An Improved Empirical Mode Decomposition Based on Local Integral Mean and Its Application in Signal Processing

Niu, Xiao-dong; Lu, Li-rong; Wang, Jian; Han, Xing-cheng; Li, Xuan; Wang, Li-ming

doi:https://doi.org/10.1155/2021/8891217

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2021 | Article ID 8891217 | https://doi.org/10.1155/2021/8891217

An Improved Empirical Mode Decomposition Based on Local Integral Mean and Its Application in Signal Processing

Xiao-dong Niu,^1,2Li-rong Lu,³Jian Wang,¹Xing-cheng Han,¹Xuan Li,¹and Li-ming Wang¹

Academic Editor: Volodymyr Ponomaryov

Received29 Sept 2020

Revised18 Dec 2020

Accepted03 Jan 2021

Published02 Feb 2021

Abstract

Empirical mode decomposition (EMD) is an effective method to deal with nonlinear nonstationary data, but the lack of orthogonal decomposition theory and mode-mixing are the main problems that limit the application of EMD. In order to solve these two problems, we propose an improved method of EMD. The most important part of this improved method is to change the mean value by envelopes of signal in EMD to the mean value by the definite integral, which enables the mean value to be mathematically expressed strictly. Firstly, we prove that the signal is orthogonally decomposed by the improved method. Secondly, the Monte Carlo method of white noise is used to explain that the improved method can effectively alleviate mode-mixing. In addition, the improved method is adaptive and does not need any input parameters, and the intrinsic mode functions (IMFs) generated from it is robust to sifting. We have carried out experiments on a series of artificial and real data, the results show that the improved method is the orthogonal decomposition method and can effectively alleviate mode-mixing, and it has better decomposition performance and physical meaning than EMD, ensemble EMD (EEMD), and complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN). In addition, the improved method is generally more time-consuming than EMD, but far less than EEMD and CEEMDAN.

1. Introduction

Empirical mode decomposition (EMD) [1] is a local, data-driven, and adaptive method in processing nonlinear and nonstationary signals and has been widely used in machinery, voice, geography, medicine, and other fields [2–10].

However, because EMD is an experience-based method, there are many problems [11]. These problems include the mixing of multiscale modes, and the sifting process for intrinsic mode functions (IMFs) requires an appropriate stopping criterion, the end effect, and the orthogonality of IMFs. Among them, EMD has been criticized by signal processing experts because of the lack of orthogonal decomposition theory, which has become a major problem restricting the application of EMD [12, 13]. Various research studies based on the orthogonal or approximate orthogonal decomposition methods of EMD [12, 14–17] are not convincing. References [18, 19] use Schmidtʼs formula to make IMFs strictly orthogonal, but it cannot guarantee that the IMF is obtained from the orthogonal decomposition of the signal. Mode-mixing between IMFs is another major problem that limits EMD applications. Although a large number of improved methods of EMD are devoted to solving this problem, they often need to determine the input parameters a priori. Inappropriate parameters will reduce the accuracy of decomposition [20] and even lead to decomposition failure. For example, ensemble EMD (EEMD) [21], complementary EEMD (CEEMD) [22], and complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) [23] effectively alleviate mode-mixing, but need to set parameters of the amplitude and quantity of auxiliary noise; moreover, their related procedures are time-consuming, which is often intolerable for large-length signal; empirical wavelet transform (EWT) [24], variational mode decomposition (VMD) [25], and the methods based on Hankel matrix eigenvalue decomposition (EVD) [26] build EMD on additional classical theoretical methods to alleviate mode-mixing, but they also need to carefully select appropriate parameters, especially the number of mode components.

In fact, the orthogonality and mode-mixing between IMFs are related. If the signal will be orthogonally decomposed and the mode components will orthogonal, mode-mixing will be effectively alleviated. For this reason, an improved EMD method is proposed in this study. The most important change in the improved method is to change the envelope mean method (in EMD) [1] to the integral mean value theorem (IMVT) to find the data’s mean, which makes the mean be strictly expressed mathematically. Although some EMD improvement methods have tried to change the mean method, none of these methods can be expressed in a strict analytical formula [27–30]. Local mean decomposition (LMD) constructs a mean curve with the mean of adjacent extreme points of signal, but this will lose the detailed fluctuation information between these two adjacent extreme points [31]. We call the improved method the integral mean mode decomposition (IMMD). In the IMMD method, we try to prove the orthogonality of IMFs and show that mode-mixing is effectively alleviated. In addition, IMMD uses spline interpolation to directly predict the mean value at both ends of the data and uses a stop criterion with a fixed number of sifting [32] to reduce complexity.

The rest of the manuscript is organized as follows: in Section 2, we give in detail the steps of the IMMD method and the corresponding flowchart (Figure 1), the stopping criterion of IMF, and the method of restraining the end effect. And in the part of the stopping criterion, we will take Gaussian white noise as an example to analyze the robustness of IMFs from IMMD to sifting. In Section 3, a widely used nonlinear nonstationary signal is used as an example to prove the orthogonality of IMMD decomposition in three steps. In Section 4, the Monte Carlo method is used to show that IMMD can alleviate mode-mixing than EMD. In Section 5, a series of artificial and real signals are used to verify that the IMMD method is the orthogonal decomposition method and can alleviate mode-mixing. Finally, discussion and conclusion are given in Section 6 (Figure 1).

2. IMMD Method

2.1. Steps

(1)The original signal (or protomode function (PMF) [33]) is divided into many local parts by all extrema.(2)For each local part, the value of the local mean is calculated by IMVT, and its position is fixed at the midpoint of the local time series. All the local means are interpolated by cubic splines to get the mean curve (in EMD, the mean curve of the signal is obtained by the difference between the upper and lower envelopes, and the upper (lower) envelope is obtained by smoothing the maxima (minima) of the signal. The remaining steps of EMD are the same as those of IMMD (see Figure 1)).(3)PMF is obtained by subtracting the mean curve from the signal.(4)Repeat steps (1)–(3) on PMF (as a signal) iteratively, until PMF satisfies the stoppage criterion. This process of subtracting the mean iteratively from the signal is called sifting. When the sifting is over, PMF is IMF₁.(5)New signal is obtained by subtracting IMF₁ from the original signal, and IMF₂ is obtained by repeating steps (1)–(4) on the new signal.(6)Repeat steps (1)–(5) to get other IMFs of the original signal.

The flowchart of IMMD based on EMD is shown in Figure 1. The method to construct the mean curve of signal in [34] is the same as that in IMMD. However, in [34], the definition of “centroid” is used to determine the time series position of the local mean , which makes the time series position of be restricted by the signal amplitude. Therefore, IMMD uses the simplest time series midpoint as the time series position of .

2.2. The Stoppage Criterion

There are two purposes of sifting: to eliminate riding waves and to make the wave profiles more symmetric [1]. The commonly used sifting stoppage criterion was called the Cauchy type of stoppage criterion [11], and the equation is as follows:

Among them, PMF_k is the PMF obtained by k-th sifting. If SD < 0.2∼0.3, sifting stops. In fact, although 0.2∼0.3 as the SD threshold is accepted by most users of EMD, it is controversial [11]. It is generally accepted that, under a limited sifting number, the smaller the SD value, the better. In order to reduce the complexity, EMD uses the stoppage criterion of fixing sifting number to 10, which also satisfies the Cauchy type of stoppage criterion [32].

Let IMMD adopt the stoppage criterion of fixed sifting number and decompose a randomly generated Gaussian white noise. The SDs of all IMF vary with the sifting number, as shown in Figure 2(b). It can be seen that SD₁∼SD₁₄ shows fast and stable strong convergence. When the sifting number >6, all SD values <0.01; when the sifting number >9, all SD values <0.001; and when 100 < the sifting number <5000, almost all SD values <1 × 10⁻³⁰.

(a)

(b)

Let EMD decompose the same Gaussian white noise, and the SDs of IMF vary with the sifting number, as shown in Figure 2(a). Almost all SDs show weak convergence of slow oscillation. Except for SD₈ and SD₉ (the sifting number > 47, and SD₈ and SD₉ < 0.1), when the sifting number > 3, all SD values < 0.01; when the sifting number > 19, all SD values < 0.001; and when the sifting number < 5000, almost all SD values are still more than 1 × 10⁻¹⁰.

The results of the two methods show that the sifting process of IMFs in IMMD is stable and can meet the Cauchy stop criterion very well. In addition, we have tested all the signals decomposed by IMMD in this study about sifting, and the SD curve of them is almost exactly similar to that of Gaussian white noise, and there is no qualitative difference. Therefore, we recommend that the IMMD method adopts the stop criterion of fixed sifting number.

2.3. Method of Alleviating End Effects

End-effect processing methods are essentially predicting the end mean of data [21]. The error caused by prediction will spread from the end to the interior of the data with the sifting, so it will affect the stability of IMF. We propose a method of predicting the end mean which is suitable for IMMD: using the spline interpolation of the data’s mean to directly predict the end mean of the data.

Compared with the other prediction methods, the proposed method has the following characteristics: (1) it directly predicts the end mean; (2) it does not need to introduce additional end-effect processing methods; and (3) it ensures that IMFs have good robustness to the sifting. We present the SD curves (see Figure 3) obtained when Gaussian white noise is decomposed by IMMD with enlargement of the ends (one common method to alleviate end effects in EMD) [21], and make a simple visual comparison between Figures 3 and 2(b). The details of the comparison are similar to that in Section 2.2, and there is no essential difference, so we will not repeat it here. The comparative analysis between the proposed method and other methods is beyond the scope of this study.

3. The Proof of IMMD Orthogonal Decomposition

Let IMMD decompose the signal into a total of n components of IMF₁, …, IMF_j, …, IMF (n − 1), r_n, where IMF_j is the j-th mode component and its corresponding residue is r_j. The proof of the orthogonal decomposition of IMMD is divided into three steps: first, IMF_j is orthogonal to r_j; second, IMF_j is orthogonal to r_k (j < k ≤ n); and third, IMF_j is orthogonal to IMF_k.

3.1. The Proof of Orthogonality of IMF_j and r_j

3.1.1. The Proof

Take IMF₁ as an example. Suppose that PMF_k (k = 0, 1, …, ) is obtained after the k-th sifting and the number of its extrema is n, let , can be described as follows:where is the local part of from to , is the position (on the time series) of the i-th extremum of (see Figure 4). Let is the mean curve of ; can also be described as follows:

According to the IMVT, we havewhere is a constant, described as (see Figure 4). After one shifting on , we get , and can also be described as follows:

From equation (4), we have

is obtained after the l-th (l ≤ k) sifting, and its mean curve is . can also be described as , where is also a constant, described as (see Figure 4). Then, we have

Therefore, we havei.e., , (l = 0, 1, …, k; is the mean curve of PMF₀).

Here, it should be noted that the sifting will produce new extrema (the result of overshoot and undershoot), but the new extrema have no effect on the above conclusion. The reason is that the mean curve is a constant in any local part of (take the annual global surface temperature anomaly (GSTA) [35] as an example (Figure 5): the sifting produced a new extremum at 1912 in PMF₃, but in the local parts [1911, 1912] and [1912, 1913], , , and are all constant).

Assuming that satisfies the stoppage criterion, so . The difference between and IMF₁ is the corresponding residue of IMF₁, and . So we get

In the same way, it is easy to get

Therefore, any IMF is orthogonal to its corresponding residue.

3.1.2. The Effect of Sifting on the Orthogonality of IMF_j and r_j

Although the new extremum produced by sifting has no effect on the above conclusion, the spline interpolation of all local means in the sifting process will affect the orthogonality of IMF and its corresponding residue. In the above proof, the spline interpolation of all local means is ignored. The explanation is as follows: suppose is the mean curve of the signal obtained by spline interpolating all local means of the signal. The error of the mean curve is the difference between and the stair step curve of mean, (the default mean curve in the above proof). Similar to the SD of two continuous PMFs, the standard deviation of mean is used to evaluate this error, which is defined as follows:

Take the IMF₁ of GSTA as an example (see Figure 6). With the increase of the sifting number, the SD value increases linearly and reaches the maximum 0.028 after 20 siftings. For GSTA with low sampling frequency, 0.028 is enough small (in practice, the error can be reduced by increasing the sampling frequency). Therefore, the error of the mean curve has a small effect on the orthogonality, and it can be ignored in theoretical proof.

3.2. The Proof of Orthogonality of IMF_j and r_k (j < k ≤ n)

3.2.1. The Proof

The average period of the IMF_j (j = 1, 2, ...) from EMD and its improved methods will increase with the increase of order j. which means

In particular, for Gaussian white noise, [32, 36, 37]. Let L_j be any local time region of IMF_j ( is the time series point of the i-th extremum of IMF_j) and L_k be any local time region of IMF_k ( is the time series point of the i′-th extremum of IMF_j). Generally, if , then according to equation (12), and , that is, . Taking GSTA as an example, when the fixed sifting number is 1, IMMD decomposes into IMF₁∼IMF₇ of GSTA, as shown in Figure 7. In most local parts, L_j ≤ L_k (a typical part is marked in the black dotted box: L₁ < L₂ <⋯<L₇), but there are a few local parts (marked by red box), L₁ > L₂ or L₁ > L₃.

Section 3.1 has proved that ignoring the smoothing of the mean, the residue r_k of IMF_k is a constant within L_k. Because , r_k should also be a constant in L_j. Due to the importance of this conclusion, we take GSTA as an example to explain and verify. First of all, in order to reduce the influence of the spline smoothing mean curve, and to better illustrate the above conclusion, the sifting number is fixed at 1. Figure 8 gives all the IMF_j curves and r_k (k > j; j, k = 1∼6) step curves of GSTA. It can be seen in Figure 8 that r_k is a constant in most locals except for a few locals (marked by color dotted line in Figure 8) of IMF_j.

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(b)

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Data sampling and spline interpolation produce a few locals of IMF_j where r_k is not constant. Discrete sampling of data and local definite integral to obtain the mean value makes the mean curve of the data discontinuous and needs to be smoothed. Smoothing may cause a small number of extrema to move or even generate new extrema. For example, in Figures 8(e) and 8(f), r₅ becomes rs₅ after smoothing. As a new data, rs₅ is decomposed by IMMD to get IMF₆. Let r₆ be the step mean curve of rs₅. If r₆ is not smooth, r₆ is constant in the two parts of rs₅-r₆ (should be equal to IMF₆); however, r₆ must be smoothed, so the extremum of real IMF₆ has shifted slightly to the right, which causes r₆ to be no longer a constant in the first local of IMF₆.

After the above analysis, it can be determined that if the smoothing of the mean curve is ignored, not only r_j but also r_k (k > j) are constant in any local of IMF_j (j = 1, 2, ..., n). In addition, according to equation (6), in any local of IMF_j, the definite integral of IMF_j is 0, so the inner product of IMF_j and r_k is 0. Therefore, globally, IMF_j and r_k are orthogonal, which is

3.2.2. Index of Orthogonality of IMF_j and r_k

Because of the smoothing of the mean curve, the finite length of the data will cause leakage, and equations (10) and (13) are not strictly equal to 0. Therefore, define an index for checking the orthogonality of IMF_j and r_k (k ≥ j) as

When the sifting number is fixed to 1, the IRO_jk of IMF₁∼IMF₆ and r₁^′∼r₆^′ (sum of step mean curves) and the IRO_jk of IMF₁∼IMF₆ and r₁∼r₆ (sum of smooth mean curves obtained by smoothing the step mean curves) of GSTA are shown in Table 1. Overall, these values verify IMF_j and r_k is orthogonal, and the smoothing of the mean curve reduces the orthogonality of IMF_j and r_k.

3.2.3. The Effect of Sifting on the Orthogonality of IMF_j and r_k

In the process of sifting, the influence on the orthogonality of IMF_j and r_k includes the smoothing of the mean curve of PMF and the symmetry of PMF with respect to the time axis. It has been explained and verified in Sections 3.2.1 and 3.2.2 that smoothing the mean of PMF will reduce the orthogonality of IMF_j and r_k. However, when the PMF is more symmetric about the time axis, its local integral tends to 0, so the orthogonality is stronger. As the sifting number increases, the mean value of PMF and the change of PMF both tend to 0, and the orthogonality no longer changes. Therefore, sifting makes the orthogonality index oscillate and eventually tend to be constant.

Figure 9 shows the curve of the IRO_1k value of IMF₁ and r₁∼r₆ of GSTA with the change of fixed sifting number (1∼5000). It can be seen visually that when the fixed sifting number is less than 20, the IRO_1k value oscillates; when it is greater than 20, it quickly converges to a fixed value. Table 2 shows the IRO_jk value of IMF_j and r_k (k ≥ j; j, k = 1∼9) when the fixed sifting number = 10. Compared with Table 1, it can be seen that sifting enhances orthogonality.

3.3. The Proof of Orthogonality of IMF_j and IMF_k

3.3.1. The Proof

According to equations (10) and (13), it can be obtained that

And it is equivalent to:

So far, we have proved that any two IMF components are orthogonal.

3.3.2. Index of Orthogonality of IMF_j and IMF_k

Similar to equation (13), equation (16) is not strictly equal to zero. Reference [1] has defined an index to check the orthogonality of any two IMFs:

When the fixed sifting number is 1, the IO_jk values of IMF₁∼IMF₇ of GSTA are shown in Table 3. There are 5 values with the order of magnitude of 10⁻¹ in Table 3, mainly due to the poor symmetry of the IMFs obtained by sifting once (especially IMF₄ and IMF₅ in Figures 8(d) and 8(e)). Therefore, the integral of IMF in each local is 0 and cannot be satisfied very well.

3.3.3. The Effect of Sifting on the Orthogonality of IMF_j and IMF_k

With the increase of sifting number, the symmetry of IMF and the orthogonality of IMFs increased. Figure 10 shows that the IO_1k of GSTA varies with the fixed sifting number (1∼5000). It can be seen that, with the increase of sifting number, the values of IO_1k oscillate but converge rapidly to the constant. In fact, Figure 10 is consistent with Figure 9, and it shows that IMF from IMMD has good robustness to sifting.

When the fixed sifting number is 10, the IO_jk values of IMF₁∼IMF₁₀ of GSTA are shown in Table 4. In Table 4, the five IO_jk values with an order of magnitude of 10⁻¹ (in Table 3) do decrease. In addition, the increase in the number of IMFs also shows the performance of orthogonal decomposition. Therefore, sifting can enhance the orthogonality, which is consistent with the conclusion of Section 3.2.3.

In addition, reference [1] also has defined an overall index of orthogonality (IO):

Figure 11(a) shows the IO value of GSTA, which varies with the fixed sifting number (1∼5000). When sifting number <20, IO value oscillates; when sifting number ≥20, IO = 5.1 × 10⁻². Figure 11(b) shows the number of IMFs of GSTA, which varies with the fixed sifting number (1∼5000). Compared Figure 11(a) with Figure 11(b), the IO value is approximately inversely proportional to the number of IMFs, which indicates that the number of IMFs increases, the IO value is smaller, and the orthogonality is stronger.

(a)

(b)

4. Alleviation of Mode-Mixing

IMFs from IMMD are orthogonal to each other, so it can effectively alleviate the mode-mixing. Using the Monte Carlo method of Gaussian white noise, references [32, 36, 37] indirectly explain that EMD, EEMD, B-spline interpolation-based EMD (B-EMD) [38], and trigonometric cardinal spline interpolation-based EMD (C-EMD) [39] are all equivalent to a set of binary filter banks with a constant quality factor Q. This section will use the same method to illustrate that IMMD can effectively alleviate the mode-mixing.

Let IMMD and EMD decompose 5000 samples of Gaussian white noise with length = 512, mean = 0, and variance = 1, respectively. IMMD (EMD) decomposes each Gaussian white noise into at least 10 (8) IMFs. The average Fourier power spectrum of the corresponding order IMF of all samples is shown in Figure 12. The IMF₁ from IMMD or EMD is equivalent to a high-pass filter, and the other IMF is equivalent to a set of overlapping band-pass filters. The center frequency of the latter band-pass filter from IMMD (EMD) is approximately 2/3 (1/2) of the center frequency of the previous band-pass filter. Therefore, the decomposition of Gaussian white noise by the IMMD method is also equivalent to filter banks and has the characteristic of trisection.

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Except for IMF₁, the bandwidth of the equivalent band-pass filter of the latter IMF from IMMD (EMD) is roughly 2/3 (1/2) times that of the previous IMF, so the equivalent band-pass filter bank has a constant-Q property. It means that the power spectra of IMFs have self-similarity [36]:

Among them, () is the power spectrum of the -th (k-th) IMF; ρ is a constant, for IMMD, ; for EMD, . Based on equation (5), the power spectra of all IMFs are standardized and collapsed and coincident into a single curve [36]. For EMD, the mode-mixing causes the band-pass filter to mix more low-frequency components. Therefore, the filter has poor symmetry about the center frequency and poor narrow-band characteristics, and the ρ value needs to be adjusted to 2.01 [40]. IMMD does not have these problems, and the power spectra normalization result of IMFs is shown in Figure 13.

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When IMMD decomposes Gaussian white noise, its equivalent filter bank has the characteristic of trisection. The number of equivalent filters from IMMD is more than that from EMD, and the narrow-band characteristics and constant-Q properties of the filter bank are better than EMD. Therefore, it has better multiresolution analysis capabilities and can effectively alleviate mode-mixing.

5. Experiments and Results

As mentioned in the Introduction, the orthogonality of signal decomposition will lead to the alleviation of mode-mixing, and we believe that they are unified. They are not only jointly displayed in the orthogonal index, but more importantly reflected in the decomposition capabilities of good decorrelation, energy concentration, and better physical meaning. IMMD has a good orthogonal decomposition theory, which can effectively alleviate mode-mixing. Therefore, IMMD has good orthogonal decomposition capabilities.

In this section, we will conduct experiments on the decomposition of a series of test signals. Firstly, we start with multicomponent nonstationary signals. Secondly, in order to highlight the capabilities and shortcomings of EMD, the authors in [41] successfully used a two-tone signal, and it is often used in many literature studies [15, 25, 27, 28, 41]. Therefore, the two-tone signal is used to evaluate the IMMD method. In addition, EMD has a critical frequency ratio of 0.67 that cannot successfully decompose two-tone signal [41], so we specially construct a three-sine superimposed signal with a frequency ratio of 1 : 0.67 : 0.5 and compare IMMD with EMD to evaluate the IMMD method. Finally, two real complex signals are used to test and evaluate IMMD. One is the annual GSTA data, which is one of the most widely studied nonlinear nonstationary climate time series. It can explain the general method well and is often used by Huang to illustrate the performance of EMD and EEMD methods [42–44]. The other is the electrocardiogram (ECG) signal provided by Moody and Mark [45], which is often used by researchers to illustrate the performance of signal processing methods [46–51]. For these two realistic nonstationary and nonlinear signals, we compare the performance of EMD, EEMD, CEEMDAN, and IMMD methods at the same time and demonstrate the ability of the IMMD method to orthogonally decompose and alleviate mode-mixing.

5.1. Multicomponent Nonstationary Signals

5.1.1. Example 1

Signal₁ (Sig₁) is the superposition of two different frequency-modulation (FM) signals:

Let IMMD and EMD decompose Sig₁ into Sig₃ mode components. The waveforms and time-frequency spectra of these components are shown in Figure 14. Figure 14(a) shows that IMF₁ and IMF₂ from EMD have severe waveform distortion. In the frequency domain (see Figure 14(c)), IMF₁ and IMF₂ from EMD are mixed with more frequency components; at t = 0.7 s, mode-mixing occurs between these two IMFs. IMMD can separate mode components well, and only the frequency mixing occurs at the end of IMF₂ (see Figures 14(b) and 14(d)). Compared with EMD, IMMD alleviates mode-mixing, separates two FM signals well, and shows better orthogonal decomposition capability than EMD.

(a)

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Table 5 shows the values of orthogonal indexes of Sig₁ decomposed by two methods. These values verified the theory that IMFs from IMMD are orthogonal, and IMFs from EMD are posteriori orthogonal (the orthogonality of the EMD method cannot be proved, and it can only be verified by the IO value of data [1]). In addition, the three IO_jk values of IMMD are all smaller than those of EMD, but the IO value is larger than that of EMD. Overall, IMMD has better orthogonality than EMD.

5.1.2. Example 2

Signal₂ (Sig₂) is the superposition of three monocomponent signals including a FM signal, an amplitude-modulated (AM) signal, and a linear signal:

Let IMMD and EMD decompose Sig₂ into Sig₃ IMFs, respectively, and waveforms and time-frequency spectra of the components are shown in Figure 15. The characteristic of Sig₂ is that the frequency of the FM signal is equal to the frequency of amplitude change of the AM signal when . When 0 < t < 2, two mode components from EMD have serious waveform distortion (see Figure 15(a)). In the frequency domain, when t < 2, IMF₁ and IMF₂ from EMD are mixed with more frequency components; when t = 0.17 s and 0.6 s, mode-mixing occurs between IMF₁ and IMF₂ (see Figure 15(c)). IMF₁ from IMMD restores the FM signal well; except for the end, IMF₂ also restores the AM signal well, but more frequency components are mixed (see Figure 15(d)). In addition, the residual components from the two methods restore the linear signal well. Therefore, IMMD alleviates mode-mixing.

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All IO_jk and IO values of Sig₂ decomposed by IMMD are smaller than that by EMD (see Table 6). It is verified that IMFs from IMMD are orthogonal, and IMMD has better orthogonality than EMD.

5.1.3. Two-Tone Signal

Compared with EMD, this section uses a commonly used discrete-time two-tone signal to evaluate the decomposition capability of IMMD. The two-tone signal is as follows:where a ∈ [0.01, 100] is the amplitude ratio and f ∈ (0, 1) is the frequency ratio of low-frequency (LF) component, , and high-frequency (HF) component, . For the convenience of discussion, let [28]. The criterion for judging the correct separation of two components is as follows:

IMF₁ is the first IMF extracted from Sig₃ with n-th siftings and stands for the Euclidean norm on functions defined over [0, T]. The value of equation (23) equals 0 when the two components are correctly separated; it is close to 1 when the two components are badly separated.

However, this criterion has one obvious drawback: there may be some discrete data points with large errors at the ends of IMF₁, which makes the value of equation (23) far greater than 1, especially when a is small (for example, in Figure 14(b), the error between IMF and the FM signal is great only at the ends, so that the value of equation (23) is much greater than 1, but the high- and low-frequency components are well separated). Therefore, we change the criterion to the correlation coefficients described in the following equation:

If the HF and LF components are separated correctly, the difference between IMF₁ and HF does not contain any information of LF, and the value of ρ is 0. When they are not well separated, the difference between IMF₁ and HF is close to LF and the value of ρ is close to 1. Threshold 0.5 is used to determine whether HF and LF are separated.

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Based on equation (24), Figures 16(c) and 16(d) show the performance of the EMD method for separating two-tone signals. It is almost identical with the performance measure figures (see Figures 16(a) and 16(b)) using equation (23) in [41], but Figures 16(c) and 16(d) have more high and low peaks, representing more details of decomposition. Equation (24) is better as a new criterion.

When EMD and IMMD decompose Sig₃, there is a critical cutoff frequency ratio (see Figure 16). If f exceeds the cutoff frequency ratio, no matter what the amplitude ratio is, the two components cannot be separated. The cutoff frequency ratio from EMD is about 0.65 (0.67 [41] and 0.65 [28]), and that from IMMD is about 0.95. The transition areas (; ) [41] of the two methods are almost identical. Obviously, in the area where the critical frequency ratio is 0.65∼0.95, IMMD can successfully separate two-tone signals against mode-mixing. Therefore, IMMD shows superior orthogonal decomposition capability than EMD.

5.1.4. Three-Tone Signal

The cutoff frequency ratio for two-tone signals decomposed by IMMD is greater than that by EMD. Therefore, we particularly construct the two-tone signal with a frequency ratio of 1 : 0.67, to evaluate the decomposition performance of the IMMD method in detail. Furthermore, as there is a dyadic filter bank of EMD, a signal whose frequency is 0.5 times of frequency of the HF signal is superposed on the signal, which increases the difficulty of separation. The three-tone signal model is as follows:

Let two methods decompose Sig₄ into Sig₃ IMFs and a residual component r. The Fourier spectra of three IMFs are shown in Figure 17.

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(c)

(d)

In the results of Sig₄ decomposed by EMD (see Figures 17(a) and 17(c)), tone with a frequency of 10 Hz only exists in IMF₁. Undoubtedly, EMD breaks the tone with a frequency of 6.7 Hz into two parts, one in IMF₁ and the other in IMF₂. To our surprise, the tone with a frequency of 5 Hz does not exist only in IMF₂, but is broken in both IMF₁ and IMF₂. So there is severe mode-mixing between IMF₁ and IMF₂ from EMD. IMF₃ is a pseudocomponent from EMD. On the contrary, although the three sinusoidal signals in the frequency domain have mode-mixing, the amount of mixing is very small. IMMD separates three-tone signal well (see Figures 17(b) and 17(d)).

Finally, Table 7 shows the orthogonal values of Sig₄ decomposed by two methods. It is verified that IMFs from IMMD are orthogonal. Although some IO_jk and IO values from EMD are better than those from IMMD, the decomposition of Sig₄ by the EMD method is a failure. Therefore, there is no doubt that the orthogonal decomposition capability of IMMD is better than that of EMD about Sig₄.

5.2. Two Commonly Used Realistic Nonstationary and Nonlinear Signals

In this section, we will conduct experiments on two commonly used real-world signals to compare the decomposition performance of the three common EMD domain methods of EMD, EEMD, and CEEMDAN with the IMMD method, including orthogonal performance and mode-mixing.

5.2.1. GSTA

The IMFs of GSTA decomposed by EMD, EEMD, CEEMDAN, and IMMD methods are shown in Figure 18.

(a)

(b)

(c)

(d)

We carry out a statistical significance test on IMFs of annual GSTA with the posteriori test method proposed in [42–44]. The premise of applying the posteriori test method is that IMF₁ of any well-sampled data is almost always the noise, so the noise contained in the data was estimated based on the IMF₁ [42]. In Figure 19, the solid line is the expectation of variance of IMFs of the white noise, and IMF₁ of white noise contains the same variance as the IMF₁ of GSTA (the expected case); the upper (lower) dotted line is the expectation of variance of IMFs of white noise of three (one-third) times that of the expected case [42]; the upper (lower) dash-dotted line is the expectation of variance of IMFs of white noise of two (one-second) times that of the expected case. The results show that IMMD and EMD each have 4 IMFs and EEMD and CEEMDAN each have 3 IMFs which are useful information of the data because they are beyond the three-time variance of the noise. EEMDʼs IMF₄ and IMMDʼs IMF₅ are also useful information for data, but it is beyond the twice variance of noise. In addition, there is a pseudocomponent IMF₆ (with minimal energy) from EMD.

(a)

(b)

(c)

(d)

For IMFs with insignificant noise in the statistical sense, it is indicated that these IMFs contain physically meaningful information, that is, various trends of GSTA [42] (see Figure 20). EMD obtains the overall trend and the change trend about 60 years of GSTA [42]; in addition to the above two trends, EEMD also obtains a trend of about 20 years of GSTA [42]; CEEMDAN obtains a suspected noised 60-year trend of GSTA; in addition to the above three different trends, IMMD also obtains a linear trend of GSTA [42]. Because the decomposition of GSTA by IMMD is adaptive and orthogonal, the number of mode components obtained is more, so it can effectively alleviate mode-mixing (remarkably, the mixing of IMF₉ and r of IMMD is approximately the r component of the other three methods), making GSTA have more physical meaning.

(a)

(b)

(c)

(d)

Tables 8 and 9 give the IO_jk values of the first 7 IMFs of GSTA, which verifies that GSTA is orthogonally decomposed by IMMD and posteriori orthogonally decomposed by the other three methods. IO_j6 and IO₆₇ from EMD are very small, which are caused by the pseudocomponent IMF₆, and have no practical significance. The number of IMFs from IMMD is more than that from the other three methods, resulting in more leakage, so most of the IO_jk values from IMMD are greater than that from the other three methods. Table 10 shows the IO value of GSTA. Among them, EEMD has the best IO value, followed by EMD, IMMD, and CEEMDAN. Except for IMMD, the order of the magnitude of the IO value corresponds to the ability of the other three methods to decompose GSTA (the number of trends). We believe that if we evaluate the decomposition orthogonality of the signal, then the decomposition result with better practical physical meaning should be the most important. Therefore, IMMD has better orthogonal decomposition capability than the other three methods.

In addition, Table 10 also shows the time-consuming of GSTA decomposed by four methods related procedures; EMD has the least time-consuming, followed by IMMD, EEMD, and CEEMDAN.

5.2.2. Electrocardiogram (ECG) Signal

ECG signal is a typical nonlinear and nonstationary weak signal. Muscle artifact (MA) and baseline wander (BW) are the main noises of ECG signal. Figure 21 shows the original ECG100 (numbered as 100 in the database), MA, BW waveforms [52], and the noised ECG100 waveform formed by superimposing these three signals, and Figure 22 shows all IMFs of the noised ECG100 decomposed by EMD, EEMD, CEEMDAN, and IMMD. In ECG reconstruction, IMF₁ contains a lot of high-frequency noise, which is generally recognized as MA, so it is often removed [53–56]. For the remaining IMFs, the QRS characteristic wave is used to identify the mode components of ECG. IMF₂∼IMF₅ from EMD, IMF₂∼IMF₅ from EEMD, IMF₂∼IMF₇ from CEEMDAN, and IMF₂∼IMF₁₁ from IMMD contain obvious QRS waves, which are identified as mode components of ECG100 and used to reconstruct ECG100.

(a)

(b)

(c)

(d)

Figure 23 shows the waveform and spectrum of the ECG100 reconstructed by four methods. In the frequency domain, the four methods have removed BW well, and for low-frequency BW with f < 1 Hz, IMMD has the best elimination effect; the best method to eliminate MA is EMD, and the worst is CEEMDAN, which may be caused by the largest amount of auxiliary noise in CEEMDAN. In the time domain, except for the IMMD, the first, second, third, and sixth characteristic T waves of ECG100 reconstructed by the other three methods have all moved forward, so pseudo-T waves are generated, which may cause doctors to misdiagnose.

In order to quantitatively evaluate the performance of the four methods, the correlation coefficient R is used to quantitatively describe the reconstruction accuracy, the signal-to-noise ratio (SNR) and the mean square error (MSE) are used to quantitatively describe the ability of denoising, and the corresponding formulas are as follows:where x [n] is ECG100 and y [n] is ECG100 reconstructed by methods. The three characteristic quantities of each of the four methods are shown in Table 11. The ECG100 reconstructed by IMMD has the largest ρ and SNR values and the smallest MSE value. Therefore, IMMD causes the smallest ECG100 distortion, its denoising performance is the best, and it shows the best reconstruction performance, followed by CEEMDAN, EEMD, and EMD. IMMD has better orthogonal decomposition capability than the other three methods, noise can be well separated from ECG100, and mode-mixing can be alleviated.

In addition, the related procedure of noised ECG100 decomposed by EMD has the least time-consuming, followed by IMMD, EEMD, and CEEMDAN.

Tables 12–15 show the IO_jk values of the first 11 IMF components of noised ECG100, which verifies that the IMMD method decomposing complex ECG is orthogonal and that the other three methods decomposing ECG have posterior orthogonality. In addition, most of the IO_jk values of IMMD are larger than those of the other three methods. This is because the number of IMFs from IMMD is nearly 1.7 times more than that from the other three methods, which causes more leakage. Table 11 shows the IO value of ECG100 decomposed by the four methods. IMMD has the best IO value, followed by EMD, CEEMDAN, and EEMD.

6. Discussions and Conclusions

6.1. Discussions

In theory, the signal is orthogonally decomposed by IMMD. The mean value by the envelope used in EMD and its existing improved methods is difficult to be expressed mathematically, so the decomposition orthogonality cannot be proved. IMMD overcomes this problem by using local integral to calculate the mean. Theoretical proof and experiments involving signals in manuscripts verify that the decomposition of the signal by IMMD is orthogonal, but this needs to be verified by more signals, especially real signals.

Sifting affects the orthogonal decomposition of the signal by IMMD method. With the increase of the sifting number, (1) the smoothing of the mean curve leads to the decrease of orthogonality and (2) IMF symmetry leads to the enhancement of orthogonality. For a certain two mode components, the orthogonality may decrease with the increase of sifting number, or even to the extent that the two mode components are no longer considered to be orthogonal, but it is precisely this way that the IMMD (or EMD) method can separate two mode components that are not strictly orthogonal (for example, signal Sig₂). Therefore, the role of sifting needs to be studied more deeply.

Because the signal is orthogonally decomposed by IMMD, mode-mixing can be alleviated. If the signal is orthogonally decomposed, the mode components of the signal are orthogonal, and there is no mode-mixing between them, so IMMD can alleviate mode-mixing. The Monte Carlo method of Gaussian white noise and the signal experiments given in this study show that IMMD effectively alleviates mode-mixing and has better multiresolution analysis characteristics. In particular, the characteristic of IMMD equivalent constant-Q filter bank is trisection, which is different from EMD, EEMD, etc.

PMF from IMMD has good robustness to the sifting, but it also needs to be verified by more signals, especially real signals. Due to the envelope mean method, the PMF from EMD, EEMD, and CEEMDAN is weakly convergent to the sifting, and the PMF will continue to change with the increase of sifting number and eventually become uniform in amplitude (the problem of oversifting), and IMF loses the substantial significance of amplitude modulation [57]. In addition, due to the influence of auxiliary noise, any two decomposition experiments of the same signal by EEMD or CEEMDAN have small differences in the two decomposition results [20]. If the signal needs to be decomposed very accurately, this small difference is very likely to cause wrong conclusions.

In short, compared with the EMD, EEMD, and CEEMDAN methods, the decomposition of signal by the IMMD method is orthogonal, and the number of IMFs from IMMD is more, so it has better multiresolution analysis capabilities, and the decomposition has better physical meaning; IMF from IMMD is robust to the sifting and will not cause oversifting problem; in addition, IMMD can effectively alleviate mode-mixing; compared to EEMD and CEEMDAN, IMMD does not require any parameter settings, and the decomposition is data-driven and adaptable; due to the large number of IMFs produced by IMMD, the time-consuming of IMMD-related procedures is greater than that of EMD, but it is much less than that of EEMD and CEEMDAN procedures, which is caused by the hundreds of auxiliary noises of them.

6.2. Conclusions

An improved method of EMD, IMMD, is proposed in this study. Compared with EMD, EEMD, and CEEMDAN, IMMD has the following advantages: smoothing of the mean curve is ignored, IMMD method decomposes the signal orthogonally; without any input parameters, IMMD can effectively alleviate mode-mixing; and IMMD method has good robustness to sifting.

We suggest that IMMD is suitable for nonlinear and nonstationary signal research fields that require multiresolution and high stability, such as geophysics and bioelectric signal processing.

Data Availability

The data used to support the findings of this study are publicly available on the Internet, and the detailed website address or information is given in the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundations of China (Grant nos. 61571404, 61842103, 61871351, and 61801437) and Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi, China (Grant nos. 2020L0301 and 2020L0389).

References

N. E. Huang, Z. Shen, and S. R. Long, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proceedings of Royal Society A, vol. 454, pp. 903–995, 1998.
View at: Publisher Site | Google Scholar
X.-D. Niu, L.-R. Lu, J. Wang, X.-C. Han, S-Y. Guo, and L.-M. Wang, “Electrocardiogram signal reconstruction based on mode component identification by heartbeat physical feature in an improved emperical mode decomposition domain,” Acta Physica Sinica, vol. 70, no. 3, Article ID 038702, 2021.
View at: Google Scholar
J. Zheng, H. Chu, D. Struppa et al., “Optimal multi-stage arrhythmia classification approach,” Scientific Reports, vol. 10, no. 1, 2020.
View at: Publisher Site | Google Scholar
C. Varon, J. Morales, J. Lázaro et al., “A comparative study of ECG-derived respiration in ambulatory monitoring using the single-lead ECG,” Scientific Reports, vol. 10, 2020.
View at: Google Scholar
A. F. T. Borges, M. Irrmischer, T. Brockmeier, D. J A. Smit, H. D Mansvelder, and K. Linkenkaer-Hansen, “Scaling behaviour in music and cortical dynamics interplay to mediate music listening pleasure,” Scientific Reports, vol. 9, no. 1, 2019.
View at: Publisher Site | Google Scholar
D. Abboud, Y. Marnissi, and M. Elbadaoui, “Optimal filtering of angle-time cyclostationary signals: application to vibrations recorded under nonstationary regimes,” Mechanical Systems and Signal Processing, vol. 145, Article ID 106919, 2020.
View at: Publisher Site | Google Scholar
H. Sun, “High-resolution anisotropic prestack Kirchhoff dynamic focused beam migration,” IEEE Sensors Journal, vol. 99, no. 1, 2019.
View at: Google Scholar
B. Karan, S. S. Sahu, and K. Mahto, “Parkinson disease prediction using intrinsic mode function based features from speech signal,” Biocybernetics and Biomedical Engineering, vol. 40, no. 1, pp. 249–264, 2020.
View at: Publisher Site | Google Scholar
A. T. Eseye and M. Lehtonen, “Short-term forecasting of heat demand of buildings for efficient and optimal energy management based on integrated machine learning models,” IEEE Transactions on Industrial Informatics, vol. 99, no. 1, 2020.
View at: Google Scholar
Z. Jin, Y. Yang, and Y. Liu, “Stock closing price prediction based on sentiment analysis and LSTM,” Neural Computing and Applications, vol. 32, no. 3, 2020.
View at: Publisher Site | Google Scholar
N. E. Huang and Z. Wu, “A review on Hilbert Huang transform: method and its applications to geophysical studies,” Reviews of Geophysics, vol. 46, no. 2, 2008.
View at: Publisher Site | Google Scholar
K. J. Moore, M. Kurt, M. Eriten et al., “Wavelet-bounded empirical mode decomposition for vibro-impact analysis,” Nonlinear Dynamics, vol. 93, no. 3, pp. 1559–1577, 2018.
View at: Google Scholar
S. Kizhner, K. Blank, T. Flatley et al., “On certain theoretical developments underlying the hilbert-huang transform,” in Proceedings of the IEEE Conference on Aerospace, 2006.
View at: Google Scholar
R. Hayashi, “Subharmonic response of polymer contrast agents based on the empirical mode decomposition,” IEEE Transactions on Ultrasonics Ferroelectrics & Frequency Control, vol. 99, p. 1, 2016.
View at: Google Scholar
C. Wang, Q. Kemao, and F. Da, “Regenerated phase-shifted sinusoid-assisted empirical mode decomposition,” IEEE Signal Processing Letters, vol. 23, no. 4, pp. 556–560, 2016.
View at: Publisher Site | Google Scholar
W. Rongxi, “Hilbert-Huang transform based pseudo-periodic feature extraction of nonlinear time series,” in Proceedings of the 2015 7th International Conference on Measuring Technology and Mechatronics Automation IEEE, Nanchang, China, June 2015.
View at: Google Scholar
T.-F. Hsu, K.-H. Jao, and Y.-C. Hung, “Phase synchronization in a two-mode solid state laser: periodic modulations with the second relaxation oscillation frequency of the laser output,” Physics Letters A, vol. 378, no. 44, pp. 3269–3273, 2014.
View at: Publisher Site | Google Scholar
H. R. Singh, “Real-time implementation of signal processing techniques for disturbances detection,” IEEE Transactions on Industrial Electronics, no. 1, 2018.
View at: Google Scholar
D. Zhang, H. Y. Zhou, and Y. Xue, “A novel facial feature extraction method based on Empirical Mode Decomposition,” in Proceedings of the International Conference on Machine Learning & Cybernetics IEEE, Lanzhou, China, July 2014.
View at: Google Scholar
M. A. Motin, C. K. Karmakar, and M. Palaniswami, “Selection of empirical mode decomposition techniques for extracting breathing rate from PPG,” IEEE Signal Processing Letters, vol. 26, no. 4, pp. 592–596, 2019.
View at: Publisher Site | Google Scholar
W. U. Zhaohua and N. E. Huang, “Ensemble empirical mode decomposition: a noise-assisted data analysis method,” Advances in Adaptive Data Analysis, vol. 1, no. 1, pp. 1–41, 2009.
View at: Google Scholar
J.-R. Yeh, J.-S. Shieh, and N. E. Huang, “Complementary ensemble empirical mode decomposition: a novel noise enhanced data analysis method,” Advances in Adaptive Data Analysis, vol. 2, no. 2, pp. 135–156, 2010.
View at: Publisher Site | Google Scholar
M. E. Torres, M. A. Colominas, G. Schlotthauer, and P. Flandrin, “A complete ensemble empirical mode decomposition with adaptive noise,” in Proceedings of the IEEE International Conference on Acoustics IEEE, Prague, Czech Republic, May 2011.
View at: Google Scholar
J. Gilles, “Empirical wavelet transform,” IEEE Transactions on Signal Processing, vol. 61, no. 16, pp. 3999–4010, 2013.
View at: Publisher Site | Google Scholar
K. Dragomiretskiy and D. Zosso, “Variational mode decomposition,” IEEE Transactions on Signal Processing, vol. 62, no. 3, pp. 531–544, 2014.
View at: Publisher Site | Google Scholar
P. Jain and R. B. Pachori, “An iterative approach for decomposition of multi-component non-stationary signals based on eigenvalue decomposition of the Hankel matrix,” Journal of the Franklin Institute, vol. 352, no. 10, 2015.
View at: Publisher Site | Google Scholar
H. Li, Z. Li, and W. Mo, “A time varying filter approach for empirical mode decomposition,” Signal Processing, vol. 138, pp. 146–158, 2017.
View at: Publisher Site | Google Scholar
J. Zheng, H. Pan, T. Liu, and Q. Liu, “Extreme-point weighted mode decomposition,” Signal Processing, vol. 142, pp. 366–374, 2018.
View at: Publisher Site | Google Scholar
J. Zheng, J. Cheng, and Y. Yang, “A rolling bearing fault diagnosis approach based on LCD and fuzzy entropy,” Mechanism and Machine Theory, vol. 70, no. 6, pp. 441–453, 2013.
View at: Publisher Site | Google Scholar
Y. Kopsinis and S. Mclaughlin, “Investigation and performance enhancement of the empirical mode decomposition method based on a heuristic search optimization approach,” IEEE Transactions on Signal Processing, vol. 56, no. 1, pp. 1–13, 2008.
View at: Publisher Site | Google Scholar
J. S. Smith and S. Jonathan, “The local mean decomposition and its application to EEG perception data,” Journal of the Royal Society Interface, vol. 2, no. 5, pp. 443–454, 2005.
View at: Publisher Site | Google Scholar
Z. Wu and N. E. Huang, “A study of the characteristics of white noise using the empirical mode decomposition method,” Proceedings Mathematical Physical & Engineering ences, vol. 460, no. 2046, pp. 1597–1611, 2004.
View at: Publisher Site | Google Scholar
N. E. Huang, E. Huang, M. L. C. Wu et al., “A confidence limit for the empirical mode decomposition and Hilbert spectral analysis,” Proceedings Mathematical Physical & Engineering ences, vol. 459, no. 2037, pp. 2317–2345, 2003.
View at: Publisher Site | Google Scholar
H. Hong, X. Wang, and Z. Tao, “Local integral mean-based sifting for empirical mode decomposition,” IEEE Signal Processing Letters, vol. 16, no. 10, pp. 841–844, 2009.
View at: Google Scholar
Intergovernmental Panel on Climate Change, Climate Change 2001: The Scientific Basis–Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge Univ Press, Cambridge, UK, 2001.
P. Flandrin, G. Rilling, and P. Goncalves, “Empirical mode decomposition as a filter bank,” IEEE Signal Processing Letters, vol. 11, no. 2, pp. 112–114, 2004.
View at: Publisher Site | Google Scholar
H. Li, C. Wang, and D. Zhao, “Filter bank properties of envelope modified EMD methods,” Iet Signal Processing, vol. 12, no. 7, pp. 844–851, 2018.
View at: Publisher Site | Google Scholar
H. Li, J. Zhao, X. Zhang, and H. Teng, “Gear crack level classification based on EMD and EDT,” Mathematical Problems in Engineering, vol. 2015, Article ID 150127, 10 pages, 2015.
View at: Publisher Site | Google Scholar
D. Zhao, Z. Huang, H. Li, J. Chen, and P. Wang, “An improved EEMD method based on the adjustable cubic trigonometric cardinal spline interpolation,” Digital Signal Processing, vol. 64, pp. 41–48, 2017.
View at: Publisher Site | Google Scholar
P. Flandrin and G. O. Paulo, “Empirical mode decompositions as data-driven wavelet-like expansions,” International Journal of Wavelets Multiresolution & Information Processing, vol. 2, no. 4, pp. 477–496, 2008.
View at: Google Scholar
G. Rilling and P. Flandrin, “One or two frequencies−the empirical mode decomposition answers,” IEEE Transactions on Signal Processing, vol. 56, no. 1, pp. 85–95, 2008.
View at: Publisher Site | Google Scholar
Z. Wu, N. E. Huang, S. R. Long, and C.-K. Peng, “On the trend, detrending, and variability of nonlinear and nonstationary time series,” Proceedings of the National Academy of Sciences, vol. 104, no. 38, pp. 14889–14894, 2007.
View at: Publisher Site | Google Scholar
N. E. Huang, Z. Wu, J. E. Pinzón et al., “Reductions OF noise and uncertainty IN annual global surface temperature anomaly data,” Advances in Adaptive Data Analysis, vol. 1, no. 3, pp. 447–460, 2009.
View at: Publisher Site | Google Scholar
N. E. Huang and S. S. P. Shen, Hilbert-Huang Transform and its Applications, World Scientific, Singapore, 2005.
G. B. Moody and R. G. Mark, “The impact of the MIT-BIH arrhythmia database,” IEEE Engineering in Medicine and Biology Magazine, vol. 20, no. 3, pp. 45–50, 2001.
View at: Publisher Site | Google Scholar
H.-T. Chiang, Y.-Y. Hsieh, S.-W. Fu, K.-H. Hung, Y. Tsao, and S.-Y. Chien, “Noise reduction in ECG signals using fully convolutional denoising autoencoders,” IEEE Access, vol. 7, pp. 60806–60813, 2019.
View at: Publisher Site | Google Scholar
R. N. Vargas and A. C. Paschoarelli Veiga, “Electrocardiogram signal denoising by clustering and soft thresholding,” IET Signal Processing, vol. 12, no. 9, 2018.
View at: Publisher Site | Google Scholar
U. Satija, B. Ramkumar, and M. S. Manikandan, “Automated ECG noise detection and classification system for unsupervised healthcare monitoring,” IEEE Journal of Biomedical and Health Informatics, vol. 22, no. 3, pp. 722–732, 2018.
View at: Publisher Site | Google Scholar
R. Arumugam and J. Jebaraj, “Ensemble empirical mode decomposition-based optimised power line interference removal algorithm for electrocardiogram signal,” Iet Signal Processing, vol. 10, pp. 583–591, 2016.
View at: Google Scholar
B. H. Tracey and E. L. Miller, “Nonlocal means denoising of ECG signals,” IEEE Transactions on Biomedical Engineering, vol. 59, no. 9, p. 2383, 2012.
View at: Publisher Site | Google Scholar
A. Karagiannis and P. Constantinou, “Noise components identification in biomedical signals based on empirical mode decomposition,” in Proceedings of the 9th International Conference on IEEE Information Technology and Applications in Biomedicine (ITAB 2009), 2009.
View at: Google Scholar
G. B. Moody, W. E. Muldrow, and R. G. Mark, “A noise stress test for arrhythmia detectors,” Computers in Cardiology, vol. 11, pp. 381–384, 1984.
View at: Google Scholar
N. Ibtehaz, M. S. Rahman, and M. S. Rahman, “VFPred: a fusion of signal processing and machine learning techniques in detecting ventricular fibrillation from ECG signals,” Biomedical Signal Processing and Control, vol. 49, pp. 349–359, 2019.
View at: Google Scholar
D. Jarchi and A. J. Casson, “Towards photoplethysmography-based estimation of instantaneous heart rate during physical activity,” IEEE Transactions on Biomedical Engineering, vol. 64, no. 9, pp. 2042–2053, 2017.
View at: Publisher Site | Google Scholar
A. Karagiannis and P. Constantinou, “Noise-assisted data processing with empirical mode decomposition in biomedical signals,” IEEE Transactions on Information Technology in Biomedicine, vol. 15, no. 1, pp. 11–18, 2011.
View at: Publisher Site | Google Scholar
M. Blanco-Velasco, F. Cruz-Roldán, J. I. Godino-Llorente, and K. E. Barner, “Nonlinear trend estimation of the ventricular repolarization segment for T-wave alternans detection,” IEEE Transactions on Biomedical Engineering, vol. 57, no. 10, pp. 2402–2412, 2010.
View at: Publisher Site | Google Scholar
G. Wang, “ON intrinsic mode function,” Advances in Adaptive Data Analysis, vol. 2, no. 3, pp. 277–293, 2010.
View at: Google Scholar

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Copyright © 2021 Xiao-dong Niu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Mathematical Problems in Engineering

An Improved Empirical Mode Decomposition Based on Local Integral Mean and Its Application in Signal Processing

Abstract

1. Introduction

2. IMMD Method

2.1. Steps

2.2. The Stoppage Criterion

2.3. Method of Alleviating End Effects

3. The Proof of IMMD Orthogonal Decomposition

3.1. The Proof of Orthogonality of IMFj and rj

3.1.1. The Proof

3.1.2. The Effect of Sifting on the Orthogonality of IMFj and rj

3.2. The Proof of Orthogonality of IMFj and rk (j < k ≤ n)

3.2.1. The Proof

3.2.2. Index of Orthogonality of IMFj and rk

3.2.3. The Effect of Sifting on the Orthogonality of IMFj and rk

3.3. The Proof of Orthogonality of IMFj and IMFk

3.3.1. The Proof

3.3.2. Index of Orthogonality of IMFj and IMFk

3.3.3. The Effect of Sifting on the Orthogonality of IMFj and IMFk

4. Alleviation of Mode-Mixing

5. Experiments and Results

5.1. Multicomponent Nonstationary Signals

5.1.1. Example 1

5.1.2. Example 2

5.1.3. Two-Tone Signal

5.1.4. Three-Tone Signal

5.2. Two Commonly Used Realistic Nonstationary and Nonlinear Signals

5.2.1. GSTA

5.2.2. Electrocardiogram (ECG) Signal

6. Discussions and Conclusions

6.1. Discussions

6.2. Conclusions

Data Availability

Conflicts of Interest

Acknowledgments

References

Copyright

3.1. The Proof of Orthogonality of IMF_j and r_j

3.1.2. The Effect of Sifting on the Orthogonality of IMF_j and r_j

3.2. The Proof of Orthogonality of IMF_j and r_k (j < k ≤ n)

3.2.2. Index of Orthogonality of IMF_j and r_k

3.2.3. The Effect of Sifting on the Orthogonality of IMF_j and r_k

3.3. The Proof of Orthogonality of IMF_j and IMF_k

3.3.2. Index of Orthogonality of IMF_j and IMF_k

3.3.3. The Effect of Sifting on the Orthogonality of IMF_j and IMF_k