are symmetric with respect to zero (i.e., the mean value of the two envelopes at any point equals zero). The extraction of IMF from the original signal or data set can be done using a sifting process that can be explained by the following steps [6]: ◗◗ Consider that the original data to be decomposed is X(t). ◗◗ Determine the maximum points in the data set, as shown in Fig. 1a. ◗◗ Connect all maxima by the cubic spline line interpolation method to yield the upper envelope, as shown in Fig. 1b. This interpolation method is chosen for EMD because it is optimal [6]. ◗◗ Repeat these two steps for the minimum points to obtain the lower envelope, as shown in Fig. 1b. All data points should be covered by the upper and lower envelopes. ◗◗ Determine the mean value of the two envelopes, m1(t). This mean will yield the lower frequency component than the original signal, as shown in Fig. 1c. ◗◗ Compute the difference between X(t) and m1(t). The result is the first component h1(t), as given by (1) and shown in Fig. 1d [6]. By this step, the highly oscillated pattern, h1(t) is separated from the data, and this is the first round of sifting. Ideally, the residue (h1(t)) should meet the conditions of IMF defined above, but one sifting process does not guarantee that the residue is an IMF. So, the sifting process is repeated to h1(t) until the definition of IMF is satisfied. The repeated sifting makes the remaining signal more symmetric around zero-mean value, according to the definition of IMF. ◗◗ In the next sifting step, h1(t) is treated as data and the first six steps are repeated again, as shown in Figs. 2a, 2b, 2c and 2d, to produce the second residue, h11(t), according to (2) [6]. Extracting IMFs from Non-Stationary and Non-Linear Signals Using EMD The EMD technique is the fundamental part of the HHT. By using it, any complicated data set or non-stationary signal can be analyzed into a finite and a small number of components known as IMFs. The first IMF contains the high-frequency component of the signal and the last one contains the lowfrequency component. This fact is used for signal smoothing using EMD where the high-frequency random noise components can be removed by rejecting the first IMFs [9]. The decomposition occurs in time domain which is the same scale as the original signal and this keeps the non-stationary nature of the signal. For this reason, the EMD is adaptive and efficient for non-stationary and non-linear signal de-noising. 54 h1 ( t ) = X ( t ) − m1 ( t ) (1) h11 ( t ) = h1 ( t ) − m11 ( t ) (2) ◗◗ After repeated sifting up to k times, the first IMF component of the data (IMF1), c1(t) results, as given by (3) [6]. c1 ( t ) = h1k ( t ) = h1( k −1) ( t ) − m1k ( t ) (3) The S-number criterion determines when the sifting process is stopped. This number is pre-selected and is defined as the number of consecutive siftings for which IMF conditions IEEE Instrumentation & Measurement Magazine April 2020

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