A novel signal feature extraction technology based on empirical wavelet transform and reverse dispersion entropy

Yu-xing Li ,Shng-in Jio ,* ,Xing Go

a School of Automation and Information Engineering,Xi’an University of Technology,Xi’an,710048,China

b School of Electrical Engineering,Xi’an University of Technology,Xi’an,710048,China

Keywords: Feature extraction Empirical mode decomposition mpirical wavelet transform Permutation entropy Reverse dispersion entropy

ABSTRACT Feature extraction is an important part of signal processing,which is significant for signal detection,classification,and recognition.The nonlinear dynamic analysis method can extract the nonlinear characteristics of signals and is widely used in different fields.Reverse dispersion entropy(RDE)proposed by us recently,as a nonlinear dynamic analysis method,has the advantages of fast computing speed and strong anti-noise ability,which is more suitable for measuring the complexity of signal than traditional permutation entropy(PE)and dispersion entropy(DE).Empirical wavelet transform(EWT),based on the theory of wavelet analysis,can decompose a complex non-stationary signal into a number of empirical wavelet functions (EWFs) with compact support set spectrum,which has better decomposition performance than empirical mode decomposition (EMD) and its improved algorithms.Considering the advantages of RDE and EWT,on the one hand,we introduce EWT into the field of underwater acoustic signal processing and fault diagnosis to improve the signal decomposition accuracy;on the other hand,we use RDE as the features of EWFs to improve the signal separability and stability.Finally,we propose a novel signal feature extraction technology based on EWT and RDE in this paper.Experimental results show that the proposed feature extraction technology can effectively extract the complexity features of actual signals.Moreover,it also has higher distinguishing ability for different types of signals than five latest feature extraction technologies.

1.Introduction

With the rapid development of signal decomposition algorithm and nonlinear dynamic analysis method,they greatly promote the update and improvement of signal processing technology,especially the signal feature extraction technology[1].On the one hand,the nonlinear dynamic analysis method can provide the nonlinear characteristics needed by signal processing technology,many of which are proved to be more effective than the traditional energy and frequency characteristics.On the other hand,the signal decomposition algorithm can decompose complex non-stationary signals into a series of sub-signals based on different criteria.It not only enriches the number of signal characteristics,but also highlights the difference of signals in the sub-signals.

As a nonlinear dynamic analysis method,permutation entropy(PE)[2],is widely used in many industries,especially in the medical and mechanical industries [3,4].In order to enhance its performance,many scholars proposed some improved PE methods.Weighted permutation entropy (WPE),as an improvement of PE,was brought forward by Fadlallah et al.,in 2013 [5],because of considering the amplitude information by weight arrangement pattern,it has many advantages of good noise robustness and high mutation detection ability.Dispersion entropy (DE),as an important technical breakthrough of PE,was put forward by Rostaghi and Azami in 2016[6],due to the introduction of mapping technology,DE has lower operation time and higher discrimination ability of different datasets.With a new definition of PE,reverse permutation entropy (RPE) was proposed by Bandt in 2017 [7],which has the opposite trend to the traditional PE and improved ones,and has better stability for time series with different lengths.By combining DE and RPE,reverse dispersion entropy(RDE)was put forward by Li et al.,in 2019 [8].RDE not only retains the advantages of DE and RPE,but also has better robustness to noise,stability and discrimination ability to target signals.

As a classical method of signal decomposition,empirical mode decomposition(EMD)attracts many scholars’attention[9].In order to suppress the mode mixing of EMD,many improved algorithms have been proposed,such as ensemble EMD(EEMD) [10],uniform phase EMD(UPEMD)[11]and complete EEMD with adaptive noise(CEEMDAN) [12].However,there is still a certain degree of mode mixing in these algorithms,which are based on experience and lack of theory [13].Empirical wavelet transform (EWT) with strict theoretical basis was proposed by Gilles in 2013 [14].EWT can adaptively decompose a complex non-stationary signal into a few empirical wavelet functions (EWFs) with compact support set spectrum by segmenting the Fourier spectrum of the target signal.

A class of effective signal feature extraction techniques are applied in different fields by combining nonlinear dynamic analysis method and signal decomposition algorithm.In the field of fault diagnosis [15],a fault feature extraction method is proposed for vibration signals of rolling bearing by combining multi-scale PE(MPE)and local mean decomposition(LMD)in Ref.[16].In Ref.[17],a fault feature extraction method for high-speed spindle is proposed by combining EMD and multiscale entropy (MSE).In Ref.[18],an early fault feature extraction method for rolling bearing is proposed by combining CEEMDAN and improved multivariate multi-scale sample entropy (MMSE).In Ref.[19],a fault feature extraction method for motor bearing is proposed by combining EWT and fuzzy entropy (FE).However,a feature extraction technology based on EWT and DE or based on EWT and RDE has yet been found in the field of fault diagnosis.

In the field of underwater acoustic signal processing,Li et al.proposed some feature extraction technologies for underwater acoustic signals by using EMD and PE [20],variational mode decomposition(VMD)and MPE[21],VMD and RPE[22],CEEMDAN and energy entropy [23],improved VMD and sample entropy (SE)[24],respectively.DE has been used in underwater acoustic signal processing since 2019.Two feature extraction techniques for underwater acoustic signals were proposed by Li et al.combined intrinsic time-scale decomposition(ITD)with fluctuation-based DE(FDE)[25],and combined improved ITD with multi-scale DE(MDE)[26],respectively.However,as far as I know,EWT has not been used for feature extraction in the field of underwater acoustic signal processing.

In view of the good performance of EWT and RDE,we propose a novel signal feature extraction technology based on EWT and RDE in this paper,EWT is used to decompose a complex non-stationary signal into a series of EWFs,and RDE can represent the complexity features of each EWF.The contributions and innovations of this paper are as follows:(i) EWT is first applied to the field of underwater acoustic signal processing,and its superiority over traditional mode decomposition methods has been proved;(ii) RDE is first applied to the field of fault diagnosis combined with EWT;(iii)the proposed signal feature extraction technology has better performance than the latest feature extraction technologies,which has potential application value in other fields[27-30],such as the field of image processing[31-33].This paper is organized as follows:the proposed signal feature extraction technology and the details of EWT and RDE are described in Section 2;we compared the proposed signal feature extraction technology with the existing six signal feature extraction ones in Section 3;in Section 4,several main conclusions are obtained.

2.The proposed signal feature extraction technology and basic theory

2.1.The proposed signal feature extraction technology

Signal decomposition algorithm is the basis and key to signal feature extraction technology,which can provide a set of effective sub-signals for further study.EWT has better decomposition performance than EMD and its improved ones,which is more suitable to decompose complex non-stationary signals.As a new nonlinear dynamic analysis method proposed by us recently,RDE has also been proved to be more stable and distinguishable than PE,WPE,RPE and DE.Combining the advantages of EWT and RDE,we propose a novel signal feature extraction technology.A flow chart of the signal feature extraction technology is shown in Fig.1.The main research contents and steps are as follows:

Step 1:EWT.

(1) Set a decomposition number of EWT for original signal.

(2) Decompose original signal into EWFs by EWT.The details of EWT can be found in Section 2.2.

Step 2:determine an EWF for further study.

(1) Calculate the average energy intensity of each EWF.The calculation of the average energy intensity can be found in Ref.[20].

(2) Get an EWF with the most energy (EWF-ME).EWF-ME has the highest average energy intensity among all EMFs.

Step 3:feature extraction.

(1) Calculate the RDE of EWF-ME.The details of RDE can be found in Section 2.3.

(2) Obtain the feature of EWF-ME,which can also represent the feature of the original signal.

Fig.1.A flow chart of the signal feature extraction technology.

Fig.2.A flow chart of EWT.

2.2.EWT

As an effective signal decomposition method,EWT can decompose a complex non-stationary signal into a number of EWFs.A flow chart of EWT is shown in Fig.2.The main steps and guidelines are as follows:

Step 1:Calculate the frequency spectrum of original signal by Fourier transform.

Step 2:Segment the frequency spectrum into M continuous segments.

(1) Reset the frequency range.The frequency spectrum range of the original signalf（t）is converted to[0,π].

(2) Search the local maxima.We assume that the number of local maxima isK.

(3) Determine the frequency boundaries ωm.Mcontinuous segments includeM+1 boundaries,which include ω0=0 and ωm=π.That means we need to findM-1 extra boundaries.IfKis greater than or equal toM,we arrange all local maxima in descending order and regard the firstM-1 local maxima as boundaries (excluding 0 and π),otherwise we resetMaccording to the number of local maxima.Then we can obtainMcontinuous segments,each segment can be represented asso we can also get

Step 3:Construct band-pass wavelet-based filters.

To construct band-pass wavelet-based filters in each segment,empirical scaling functionsand wavelet functionsare defined as:

where the function β（x）is represented as:

More details about Formula 3 can be found in Ref.[34].In order to ensure the compact support frames of empirical scaling functionsand wavelet functionsthe parameter ξ need to meet the condition as follows:

Step 4:Calculate details and approximate coefficients.

The detail coefficientsWf（f，t）can be obtained by inner product of the original signal and wavelet functions,and the approximate coefficientWf（0，t）can be obtained by inner product of the original signal and empirical scaling function φ1（t）.They can be expressed as follows:

Step 5:Calculate the EWFs.

The approximation sub-signalf0（t）and detail sub-signalfi（t）can be defined as:

where*represents convolution,andfi（t）is thei-th EWF forf（t）by EWT.

To better understand EWT,let us take an example,for an artificial signaly（t）,y1（t）,y2（t）andy3（t）are its distinct components as follows:

wheret∈［0，1］.The waveform ofy（t）and its distinct components are shown in Fig.3.

The frequency spectrum ofy（t）is shown in Fig.4,where the red dashed line is used to segment adjacent frequency spectrum of EWFs.We can observe that the frequency spectrum range ofy（t）is converted to [0,π],and the frequency spectrum is divided into three continuous segments by two red dashed line.

Fig.5 is the EWT results ofy（t）.As shown in Fig.5,we can obtain three EWFs by EWT,EWF1,EWF2 and EWF3 correspond toy1（t）,y2（t）andy3（t）,respectively.The reconstruction error ofy（t）and the sum of three EWFs is 3.1× 10-15,which shows a high reconstruction accuracy.It proves that EWT can decompose the original signaly（t）accurately.

2.3.A comparative study of different signal decomposition algorithms

In order to verify the effectiveness of EWT,a comparative study of EMD,EEMD,VMD and EWT was conducted on hybrid simulated signal.Without loss of generality,the hybrid simulated signalx（t）and its sub-signals (x1（t）,x2（t），x3（t）andx4（t）) are as follows:

Fig.3.The waveform of y（t）and its distinct components.

Fig.4.The frequency spectrum of y（t）.

Fig.5.The EWT results of y（t）.

Fig.7.The decomposition results of EMD,EEMD,VMD and EWT.

wheret∈［0，1］.The waveform ofx（t）and its sub-signals are shown in Fig.6.The decomposition results of EMD,EEMD,VMD and EWT are shown in Fig.7.

Fig.6.The waveform of x（t）and its sub-signals.

As shown in Fig.7,for the decomposition result of EMD,it is easy to see that IMF1 and IMF3 correspond tox1（t）andx4（t）;for the decomposition result of EEMD,IMF2 and IMF4 correspond tox1（t）andx4（t）;for the decomposition result of VMD,IMF3 and IMF4 correspond tox2（t）andx4（t）;however,for the decomposition result of EWT,EWF1,EWF2,EWF3 and EWF4 correspond tox1（t）,x2（t），x3（t）andx4（t）,the IMFs decomposed by EWT are closer to the sub-signals ofx（t）.Simulation results show that EWT has better decomposition performance than EMD,EEMD and VMD.

2.4.RDE

RDE is a new nonlinear dynamic analysis method.It is an important improvement of PE by combining the key technologies of DE and RPE.RDE not only has the high distinguishing ability of DE,but also has the high stability and accuracy of RPE.The detailed process of RDE can be described as follows.

One-dimension time seriesY=｛y（i），i=1，2，…，K｝ is mapped toZ=｛z（i），i=1，2，…，K｝ through normal cumulative distribution function (NCDF),which results in the distribution ofZin[0,1].To obtain a sequence of positive integers,Zis mapped toX=｛x（i），i=1，2，…，K｝ through the function round（d.z（i）+0.5）,wheredrepresents the number of classes,we setdto 4 in this paper.Therefore,we can obtain a positive integer time seriesXwith the distribution in [1,d].

Like the phase space reconstruction of PE,Xcan be restructured as:

whereNis the number of embedding vectors and equal toK-（m-1）,τ andmare the time delay and embedding dimension for time seriesX.τ is usually set to 1,mis set to 2 or 3 for RDE and needs to meet the conditionT>dmaccording to Ref.[6].

Each embedding vector corresponds to a dispersion pattern πi.There aredmdispersion patterns forXwith the embedding dimensionmand the number of classesd.The frequency of dispersion pattern πican be defined asNumber｛πi｝,and its relative frequencyP（πi）can be defined as follows:

where the range ofiis 1 ≤i≤dm.For the calculation of relative frequency,RDE and DE are the same.

RDE introduces the key technology of RPE,and RDE is defined as the distance from white noise.The RDE ofYcan be represented as

hRDE（Y，m，d，τ）reaches the minimum value of 0 whenP（πi）is 1/dm,andhRDE（Y，m，d，τ）reaches the maximum value ofwhenP（πi）is 1.The normalized RDE ofYcan be represented as:

whereHRDEis in the distribution from 0 to 1.Due to different definitions,RDE and RPE have the opposite trend with DE and PE.For RDE and RPE,a higher value indicates more regular time series,while a lower value shows less regular time series.Comparing with PE,WPE,RPE and DE,RDE has better separability and stability,which is beneficial to signal feature extraction.A large number of comparative experiments and more details about RDE can be found in Ref.[8].

3.Feature extraction of real signals

In this section,we extract the features of real signals using the proposed feature extraction technology,including ship-radiated noise signals and rolling bearing fault signals.Moreover,we compare the latest five feature extraction technologies to further prove the effectiveness of the proposed feature extraction technology.

3.1.Feature extraction of ship-radiated noise signals

In order to prove the effectiveness of the proposed feature extraction technology,we carried out feature extraction for four kinds of ship-radiated noise signals,which are from the official website (https://www.nps.gov/glba/learn/nature/soundclips.htm).Four kinds of ship-radiated noise signals are named ship I,ship II,ship III and ship IV,their time domain waveforms are shown in Fig.8.Similar to Refs.[20,21],5000 points are used as a sample,and the sampling frequency of ships is 44.1 KHz.Fig.9 is the decomposition results of EWT for four kinds of ship-radiated noise signals.We calculate the average energy intensity of each EWF and obtain an EWF with the most energy (EWF-ME) for each sample.Table 1 shows the EWF-ME distribution for four kinds of ship-radiated noise signals.As shown in Table 1,the EWF-ME distribution for Ship III is EWF4,and the EWF-ME distribution for Ship I,Ship II and Ship IV is EWF8.

Table 2 The mean and standard deviation of RDE for four kinds of ship-radiated noise signals.

Table 3 The standard deviation of five entropies for EWT-MEs of ship-radiated noise signals.

Fig.8.The time domain waveforms for four kinds of ship-radiated noise signals.

Fig.9.The decomposition results of EWT for four kinds of ship-radiated noise signals.

Table 1 The EWF-ME distribution for four kinds of ship-radiated noise signals.

We calculate the RDE of EWF-ME for four kinds of ship-radiated noise signals.Fig.10 is the RDE distribution of EWF-ME for four kinds of ship-radiated noise signals.As shown in Fig.10,compared with Ship I,Ship II and Ship IV,EWF-ME of ship III has the largest RDE,and we can easily distinguish ship III among the four kinds of ship-radiated noise signals;for Ship I,Ship II and Ship IV,their EWF-MEs are all EWF8,ships of the same category have similar RDE,there are some differences in different kinds of ship-radiated noise signals.Table 2 shows the mean and standard deviation of RDE for four kinds of ship-radiated noise signals.As shown in Table 2,the mean values of the four kinds of ship-radiated noise signals are significantly different,the mean value of Ship III is the largest and Ship IV is the smallest;the standard deviations of Ship I,Ship II and Ship III are close to 0.06,and ship IV has the smallest standard deviation.In order to prove the effectiveness of RDE in feature extraction of ship-radiated noise signals,Table 3 shows the standard deviation of five entropies for EWT-MEs of ship-radiated noise signals.As shown in Table 3,the standard deviation of RDE is the smaller than that of PE,WPE,RPE and DE.It proves that RDE has better stability for four kinds of ship radiated noise signals.

Fig.10.The RDE distribution of EWF-ME for four kinds of ship-radiated noise signals.

Fig.11.The time domain waveforms for four kinds of bearing fault signals.

Table 4 The classification results of six feature extraction technologies for four kinds of ship-radiated noise signals by using KNN.

Fig.12.The decomposition results of EWT for four kinds of bearing fault signals.

In order to further prove the effectiveness of the proposed feature extraction technology,we compare the other five latest feature extraction technologies in Refs.[20-23,35],which are named EMD-PE,VMD-PE,VMD-RPE,VMD-DCO-PE and CEEMDANED-EE.Similar to the proposed feature extraction technology,EMDPE,VMD-PE,and VMD-RPE are all complexity feature extraction technologies,and the difference is that EWT is replaced by EMD and VMD,and RDE is replaced by PE and RPE.VMD-DCO-PE and CEEMDAN-ED-EE are frequency and energy feature extraction technology,where VMD and CEEMDAN are used to decompose ship-radiated noise into IMFs,the line spectrum frequency can be obtain by using duffing chaotic oscillator (DCO) and PE,and the hybrid energy feature can be obtain by using with energy difference(ED) and energy entropy (EE).K-Nearest Neighbour (KNN) is used to distinguish four kinds of ship-radiated noise signals.50 samples of each class of ships are selected as training set for classifier training,and the remaining 50 samples of each class are used as test sets for classification.Table 4 shows the classification results of six feature extraction technologies for four kinds of ship-radiated noise signals by using KNN.As shown in Table 4,the classification recognition rates of frequency feature extraction technology VMDDCO-PE and energy feature extraction technology CEEMDAN-EDEE are less than 90%;the classification recognition rates of four complexity feature extraction technologies are higher than 90%;the proposed feature extraction technology has a highestrecognition rate of up to 99.5%,which is at least 5%higher than the other five feature extraction technologies.The experimental results show that the proposed feature extraction technology has better performance for four kinds of ship-radiated noise signals.

Table 5 The EWF-ME distribution for four kinds of bearing fault signals.

3.2.Feature extraction of rolling bearing fault signals

In this section,we carried out feature extraction for four kinds of rolling bearing fault signals,which are provided by the Case Western Reverse Laboratory at the following website (http://csegroups.case.edu/bearingdatacenter/pages/download-data-file.).Four kinds of rolling bearing fault signals are named Fault I,Fault II,Fault III and Fault IV,their time domain waveforms are shown in Fig.11.Due to the limited data length,we take 2000 sampling points as a sample.Fig.12 is the decomposition results of EWT for four kinds of rolling bearing fault signals.We calculate the average energy intensity of each EWF and obtain an EWF-ME for each fault signal sample.Table 5 shows the EWF-ME distribution for four kinds of rolling bearing fault signals.As shown in Table 5,the EWFME distribution for Fault I and Fault IV is EWF5,the EWF-ME distribution for Ship II is EWF3,and the EWF-ME distribution for Ship III is EWF2.

Fig.13.The RDE distribution of EWF-ME for four kinds of bearing fault signals.

Table 6 The mean and standard deviation of RDE for four kinds of bearing fault signals.

Table 7 The standard deviation of five entropies for EWT-MEs of bearing fault signals.

We calculate the RDE of EWF-ME for four kinds of bearing fault signals.Fig.13 is the RDE distribution of EWF-ME for four kinds of bearing fault signals.As shown in Fig.13,we can easily distinguish Fault I and Fault II among four kinds of bearing fault signals;the RDEs of some samples for Fault III and Fault IV are very close,which makes it difficult to distinguish them.Table 6 shows the mean and standard deviation of RDE for four kinds of bearing fault signals.As shown in Table 6,the mean values of the four kinds of bearing fault signals are significantly different;except for Fault II,the standard deviation of the other three bearing fault signals are all lower than 0.005.In order to prove the effectiveness of RDE in feature extraction of bearing fault signals,Table 7 shows the standard deviation of five entropies for EWT-MEs of bearing fault signals.As shown in Table 7,the standard deviation of RDE is the smaller than that of the other four entropies.It proves that RDE has better stability for four kinds of bearing fault signals.

In order to further prove the advantages of the feature extraction technology proposed in this paper,similar to the feature extraction of ship-radiated noise signals,we compare the other five latest feature extraction technologies and use KNN to distinguish four kinds of bearing fault signals.30 samples of each kind of bearing fault signals are selected as training set,and the remaining 30 samples of each kind are used as test sets for classification.Table 8 shows the classification results of six feature extraction technologies for four kinds of bearing fault signals by using KNN.As shown in Table 8,the classification recognition rates of frequency feature extraction technology VMD-DCO-PE and energy feature extraction technology CEEMDAN-ED-EE are less than 85%;the classification recognition rates of four complexity feature extraction technologies are higher than 85%;the proposed feature extraction technology has a highest recognition rate of up to 95.83%,which is at least 4% higher than the other five feature extraction technologies.The experimental results further prove the superiority of the proposed feature extraction technology in bearing fault signal recognition.

Table 8 The classification results of six feature extraction technologies for four kinds of bearing fault signals by using KNN.

4.Conclusions

The key to feature extraction technology is the selection of signal processing algorithm and feature.Compared with EMD,EEMD and VMD,EWT has better decomposition performance.As a new complexity feature,RDE has better separability than PE,RPE and DE.In view of the good performance of EWT and RDE,a novel signal feature extraction technology is proposed based on EWT and RDE.The main work and contributions are as follows.

(1) We introduce EWT into the field of underwater acoustic signal processing and prove that EWT is more effective than EMD,EEMD and VMD.

(2) The experimental results show that the proposed feature extraction technology based on EWT and RDE is better thanother five feature extraction technologies,and can improve the recognition rate by at least 5% for four kinds of shipradiated noise signals.

(3) RDE,as a new nonlinear dynamic analysis method,is introduced into the field of fault diagnosis combined with EWT.

(4) The recognition rate of the proposed feature extraction technology is higher than 95%for four kinds of bearing fault signals,which is at least 4% higher than EMD-PE,VMD-PE,VMD-RPE,VMD-DCO-PE and CEEMDAN-ED-EE.

In the future research,we will apply this feature extraction technology to other fields.In addition,we will further enhance the performance of the proposed feature extraction technology by improving the decomposition accuracy of EWT and the distinguishing ability of RDE.

Declaration of competing interest

The authors declared that they have no conflicts of interest to this work.

Acknowledgements

The authors gratefully acknowledge the supported by National Natural Science Foundation of China(No.61871318 and 11574250),and Scientific Research Plan Projects of Shaanxi Education Department (No.19JK0568).