# Morphological Filter-Assisted Ensemble Empirical Mode Decomposition

Authors: Xiaohang Zhou; Deshan Shan; Qiao Li
Journal: Mathematical Problems in Engineering
 (2018)

---

## Abstract

In the ensemble empirical mode decomposition (EEMD) algorithm, different realizations of white noise are added to the original signal as dyadic filter banks to overcome the mode mixing problems of empirical mode decomposition (EMD). However, not all the components in white noise are necessary, and the superfluous components will introduce additional mode mixing problems. To address this problem, morphological filter-assisted ensemble empirical mode decomposition (MF-EEMD) was proposed in this paper. First, a new method for determining the structuring element shape and size was proposed to improve the adaptive ability of morphological filter (MF). Then, the adaptive MF was introduced into EMD to remove the superfluous white noise components to improve the decomposition results. Based on the contributions of MF in a single EMD process, the MF-EEMD was proposed by combining EEMD with MF to suppress the mode mixing problems. Finally, an analog signal and a measured signal were used to verify the feasibility of MF-EEMD. The results show that MF-EEMD significantly mitigates the mode mixing problems and achieves a higher decomposition efficiency compared to that of EEMD.


---

## Body

## 1. Introduction

Signal decomposition techniques, such as Fourier transform (FT), wavelet transform (WT), and empirical mode decomposition (EMD), are widely used in the field of structural health monitoring for denoising, parameter identification, and damage detection [1–3]. However, these techniques have considerable deficiencies. FT [4] takes the signal as a linear combination of sine waves with different frequencies, but the frequency and amplitude of the sine wave are immutable. Consequently, FT cannot analyze nonstationary or nonlinear signals [5]. Short-time Fourier transform (STFT) is a variation of FT, and it can address nonstationary signals by employing the window function technique. However, it is hard to balance the resolution between the time domain and the frequency domain [6]. WT [7] is suitable for decomposing nonlinear signals with multiple scales and multiple resolutions, but determining the appropriate mother wavelet function is difficult. Hence, the adaptive performance of WT is poor. EMD is a fully data-driven technique with the ability to decompose nonstationary and nonlinear signals; thus, it possesses strong robustness and adaptive ability [5]. However, the mode mixing problems in EMD severely restrict its application [8, 9].To overcome the mode mixing problem of EMD, Wu and Huang [10] proposed the noise-assisted ensemble empirical mode decomposition (EEMD). In the EEMD algorithm, EMD is performed repeatedly with the addition of different white noises, and an ensemble of the same order intrinsic mode functions (IMF) is made to obtain the final decomposition results. Flandrin and Rilling [11] noted that the white noise added in EEMD could be taken as a set of dyadic filter banks, and different frequency components in the original signal were projected to the corresponding banks. This characteristic of white noise helped to separate the different orders of IMFs.The valuable components in the original signal are typically limited in quantity; in other words, the number of essential dyadic filter banks for EEMD can be counted. However, the added white noise contains all the dyadic filter banks in the frequency domain. The superfluous dyadic filter banks added to the original signal will not only reduce the decomposition efficiency but also cause additional mode mixing problems [12]. Based on the above problems, it could be speculated that the accuracy and efficiency of EEMD would be improved if the superfluous dyadic filter banks were removed.The morphological filter (MF) is a type of nonlinear filter that is particularly sensitive to impulse noise and white noise; moreover, it will not cause sudden truncation in the frequency domain or phase delay in the time domain [13, 14]. Deng [15] adopted the MF along with other traditional filters to filter impulse noise, and the results demonstrated that the MF was superior to the other filters, especially when the noise level was high. Yuan [16] adopted the MF and a hybrid filter to detect and eliminate impulse noise in images, and superb denoising results were achieved for different signal-to-noise ratios. Lin [17] used the MF to eliminate the high-ratio impulse noise, and the MF achieved a better performance than other traditional digital filters, such as the mean filter and the median filter. Saniie [14] adopted the MF to suppress the impulse noise and the white noise in an ultrasonic signal; the MF could significantly improve the signal-to-noise ratio and maintain an undistorted signal. Mukhopadhyay [18] adopted the MF to filter different types of noise, and the results showed that the MF was effective for all types of noise and superior to other standard noise removal algorithms. The above literature review illustrates that the MF is a potentially powerful tool for filtering superfluous dyadic filter banks in EEMD.This paper proposes the morphological filter-assisted ensemble empirical mode decomposition (MF-EEMD) to overcome the mode mixing problems in EEMD. First, the concepts of MF were presented, and the adaptive ability of MF was improved by proposing a new method for determining the structure element shape and size. Then, the adaptive MF was applied to remove the superfluous white noise components in EMD to improve its decomposition results. Based on the contributions of MF in a single EMD process, the MF-EEMD was proposed by introducing MF into EEMD to suppress its mode mixing problems. Finally, MF-EEMD is compared with EEMD by decomposing an analog signal and a measured signal, and the effectiveness of MF-EEMD is validated.


## 2. MF Theory

In 1986, Serra [19, 20] proposed the MF based on the theory of mathematical morphology. The MF is a type of nonlinear temporal filter that can process a signal directly in the time domain without converting it into the frequency domain. In the MF algorithm, a structuring element is used to address the data points, and its common shapes are shown in Figure 1.Figure 1
Different shapes of structuring elements.
(a)
Linear (b)
Sinusoidal (c)
TriangularIn the algorithm of MF, dilation and erosion are the two most basic operations [21]. The dilation operation on signal α with structuring element β is defined as α⊕β, and the erosion operation on signal α with structuring element β is defined as α⊝β, as shown in (1) and (2), respectively.(1)α⊕β=max⁡αi-j+βj(2)α⊝β=min⁡αi+j-βjwhere 1<i<n, 1≤j≤k, i+j≤n,n is the length of α, and k is the width of β.Based on the dilation and erosion operations, the opening operation on signalα with structuring element β is defined as α∘β, and the closing operation on signal α with structuring element β is defined as α∙β, as shown in (3) and (4), respectively.(3)α∘β=α⊝β⊕β(4)α∙β=α⊕β⊝βHowever, the opening operation can only address the local maximums, while the closing operation can only address the local minimums [22]. To address both the local maximums and local minimums of the signal, the opening and closing operations should be combined by (5).(5)α′=α∘β∙βorα′=α∙β∘βTo eliminate the effect of different operational sequences, the average of opening-closing and closing-opening is taken as the final filtering result. The complete operational flow of the MF is shown in Figure2.Figure 2
Operational flow of the MF.


## 3. Adaptive MF

The shape and size of the structuring element are the two most important parameters in MF, and they have significant influences on the filtering results. The determination of the two parameters relates to multiple factors, such as the sample frequency, the dominant frequency, and the noise type. The shape and size of the structuring element are generally determined by empirical methods [14, 15], and the disadvantages of those methods are low adaptive ability and poor versatility.To construct an adaptive MF algorithm, the determination method of the appropriate structuring element parameters were investigated in this paper.A composite signal of sinusoid and white noise is constructed based on (6)-(8).(6)Ssin=λ·sin⁡2πf×nsf(7)Swn=k·randnsf(8)Sc=Ssin+Swnwhere Ssin is a pure sinusoid, Swn is white noise, Sc is the composite signal,sf is the sample frequency, n=1,2,3…N,N is the length of Sc,f is the dominant frequency, λ is the amplitude of the pure sinusoid, and k is the amplitude of the white noise.To make an objective evaluation of the effectiveness of the MF, the signal-to-noise ratio (SNR) [23] is introduced as the evaluation index, as shown in (9).(9)SNR=10×log⁡StdSsinStdSMF-Ssinwhere the operator Std() is the expression of standard deviation and SMF is the filtering result of Sc. When SMF and Ssin are more similar, the value ofSNR is greater.


### 3.1. Shape Selection

To investigate the characteristics of different structuring element shapes, a composite signal with one dominant frequency was constructed by (6)-(8) with the parameterssf=256 Hz,N=512,f=5 Hz, λ=1, andk=0.3, as shown in Figure 3.Figure 3
Composite signal with one dominant frequency.
(a)
S s i n (b)
S cThree types of structuring elements were constructed, each with linear, sinusoidal, or triangular shapes. The range of the structuring element width (W) was set as 3-35, and the range of the structuring element height (H) was set as 0.1-5 times the standard deviation of Sc. Then, the three types of structuring elements with varying sizes were adopted to filter the signal, and the correspondingSNR of each SMF was calculated. In Figure 4, the initialSNR of Sc is shown as a plane surface with a value of 14.35, and theSNR of SMF is shown as a curved surface.Figure 4
Filtering results with different structuring element shapes.
(a)
Linear (b)
Sinusoidal (c)
TriangularIn Figure4(a), the changes inSNR values only relate to W. The appropriate distribution range of W is 3-11, and the best filtering result is achieved with anSNR value of 19.56 whenW=5. In Figure 4(b), the changes inSNR values relate to both W andH.W and H have wide appropriate ranges, and the best filtering result is achieved with anSNR value of 23.72 whenW=11 andH=0.7. The filtering result in Figure 4(c) is similar to that in Figure 4(b), but the appropriate ranges of W and H are narrower. The best filtering result is achieved with anSNR value of 22.61 whenW=9 andH=0.5.Although the filtering results with different white noises are not constant due to the white noise randomness, still some patterns can be observed as follows. The best filtering results of the sinusoidal and triangular structuring elements are similar, and they are superior to that of the linear element. Additionally, the distribution range of the appropriate element sizes of the triangular structuring element is narrower than that of the sinusoidal structuring element and, thus, the distribution law of the appropriate element size of the triangular structuring element is more obvious.According to the above patterns, the triangular structuring element was selected for the follow-up studies in this paper.


### 3.2. Size Determination

To investigate the distribution laws of the appropriateW andH, multiple composite signals with one dominant frequency were constructed using (6)-(8). The corresponding parameter settings are listed in Table 1. A triangular structuring element with varying sizes was adopted to filter the signals, and the correspondingSNR values of each SMF were calculated, as shown in Figure 5.Table 1
Parameter settings ofSc with one dominant frequency.
No. sf f λ k #1 256 5 1 0.1 #2 256 5 1 0.3 #3 256 5 1 0.5 #4 512 4 1 0.3 #5 512 8 1 0.3 #6 512 16 1 0.3 #7 64 3 1 0.3 #8 128 3 1 0.3 #9 512 3 1 0.3Figure 5
Filtering results with varying structuring element sizes.In signals #1-3,k is the only parameter that is varied. The distribution range of appropriate sizes becomes wider with increasingk; however, the ridgeline (colored yellow) of theSNR surface remains relatively unchanged. In signals #4-6,f is the only parameter being varied. When f is small, the ridgeline is close to axisW, and vice versa. In signals #7-9,sf is the only parameter being varied, and the effect ofsf on the ridgeline is opposite that off.Figure5 illustrates that the location and direction of the ridgeline ofSNR surface are mainly dominated bysf and f. The intersection point of the ridgeline andH=0.1 has a larger value with increases insfor decreases in f. The value ofW/H also has the same relationship withsf and f. The above laws can be summarized as (10)-(12).(10)WH=0.1∽sff(11)W/H∽sff(12)W∽sff+H·sffA linear regression was performed to fit (12) using 650 samples with the value ofsf/f varying from 4 to 1,024. The fitting result is shown in (13).(13)W=0.33sff+0.24H·sffIn Figure6, three examples with different values ofsf/f were selected to demonstrate the prediction precision of (13). The black points are the predictions of the appropriate structuring element sizes, and they are very close to the ridgelines of theSNR surfaces, which indicates that the prediction of appropriate structuring element sizes has a high precision.Figure 6
Fitting results of the SNR ridgeline with one dominant frequency.
(a)
sf/f=16 (b)
sf/f=64 (c)
sf/f=256To investigate the applicability of (13) to signals with multiple dominant frequencies, three composite signals with two dominant frequencies were constructed with the parameter settings listed in Table 2. TheSNR values of the filtering results along with the predictions of the appropriate structuring element sizes are shown in Figure 7, where the blue points represent the predictions with fh and the red points represent the predictions with fl.Table 2
Parameter settings ofSc with two dominant frequencies.
No. sf f l f h λ l λ h k #1 256 8 16 0.4 0.6 0.3 #2 256 2 8 0.5 0.5 0.3 #3 256 4 32 0.6 0.4 0.3Figure 7
Fitting results of the SNR ridgeline with two dominant frequencies.
(a)
f l=8, fh=16 (b)
f l=2, fh=8 (c)
f l=4, fh=32Figure7 illustrates that the ridgeline of theSNR surface is between the two predictions but very close to the blue points, indicating that the distribution of appropriate structuring element sizes is mainly dominated by fh. Therefore, the ridgeline of theSNR surface can be approximately estimated by (13) with fh.In actual operation, one pair of appropriate size (W,H) is sufficient for the MF, and the value of W for the triangular structuring element isW=2n+1, which means that the smallest value of W is 3. In this paper, a simplified and practical size determination method for the triangular structuring element was proposed as follows: (1) Extract the highest valuable frequency f of the signal. (2) Introducesf and f into (13), increase H from 0.1 with a step length of 0.1, and calculate the corresponding odd-value ofW. (3) Stop the calculation whenW ≥ 3, and obtain one pair of appropriate size (W,H).


## 3.1. Shape Selection

To investigate the characteristics of different structuring element shapes, a composite signal with one dominant frequency was constructed by (6)-(8) with the parameterssf=256 Hz,N=512,f=5 Hz, λ=1, andk=0.3, as shown in Figure 3.Figure 3
Composite signal with one dominant frequency.
(a)
S s i n (b)
S cThree types of structuring elements were constructed, each with linear, sinusoidal, or triangular shapes. The range of the structuring element width (W) was set as 3-35, and the range of the structuring element height (H) was set as 0.1-5 times the standard deviation of Sc. Then, the three types of structuring elements with varying sizes were adopted to filter the signal, and the correspondingSNR of each SMF was calculated. In Figure 4, the initialSNR of Sc is shown as a plane surface with a value of 14.35, and theSNR of SMF is shown as a curved surface.Figure 4
Filtering results with different structuring element shapes.
(a)
Linear (b)
Sinusoidal (c)
TriangularIn Figure4(a), the changes inSNR values only relate to W. The appropriate distribution range of W is 3-11, and the best filtering result is achieved with anSNR value of 19.56 whenW=5. In Figure 4(b), the changes inSNR values relate to both W andH.W and H have wide appropriate ranges, and the best filtering result is achieved with anSNR value of 23.72 whenW=11 andH=0.7. The filtering result in Figure 4(c) is similar to that in Figure 4(b), but the appropriate ranges of W and H are narrower. The best filtering result is achieved with anSNR value of 22.61 whenW=9 andH=0.5.Although the filtering results with different white noises are not constant due to the white noise randomness, still some patterns can be observed as follows. The best filtering results of the sinusoidal and triangular structuring elements are similar, and they are superior to that of the linear element. Additionally, the distribution range of the appropriate element sizes of the triangular structuring element is narrower than that of the sinusoidal structuring element and, thus, the distribution law of the appropriate element size of the triangular structuring element is more obvious.According to the above patterns, the triangular structuring element was selected for the follow-up studies in this paper.


## 3.2. Size Determination

To investigate the distribution laws of the appropriateW andH, multiple composite signals with one dominant frequency were constructed using (6)-(8). The corresponding parameter settings are listed in Table 1. A triangular structuring element with varying sizes was adopted to filter the signals, and the correspondingSNR values of each SMF were calculated, as shown in Figure 5.Table 1
Parameter settings ofSc with one dominant frequency.
No. sf f λ k #1 256 5 1 0.1 #2 256 5 1 0.3 #3 256 5 1 0.5 #4 512 4 1 0.3 #5 512 8 1 0.3 #6 512 16 1 0.3 #7 64 3 1 0.3 #8 128 3 1 0.3 #9 512 3 1 0.3Figure 5
Filtering results with varying structuring element sizes.In signals #1-3,k is the only parameter that is varied. The distribution range of appropriate sizes becomes wider with increasingk; however, the ridgeline (colored yellow) of theSNR surface remains relatively unchanged. In signals #4-6,f is the only parameter being varied. When f is small, the ridgeline is close to axisW, and vice versa. In signals #7-9,sf is the only parameter being varied, and the effect ofsf on the ridgeline is opposite that off.Figure5 illustrates that the location and direction of the ridgeline ofSNR surface are mainly dominated bysf and f. The intersection point of the ridgeline andH=0.1 has a larger value with increases insfor decreases in f. The value ofW/H also has the same relationship withsf and f. The above laws can be summarized as (10)-(12).(10)WH=0.1∽sff(11)W/H∽sff(12)W∽sff+H·sffA linear regression was performed to fit (12) using 650 samples with the value ofsf/f varying from 4 to 1,024. The fitting result is shown in (13).(13)W=0.33sff+0.24H·sffIn Figure6, three examples with different values ofsf/f were selected to demonstrate the prediction precision of (13). The black points are the predictions of the appropriate structuring element sizes, and they are very close to the ridgelines of theSNR surfaces, which indicates that the prediction of appropriate structuring element sizes has a high precision.Figure 6
Fitting results of the SNR ridgeline with one dominant frequency.
(a)
sf/f=16 (b)
sf/f=64 (c)
sf/f=256To investigate the applicability of (13) to signals with multiple dominant frequencies, three composite signals with two dominant frequencies were constructed with the parameter settings listed in Table 2. TheSNR values of the filtering results along with the predictions of the appropriate structuring element sizes are shown in Figure 7, where the blue points represent the predictions with fh and the red points represent the predictions with fl.Table 2
Parameter settings ofSc with two dominant frequencies.
No. sf f l f h λ l λ h k #1 256 8 16 0.4 0.6 0.3 #2 256 2 8 0.5 0.5 0.3 #3 256 4 32 0.6 0.4 0.3Figure 7
Fitting results of the SNR ridgeline with two dominant frequencies.
(a)
f l=8, fh=16 (b)
f l=2, fh=8 (c)
f l=4, fh=32Figure7 illustrates that the ridgeline of theSNR surface is between the two predictions but very close to the blue points, indicating that the distribution of appropriate structuring element sizes is mainly dominated by fh. Therefore, the ridgeline of theSNR surface can be approximately estimated by (13) with fh.In actual operation, one pair of appropriate size (W,H) is sufficient for the MF, and the value of W for the triangular structuring element isW=2n+1, which means that the smallest value of W is 3. In this paper, a simplified and practical size determination method for the triangular structuring element was proposed as follows: (1) Extract the highest valuable frequency f of the signal. (2) Introducesf and f into (13), increase H from 0.1 with a step length of 0.1, and calculate the corresponding odd-value ofW. (3) Stop the calculation whenW ≥ 3, and obtain one pair of appropriate size (W,H).


## 4. MF-EEMD

The superfluous white noise components in EEMD may cause additional mode mixing problems and reduce the decomposition efficiency [12]. In this paper, the MF was adopted to filter out the superfluous white noise components to improve EEMD.


### 4.1. EMD Combined with MF

A composite signal with two dominant frequencies was constructed by (6)-(8), as shown in Figure 8, and the parameters were set as f1=2, λ1=0.5, f2=5, λ2=0.5,sf=256,N=512, andk=0.3.Figure 8
Composite signal with two dominant frequencies.
(a)
S sin1 (b)
S sin2 (c)
S cEMD was directly adopted for the composite signal in Figure8(c). The stoppage criterion of EMD was that the number of zero crossing points of IMF must be larger thanN/sf, and the decomposition shall be stopped after the appearance of the first unqualified IMF. This stoppage criterion was also applicable to EEMD and MF-EEMD. There were 9 orders of IMFs in the decomposition results of EMD, as shown in Figure 9. The 1st- to 3rd-order IMFs are noise, the 4th-order IMF consists of noise and a 5 Hz component, the 5th-order IMF consists of 5 Hz and 2 Hz components, the 6th-order IMF is mainly a 2 Hz component, and the 7th- to 9th-order IMFs are trends. These results illustrate that there are obvious mode mixing problems in the EMD results.Figure 9
EMD results.There is typically no prior knowledge about the highest valuablef in the signal; thus, the application of (13) is limited. To address this problem, an extraction method for f based on the decomposition results of EMD or EEMD was proposed.First, the original signal is decomposed with EMD or EEMD. Then, the criterionET proposed by Wu and Huang [24] is used to distinguish the noise components from the valuable components. The criterionEnTn of the nth IMF is defined by (14) and (15).(14)En=1N∑i=1NIMFni2(15)Tn=2NNzcwhere En is the energy density, Tn is the averaged period,N is the length of IMFn, and Nzc is the number of zero-crossings of IMFn. When IMFn is a noise component, the value of EnTn is small; otherwise, the value of EnTn is considerably larger.By observing the values ofEnTn, the noise components and valuable components are distinguished, and the IMF with the desired f is selected. Finally, the auto-power spectrum analysis is adopted for the selected IMF, and the frequency of the peak point in the spectrum is taken asf.TheET values of the IMFs in Figure 9 are shown in Figure 10, where there is a sudden increase between the 3rd and 4thET values. This result indicates that the frequency of the 4th IMF can be taken as f. The auto-power spectrum of the 4th IMF is shown in Figure 11 withf=5 Hz.Figure 10
ET values of the IMFs.Figure 11
Auto-power spectrum of the4th IMF.The appropriate structuring element size was derived asW=3 andH=0.1 by (13) withsf=256 andf=5. Then, the composite signal in Figure 8(c) was filtered by the MF, as shown in Figure 12.Figure 12
Filtering result of the MF.
(a)
S M F (b)
Noise componentThe Fourier spectrum of the noise component is shown in Figure13. The amplitude of the low-frequency domain, where the frequencies of the valuable components are located, is smaller than that of the high-frequency domain. In other words, the superfluous dyadic filter banks are removed while the valuable ones are retained in SMF.Figure 13
Fourier spectrum of the noise component.Then, the EMD was applied again to the filtering result in Figure12(a), and 7 orders of IMFs were obtained, as shown in Figure 14. The 1st- to 3rd-order IMFs are noise, the 4th-order IMF is a 5 Hz component, the 5th-order IMF is mainly a 2 Hz component, and the 6th- and 7th-order IMFs are trends. Only slight mode mixing problems are observed in the decomposition results, indicating that the mode mixing problems are suppressed. Moreover, the number of IMFs is reduced from 9 to 7, which means that the decomposition efficiency of EMD is also improved.Figure 14
Results of the EMD combined with MF.


### 4.2. EEMD Combined with MF

The decomposition results of a single operation of EMD were improved with the addition of MF. EEMD is an ensemble of numerous EMD; therefore, the EEMD results would also be improved by the MF. Hence, MF-EEMD was proposed in this paper by introducing MF into EEMD.The main flows of MF-EEMD are as follows: (1) Decompose the original signal by traditional EMD or EEMD and determine the appropriate structuring element size based on the decomposition results. (2) Add white noise to the original signal and employ the MF to filter the polluted signal. (3) Adopt EMD to decompose the filtering signal. (4) Repeat the second and third processes by adding different series of white noise. (5) Make an ensemble of the numerous decomposition results and achieve the final decomposition results of EEMD. The above flows of MF-EEMD are shown in Figure15, where the additional operations proposed in this paper are marked with a gray shade.Figure 15
Operational flow of the MF-EEMD.


## 4.1. EMD Combined with MF

A composite signal with two dominant frequencies was constructed by (6)-(8), as shown in Figure 8, and the parameters were set as f1=2, λ1=0.5, f2=5, λ2=0.5,sf=256,N=512, andk=0.3.Figure 8
Composite signal with two dominant frequencies.
(a)
S sin1 (b)
S sin2 (c)
S cEMD was directly adopted for the composite signal in Figure8(c). The stoppage criterion of EMD was that the number of zero crossing points of IMF must be larger thanN/sf, and the decomposition shall be stopped after the appearance of the first unqualified IMF. This stoppage criterion was also applicable to EEMD and MF-EEMD. There were 9 orders of IMFs in the decomposition results of EMD, as shown in Figure 9. The 1st- to 3rd-order IMFs are noise, the 4th-order IMF consists of noise and a 5 Hz component, the 5th-order IMF consists of 5 Hz and 2 Hz components, the 6th-order IMF is mainly a 2 Hz component, and the 7th- to 9th-order IMFs are trends. These results illustrate that there are obvious mode mixing problems in the EMD results.Figure 9
EMD results.There is typically no prior knowledge about the highest valuablef in the signal; thus, the application of (13) is limited. To address this problem, an extraction method for f based on the decomposition results of EMD or EEMD was proposed.First, the original signal is decomposed with EMD or EEMD. Then, the criterionET proposed by Wu and Huang [24] is used to distinguish the noise components from the valuable components. The criterionEnTn of the nth IMF is defined by (14) and (15).(14)En=1N∑i=1NIMFni2(15)Tn=2NNzcwhere En is the energy density, Tn is the averaged period,N is the length of IMFn, and Nzc is the number of zero-crossings of IMFn. When IMFn is a noise component, the value of EnTn is small; otherwise, the value of EnTn is considerably larger.By observing the values ofEnTn, the noise components and valuable components are distinguished, and the IMF with the desired f is selected. Finally, the auto-power spectrum analysis is adopted for the selected IMF, and the frequency of the peak point in the spectrum is taken asf.TheET values of the IMFs in Figure 9 are shown in Figure 10, where there is a sudden increase between the 3rd and 4thET values. This result indicates that the frequency of the 4th IMF can be taken as f. The auto-power spectrum of the 4th IMF is shown in Figure 11 withf=5 Hz.Figure 10
ET values of the IMFs.Figure 11
Auto-power spectrum of the4th IMF.The appropriate structuring element size was derived asW=3 andH=0.1 by (13) withsf=256 andf=5. Then, the composite signal in Figure 8(c) was filtered by the MF, as shown in Figure 12.Figure 12
Filtering result of the MF.
(a)
S M F (b)
Noise componentThe Fourier spectrum of the noise component is shown in Figure13. The amplitude of the low-frequency domain, where the frequencies of the valuable components are located, is smaller than that of the high-frequency domain. In other words, the superfluous dyadic filter banks are removed while the valuable ones are retained in SMF.Figure 13
Fourier spectrum of the noise component.Then, the EMD was applied again to the filtering result in Figure12(a), and 7 orders of IMFs were obtained, as shown in Figure 14. The 1st- to 3rd-order IMFs are noise, the 4th-order IMF is a 5 Hz component, the 5th-order IMF is mainly a 2 Hz component, and the 6th- and 7th-order IMFs are trends. Only slight mode mixing problems are observed in the decomposition results, indicating that the mode mixing problems are suppressed. Moreover, the number of IMFs is reduced from 9 to 7, which means that the decomposition efficiency of EMD is also improved.Figure 14
Results of the EMD combined with MF.


## 4.2. EEMD Combined with MF

The decomposition results of a single operation of EMD were improved with the addition of MF. EEMD is an ensemble of numerous EMD; therefore, the EEMD results would also be improved by the MF. Hence, MF-EEMD was proposed in this paper by introducing MF into EEMD.The main flows of MF-EEMD are as follows: (1) Decompose the original signal by traditional EMD or EEMD and determine the appropriate structuring element size based on the decomposition results. (2) Add white noise to the original signal and employ the MF to filter the polluted signal. (3) Adopt EMD to decompose the filtering signal. (4) Repeat the second and third processes by adding different series of white noise. (5) Make an ensemble of the numerous decomposition results and achieve the final decomposition results of EEMD. The above flows of MF-EEMD are shown in Figure15, where the additional operations proposed in this paper are marked with a gray shade.Figure 15
Operational flow of the MF-EEMD.


## 5. Application Instance

### 5.1. Analog Signal Decomposition

To verify the feasibility of the proposed method, the composite signalSc in Figure 8(c) was selected again to be decomposed by EEMD and MF-EEMD. During operations of the two methods, the level of white noise added in each operation of EMD was set as 0.1 times the standard deviation of Sc, and the number of ensembles was set as 500. The appropriate size of triangular structuring element was set asW=3,H=0.1. Nine IMFs are present in the traditional EEMD results, as shown in Figure 16, while there are 7 IMFs in the MF-EEMD results, as shown in Figure 17. The corresponding Hilbert spectra of the EEMD and MF-EEMD results are shown in Figures 18 and 19, respectively.Figure 16
EEMD results.Figure 17
MF-EEMD results.Figure 18
Hilbert spectrum of EEMD.Figure 19
Hilbert spectrum of MF-EEMD.In Figure16, the 1st- to 4th-order IMFs are noise, the 5th-order IMF is a 5 Hz component, the 6th-order IMF is a 2 Hz component, and the 7th- to 9th-order IMFs are trends. However, the waveform of the 4th-order IMF from 1 s to 2 s is quite similar to that of the 5th-order IMF; there are slight mode mixing problems in the EEMD results. The mode mixing problems can also be observed in Figure 18, where there are disturbances around the 5 Hz spectrum.In Figure17, the 1st- to 3rd-order IMFs are noise, the 4th-order IMF is a 5 Hz component, the 5th-order IMF is a 2 Hz component, and the 6th- and 7th-order IMFs are trends. In Figure 19, the 5 Hz and 2 Hz spectra are straight, and the background is quite clean. No obvious mode mixing problem is found in the MF-EEMD results.


### 5.2. Measured Signal Decomposition

For further verifying the proposed method, the acceleration signal measured from a shaking table test of a cable-stayed bridge model was taken as another instance. A photograph of the cable-stayed bridge model is shown in Figure20. The input of the test was white noise with a peak value of 0.1 g, and the input direction was lateral. A set of acceleration responses of the tower top with a sample frequency of 256 Hz was selected, as shown in Figure 21.Figure 20
Cable-stayed bridge model.Figure 21
Acceleration response of the tower top.In this instance, the level of white noise added in each operation of EMD was set as 0.2 times the standard deviation of the measured signal, and the number of ensembles was set as 2,000. First, the measured signal was decomposed by EEMD, and 11 IMFs were obtained. The decomposition results along with theET values are shown in Figures 22 and 23. There is a sudden increase between the 3rd and 4thET values, indicating that the desired f is in the 4th-order IMF. The auto-power spectrum of the 4th-order IMF is shown in Figure 24, and the frequency of the peak point is 10.38 Hz. The appropriate size of the structuring element was derived asW=3 andH=0.3 by (13) withsf=256 andf=10.38. Then, the measured signal was decomposed by MF-EEMD, and 9 IMFs were obtained, as shown in Figure 25.Figure 22
EEMD results.Figure 23
ET values of IMFs.Figure 24
Auto-power spectrum of the4th IMF.Figure 25
MF-EEMD results.To compare the mode mixing problems in detail, the local waveforms of the IMFs of the two methods are presented in Figures26 and 27. Mode mixing problems are present in the 5th-order IMF of EEMD, and the waveforms marked with the red circle are similar to that of the 4th-order IMF. The waveforms marked with the blue circles are similar to that of the 6th-order IMF. However, there are no evident mode mixing problems in the IMFs of MF-EEMD.Figure 26
Local waveforms of the IMFs of EEMD.Figure 27
Local waveforms of the IMFs of MF-EEMD.The Hilbert spectra of the EEMD and MF-EEMD results are shown in Figures28 and 29, respectively. In Figure 28, there are serious mode mixing problems in the two spectra of approximately 4 Hz, and the fluctuations of their frequencies cannot be easily tracked. The spectrum of approximately 10 Hz is widely distributed in the frequency domain, which means that the corresponding waveforms of this spectrum are not smooth. In Figure 29, the mode mixing problems in the two spectra of approximately 4 Hz are mild, and the fluctuations of their frequencies are clear and easy to track. The distribution band of the spectrum of approximately 10 Hz is narrow, which means that the smoothness of its waveforms is improved. The comparison results show that the mode mixing problems in EEMD are effectively suppressed by MF-EEMD.Figure 28
Hilbert spectrum of EEMD.Figure 29
Hilbert spectrum of MF-EEMD.


## 5.1. Analog Signal Decomposition

To verify the feasibility of the proposed method, the composite signalSc in Figure 8(c) was selected again to be decomposed by EEMD and MF-EEMD. During operations of the two methods, the level of white noise added in each operation of EMD was set as 0.1 times the standard deviation of Sc, and the number of ensembles was set as 500. The appropriate size of triangular structuring element was set asW=3,H=0.1. Nine IMFs are present in the traditional EEMD results, as shown in Figure 16, while there are 7 IMFs in the MF-EEMD results, as shown in Figure 17. The corresponding Hilbert spectra of the EEMD and MF-EEMD results are shown in Figures 18 and 19, respectively.Figure 16
EEMD results.Figure 17
MF-EEMD results.Figure 18
Hilbert spectrum of EEMD.Figure 19
Hilbert spectrum of MF-EEMD.In Figure16, the 1st- to 4th-order IMFs are noise, the 5th-order IMF is a 5 Hz component, the 6th-order IMF is a 2 Hz component, and the 7th- to 9th-order IMFs are trends. However, the waveform of the 4th-order IMF from 1 s to 2 s is quite similar to that of the 5th-order IMF; there are slight mode mixing problems in the EEMD results. The mode mixing problems can also be observed in Figure 18, where there are disturbances around the 5 Hz spectrum.In Figure17, the 1st- to 3rd-order IMFs are noise, the 4th-order IMF is a 5 Hz component, the 5th-order IMF is a 2 Hz component, and the 6th- and 7th-order IMFs are trends. In Figure 19, the 5 Hz and 2 Hz spectra are straight, and the background is quite clean. No obvious mode mixing problem is found in the MF-EEMD results.


## 5.2. Measured Signal Decomposition

For further verifying the proposed method, the acceleration signal measured from a shaking table test of a cable-stayed bridge model was taken as another instance. A photograph of the cable-stayed bridge model is shown in Figure20. The input of the test was white noise with a peak value of 0.1 g, and the input direction was lateral. A set of acceleration responses of the tower top with a sample frequency of 256 Hz was selected, as shown in Figure 21.Figure 20
Cable-stayed bridge model.Figure 21
Acceleration response of the tower top.In this instance, the level of white noise added in each operation of EMD was set as 0.2 times the standard deviation of the measured signal, and the number of ensembles was set as 2,000. First, the measured signal was decomposed by EEMD, and 11 IMFs were obtained. The decomposition results along with theET values are shown in Figures 22 and 23. There is a sudden increase between the 3rd and 4thET values, indicating that the desired f is in the 4th-order IMF. The auto-power spectrum of the 4th-order IMF is shown in Figure 24, and the frequency of the peak point is 10.38 Hz. The appropriate size of the structuring element was derived asW=3 andH=0.3 by (13) withsf=256 andf=10.38. Then, the measured signal was decomposed by MF-EEMD, and 9 IMFs were obtained, as shown in Figure 25.Figure 22
EEMD results.Figure 23
ET values of IMFs.Figure 24
Auto-power spectrum of the4th IMF.Figure 25
MF-EEMD results.To compare the mode mixing problems in detail, the local waveforms of the IMFs of the two methods are presented in Figures26 and 27. Mode mixing problems are present in the 5th-order IMF of EEMD, and the waveforms marked with the red circle are similar to that of the 4th-order IMF. The waveforms marked with the blue circles are similar to that of the 6th-order IMF. However, there are no evident mode mixing problems in the IMFs of MF-EEMD.Figure 26
Local waveforms of the IMFs of EEMD.Figure 27
Local waveforms of the IMFs of MF-EEMD.The Hilbert spectra of the EEMD and MF-EEMD results are shown in Figures28 and 29, respectively. In Figure 28, there are serious mode mixing problems in the two spectra of approximately 4 Hz, and the fluctuations of their frequencies cannot be easily tracked. The spectrum of approximately 10 Hz is widely distributed in the frequency domain, which means that the corresponding waveforms of this spectrum are not smooth. In Figure 29, the mode mixing problems in the two spectra of approximately 4 Hz are mild, and the fluctuations of their frequencies are clear and easy to track. The distribution band of the spectrum of approximately 10 Hz is narrow, which means that the smoothness of its waveforms is improved. The comparison results show that the mode mixing problems in EEMD are effectively suppressed by MF-EEMD.Figure 28
Hilbert spectrum of EEMD.Figure 29
Hilbert spectrum of MF-EEMD.


## 6. Conclusion

MF-EEMD was proposed to suppress the mode mixing problems in EEMD. Structuring elements with different shapes and sizes were used in the MF to filter a series of analog signals. Based on the filtering results, the triangular shape was found to be superior to the linear and sinusoidal shapes, and an equation for calculating the appropriate sizes of the triangular structuring element was established. Then, the MF was used to filter the superfluous white noise components during the EMD operation. The MF could help suppress the mode mixing problems and improve the decomposition efficiency. Based on the above results, an improved MF-EEMD method was proposed by combining the EEMD with MF. To verify the proposed method, an analog signal and a measured signal were decomposed by EEMD and MF-EEMD, respectively. The comparison results show that the mode mixing problems in EEMD are effectively suppressed by the MF-EEMD. Moreover, under the same stoppage criterion, fewer IMFs are archived by the MF-EEMD, which means that the proposed method also has a higher decomposition efficiency.



---

*Source: 5976589-2018-09-17.xml*