Abstract
This paper proposes a novel fault feature extraction method with the aim of extracting the fault feature submerged in the singlechannel observation signal. The proposed method integrates the strengths of the constrained independent component analysis (cICA) extracting only the signals of interest (SOIs) with the advantage of ensemble empirical mode decomposition (EEMD) alleviating the mode mixing. The method, which is named EEMDbased cICA, not only enables gear fault feature extraction but also offers a new independent component analysis (ICA) mixing model with source noise and measured noise for the singlechannel observation signal. The efficiency of the proposed method is tested on simulated as well as realworld vibration signals acquired from a multistage gearbox with a missing tooth and a chipped tooth, respectively.
1. Introduction
In general, the goal of independent component analysis (ICA) [14] is to recover all the source signals from mixed signals at a time. ICA is one of the outstanding techniques for solving the signal blind source separation (BSS) problem, which has been widely applied to the source signals separation and feature extraction [3, 4] in the applications of biomedical engineering, telecommunications, mechanical engineering and audio. However, there are many problems to be solved for ICA applications: (1) classical ICA algorithm has some ambiguities, such as unknown number of source signals, undetermined the variance (energies) and the order of the independent components (ICs); (2) ICA model does not consider the source noise and measured noise simultaneously [3]; (3) It is desired to extract only the signals of interest (SOIs). (4) The difficulty of the singlechannel observation signal signature extraction based on ICA, it belongs to the extreme case of the underdetermined BBS problem [4]. Therefore, it would be important to develop approaches to extract only the desired signal with given signature instead of all source signals from the singlechannel observation signal.
ICA algorithm as the most important blind signal extraction (BSE) method has been used to extract the ICs, whose number is the same as the measured signals, but the SOIs are unknown. Hiroshi et al. [5] proposed timefrequency based ICA method to extract SOIs, but it needs some source signals to have dominant powers. W. Lu and J.C. Rajapakse [6, 7] proposed the constrained ICA (cICA) or ICA with reference (ICAR) algorithms by incorporating a prior information into the conventional ICA algorithm, which means that only a single statistical IC will be extracted from the mixed signals, but it does not specifically discuss how to generate a reference signal. ZhiLin Zhang [8] developed a morphological cICA algorithm to extract weak temporally correlated signals from a pregnant woman ECG data, this method used secondorder statistics based approach to design the suitable reference signal. ZhanLi Sun et al. [9] proposed an improved cICA by using the reference based unmixing matrix initialization, which overcame the unstable problem encountered in cICA algorithm. Changli Li et al. [10] proposed an improved ICAR algorithm for the noninvasive extraction of the fetal ECG (FECG), which alternately maximizes the negentropy contrast function for FastICA and the closeness measure function in ICAR. Xiang Wang et al. [11] extended the conventional cICA framework to the case of complexvalued mixing model and presented different prior information, the method is named as ICA with cyclostationary constraint (ICACC) and ICA with spatial constraint (ICASC). Zhiyang Wang et al. [12, 13] introduced cICA into the machine fault diagnosis, and attained some successful applications.
In practice, for most of the ICAbased methods, it should not be applied to the underdetermined BSS cases, in which the number of sensors is less than the source signals [4]. Especially in the extreme underdetermined BBS case, that is to say, singlechannel observation signal separation, the number of sensor is only one. This is a very undesirable requirement for realworld applications because the number of active source signals is unknown in advance in most practical situations. In this case, singlechannel observation signal mixing matrix is not invertible, and the traditional ICA or cICA methods fail to recover all sources, which also leads to the result that the desired signal cannot be extracted directly from the singlechannel observation signal. Therefore, singlechannel observation signal needs to be separated into several statistically independent components by using some approaches. Among these approaches, wavelet transform (WT) [14, 15] and empirical mode decomposition (EMD) [16, 17] are most usually employed to play the role of decomposing signal into various time scales. D.S Lee et al. [18] presented WT and PCAbased monitoring methods and illustrated its great potential in monitoring multiscale and multivariate processes. Wu, et al. [19, 20] combined continue WT with ICA to accomplish the early fault diagnosis of bearing. But WT requires choosing wavelet basis and decomposing layers, which makes it a nonselfadaptive signal processing method in nature. Empirical mode decomposition (EMD) algorithm [16, 17] can selfadaptively decompose any complex signal into a set of intrinsic mode functions (IMFs) according to the analyzed signal itself characteristic, and each IMF denotes a simple oscillatory mode in nature with different frequency component imbedded in the original signal. B. Mijovic et al. [21, 22] proposed a new method of sources separation from singlechannel signal based on EMD and ICA. Q. Miao et al. [23] used EMDbased ICA method to extract the bearing fault feature. But EMD still has some disadvantages, such as end effects and modes mixing. Wu and Huang [24] developed and improved the EMD algorithm substantially, and proposed the ensemble empirical mode decomposition (EEMD) algorithm, which effectively alleviates the mode mixing of EMD algorithm. M. Žvokelj et al. [25] developed a method of multivariate and multiscale monitoring of bearings using EEMD and PCA, and then proposed an approach of nonlinear multivariate and multiscale monitoring and signal denoising strategy using EEMD and KPCA [26]. Wang et al. [27] integrated EEMD and ICA to diagnosis wind turbine gearbox. After several years, Žvokelj et al. [28] again developed an EEMDbased multiscale ICA method to diagnosis the slewing bearing fault.
So far, the method of cICA combined with EEMD is seldom used to mechanical signals processing. Therefore, a socalled EEMDbased cICA method is proposed and applied to the BSE of singlechannel observation signal. The validity and practicability of this proposed method are verified through simulation and experiments of gear fault characteristics extraction with a missing tooth and a chipped tooth, respectively.
This paper is organized as follows: Section 2 introduces the ICA model and the mixing model of the singlechannel observation signal with source noise and measured noise. The singlechannel signal separation and fault feature extraction method of EEMDbased cICA are elucidated in Section 3. Then, simulation and experiments are demonstrated in Section 4 and Section 5, respectively. Finally, Section 6 provides a conclusion.
2. Mixing model of singlechannel measured signal based on ICA
2.1. Independent component analysis
In essence, ICA algorithm [14] assumes a set of $m$ observable measured signals $x\left(t\right)={\left[{x}_{1}\left(t\right),{x}_{2}\left(t\right),\dots ,{x}_{m}\left(t\right)\right]}^{T}$ to be a linear combination of $n$ unknown and statistically independent sources $s\left(t\right)={\left[{s}_{1}\left(t\right),{s}_{2}\left(t\right),\dots ,{s}_{n}\left(t\right)\right]}^{T}\left(n\le m\right)$. The time series usually have unit variance and uncorrelation by using a linear “whitening” transform. Then ICA mixing model can be expressed as:
where ${A}_{m\times n}$ is the mixing matrix, usually $m=n$.
ICA algorithm must find a separating or demixing matrix $W$ such that:
where $y\left(t\right)={\left[{y}_{1}\left(t\right),{y}_{2}\left(t\right),\dots ,{y}_{n}\left(t\right)\right]}^{T}$ is an approximate estimation of source signals $s\left(t\right)$.
2.2. Mixing model of singlechannel measured signal
In Eq. (1), if the row number $m$ of the mixing matrix $A$ is equal to 1, i.e $m=$ 1, then the classical ICA mixing model is rewritten as:
where ${A}_{1\times n}$ is an unknown nonsingular linear mixing vector, $A=\left[{a}_{1},{a}_{2},\dots ,{a}_{n}\right]$.
Consider the additional source noise and measured noise, and rewrite the Eq. (3) as:
where ${e}_{s}\left(t\right)$ and ${e}_{m}\left(t\right)$ represent the source noise and measured noise, respectively.
Eq. (4) shows the noisy ICA mixing model of the singlechannel observation signal $x\left(t\right)$. It belongs to the extreme case of the underdetermined BBS problem, and cannot be solved directly. For this reason, we developed an EEMDbased cICA method to separate fault signal from the singlechannel observation signal $x\left(t\right)$.
3. Singlechannel signal separation and fault feature extraction
3.1. Ensemble empirical mode decomposition
3.1.1. Empirical mode decomposition
Empirical mode decomposition (EMD) was pioneered by Huang et al. [16] in 1998. EMD has the ability of nonlinear multiresolution selfadaptive signal processing, and is very applicable to processing the nonstationary data. A complicated signal $x\left(t\right)$ can be decomposed into the sum of $n$ IMF components $\left\{{c}_{j}\left(t\right),j=1,2,\dots ,n\right\}$ and a residue ${r}_{n}\left(t\right)$ by EMD method:
3.1.2. EEMD algorithm
EMD method has been successfully applied to mechanical signal processing [17]. Nevertheless, EMD cannot extract mechanical fault feature accurately because of the mode mixing phenomenon, which can make physical meanings unclear. To alleviate this drawback, Wu and Huang [24] developed and improved the EMD algorithm substantially, and proposed the ensemble empirical mode decomposition (EEMD) algorithm. Y. H. Wang et al. discussed the computational complexity of EMD/EEMD algorithms [29]. The decomposition procedures of EEMD are expressed briefly as follows:
1. Add a differently generated white noise ${e}_{i}\left(t\right)$ with a different magnitude ${\sigma}_{ei}$ to the original signal $x\left(t\right)$ each time to generate a new signal:
2. Decompose the newly generated signal ${x}_{i}\left(t\right)$ into IMFs using the EMD method:
where ${c}_{i,j}\left(t\right)$, ${r}_{i,n}\left(t\right)$ and ${n}_{i}$ represent the $j$th IMF, the residue and the IMFs’ number during the $i$th trial, respectively.
3. Calculate the ensemble means of the corresponding IMFs of ${N}_{1}$ times decompositions, and take it as the final result:
4. Finally, the original signal $x\left(t\right)$ is formed as follows:
3.1.3. Criterions of IMF selection
EEMD method can effectively alleviate the mode aliasing, but it will produce false components during its decomposition procedures. Therefore, we propose the following criterions of IMFs selection in order to eliminate the influence of false IMFs.
3.1.3.1. Correlation coefficientbased
The correlation coefficient $\rho $ between IMF ${\stackrel{}{c}}_{j}\left(t\right)$and original signal $x\left(t\right)$ is as follows:
When IMF includes some fault characteristics, the correlation coefficient between the IMF and the original signal is relatively larger, on the contrary, it is much smaller.
3.1.3.2. Kurtosisbased
However, when the signaltonoise ratio (SNR) of the observation signal $x\left(t\right)$ is extremely low, that is to say, the concealed fault information is very weak. In this case, even if the IMF includes effective fault information, the correlation coefficient between the corresponding IMF and the original signal could be also very small. Therefore, we must introduce another criterion of IMF section, i.e. kurtosisbased combined with the correlation coefficientbased criterion. The kurtosis of the IMF is expressed as:
In Eqs. (1011), ${\stackrel{}{c}}_{j}\left(k\right)$ and $x\left(k\right)$ are zeromean, i.e. ${\mu}_{x}={\mu}_{{\stackrel{}{c}}_{j}}=$ 0, $\sigma $ denotes the standard deviation, and $N$ is the data length.
Usually, the larger the kurtosis value of the IMF, the more prominent the effective fault information of the corresponding IMF.
3.2. cICA principle
Constrained independent component analysis (cICA) [6, 7] method is derived from independent component analysis (ICA) algorithm. By incorporating an interesting priori information into the traditional ICA algorithm, cICA algorithm forms a constraint optimization problem, and ensures that the ICA model output is a necessarily desired independent component (IC), which is closest to a corresponding reference signal $r\left(t\right)$ [12]. The reference signal $r\left(t\right)$ with interesting fault feature denotes the inequality constrained condition but need not be a perfect match with the desired IC. We take $\epsilon \left(r,y\right)$ as the closeness measure norm between the IC $y\left(t\right)$ and the corresponding reference signal $r\left(t\right)$. Note that the desired IC, which is extracted from the new observation signal vector $\mathbf{x}\left(t\right)={\left[{x}_{1}\left(t\right),{x}_{2}\left(t\right),\dots ,{x}_{m}\left(t\right)\right]}^{T}$, is the one and only the one closest to the corresponding constructed reference signal $r\left(t\right)$, which is satisfied the following the inequality relationship:
where ${\mathbf{w}}_{*}$ is the optimum demixing vector corresponding to the desired output IC, and ${w}_{i}$, $i=1,2,\dots ,l1\left({w}_{i}\ne {w}_{0}\right)$ are any other $l1$ local optimal solutions corresponding to the undesired output ICs. Thus, an inequality constraint, only when the optimum equation $y={y}_{*}={\mathbf{w}}_{*}^{T}\mathbf{x}$ is satisfied, is expressed as follows:
where $\zeta \in \left[\epsilon \left(r,{\mathbf{w}}_{*}^{T}\mathbf{x}\right),\epsilon \left(r,{\mathbf{w}}_{1}^{T}\mathbf{x}\right)\right]$ is a threshold parameter, the closeness measure norm $\epsilon \left(r,y\right)$ is usually expressed by $\epsilon \left(r,y\right)=E\left\{{\left(ry\right)}^{2}\right\}$.
The model of cICA framework [5, 6] as a constrained optimization problem is defined as:
$\mathrm{s}.\mathrm{t}.\mathrm{}g\left(y\right)\le 0,$
$h\left(y\right)=E\left({y}^{2}\right)1=0,h\left(r\right)=E\left({r}^{2}\right)1=0,$
where $J\left(y\right)$ denotes the negentropy function, $f\left(\xb7\right)$ is an any nonquadratic function, $\rho $ is a positive constant, $\upsilon $ is a Gaussian variable with zeromean and unit variance, $g\left(y\right)$ is the closeness constraint described in Eq. (13), and the equality constraints $h\left(\xb7\right)$ ensure that the output $y\left(t\right)$ and the reference signal $r\left(t\right)$ have unitvariance.
The model of cICA algorithm is efficiently solved by the use of an augmented Lagrangian function [7]. At the same time, we use the signaltointerference ratio (SIR) index [12] to evaluate the extraction quality of the cICA algorithm. The larger the SIR, the better the extraction effect of cICA algorithm. More details about the model of cICA framework are expressed as a constrained optimization problem in Refs. [613].
3.3. Constructing reference signal for cICA in gearbox diagnostics
The faulty signal in gear transmission system mostly appears as a periodical impact sequence. Hence, we may select a series of pulses or square wave as the suitable reference signal, such as Eq. (13) below:
where ${f}_{m}$ is the gear meshing frequency, $\theta $ is the initial phase angle or timedelay and $w$ is the duty ratio or impulsewide.
3.4. Procedures of the proposed approach
The proposed method is a good candidate for extracting the desired source signal from the singlechannel measured signal with source noises and measured noise. Its procedures can be described as follows:
Step 1: Decompose the gearbox singlechannel measured signal $x\left(t\right)$ according to Eq. (6), and obtain $n$ IMF components.
Step 2: Compute the kurtosis of each IMF and correlation coefficient between each IMF and the original signal $x\left(t\right)$, select the IMF components with greater kurtosis and correlation coefficient to compose a new observation vector with the original signal $x\left(t\right)$, then take the new vector as the cICA algorithm input, given the new vector is $x\left(t\right)={\left[{x}_{1}\left(t\right),{x}_{2}\left(t\right),\dots ,{x}_{m}\left(t\right)\right]}^{T}$, $m\le n$.
Step 3: Construct the reference signal $r\left(t\right)$ with the desired fault signature, then extract the fault signal ${y}_{*}\left(t\right)$ with cICA method.
Step 4: Analyze the extracted fault signal ${y}_{*}\left(t\right)$ with Hilbert envelope spectrum and obtain the desired fault feature.
4. Simulation analysis
The aim of the simulation is to extract the desired lowfrequency weak fault signal from the mixed data set. According to Eq. (16), we generated three source signals, ${s}_{1}$, ${s}_{2}$ and ${s}_{3}$, whose time domain waveforms are shown in Fig. 1:
where signal ${s}_{2}$ is desired to be extracted, but its energy is weak. The parameter values of three simulated source signals, ${s}_{1}$, ${s}_{2}$ and ${s}_{3}$ in Eq. (16) are listed in Table 1.
Table 1Parameter values of the simulated signal
${f}_{pm}$  ${f}_{C}$  ${f}_{m}$  ${f}_{r}$  ${X}_{1}$  ${X}_{2}$  ${\theta}_{1}$, ${\theta}_{2}$  ${m}_{11}^{A}{m}_{21}^{A}$  ${m}_{12}^{A}$, ${m}_{22}^{A}$  ${f}_{1}$  ${X}_{31}$  ${X}_{32}$ 
530 Hz  5.3 Hz  46.5 Hz  1.5 Hz  6  2  0  1  0.5  25 Hz  2  1 
The source noise ${e}_{s1}$, ${e}_{s2}$ and ${e}_{s3}$ are respectively added to the three source signals ${s}_{1}$, ${s}_{2}$ and ${s}_{3}$ with SNR of –5dB. Three noisy signals are randomly mixed by a mixing vector $A$ and get a singlechannel mixed signal. Then the mixed signal is added a Gaussian white noise ${e}_{m}\left(t\right)$ with the amplitude standard deviation of 2. Finally, we obtain a singlechannel simulated signal $x\left(t\right)$, whose timedomain waveform, FFT spectrum and envelope spectrum are shown in Fig. 2. Among the three source signals, signal ${s}_{2}\left(t\right)$ without source noise ${e}_{s2}\left(t\right)$ is expected to be extracted from the mixed signal $x\left(t\right)$ by using the proposed method.
From Fig. 2, the lowfrequency modulation frequency ${f}_{r}$ (1.5 Hz) is invisible except for the frequency components 2${f}_{m}$ (93 Hz), ${f}_{pm}$ (530 Hz) and the modulated frequency ${f}_{c}$ (5.3 Hz).
Fig. 1Time domain waveforms of three simulated source signals without noise
Fig. 2Mixed signal xtand its spectrum and envelope spectrum with SNR of –5 dB
a) Mixed signal $x\left(t\right)$
b) FFT spectrum of mixed signal $x\left(t\right)$
c) Envelope spectrum of mixed signal $x\left(t\right)$
Fig. 3EEMD decomposition results of the mixed signal xt with SNR of –5 dB
a)
b)
Fig. 3 depicts the decomposition results with EEMD method for the mixed signal $x\left(t\right)$. The kurtosis of each IMF and the correlation coefficients between each IMF and the signal $x\left(t\right)$ are listed in Table 2. Among the IMFs, although the correlation coefficient value of ${c}_{1}$ is very big, it is a high frequency noise and not to be considered. So, based on the criterions of kurtosis and correlation coefficient, we select the IMFs ${c}_{2}$, ${c}_{3}$, ${c}_{4}$ and ${c}_{5}$ ($K>$ 3.0 and $\rho >$ 0.2) combined with the original signal $x\left(t\right)$ to construct a new observation vector. We generate a suitable reference signal $r\left(t\right)$ (shown in Fig. 4(a)) with frequency ${f}_{m}$ (46.5 Hz) of signal ${s}_{2}$, and then use the cICA method to successfully extract a desired source signal ${y}_{*}\left(t\right)$ (shown in Fig. 4(b)) as the closeness of the simulated signal ${s}_{2}\left(t\right)$. The SIR value of the extracted signal ${y}_{*}\left(t\right)$ is 3.16 dB.
Table 2Kurtosis and correlation coefficients of IMFs by EEMD method with SNR of –5 dB
IMFs  ${c}_{1}$  ${c}_{2}$  ${c}_{3}$  ${c}_{4}$  ${c}_{5}$  ${c}_{6}$  ${c}_{7}$  ${c}_{8}$  ${c}_{9}$  ${r}_{9}$ 
$K$  2.16  3.16  3.00  4.22  3.58  2.51  2.75  2.21  3.19  3.74 
$\rho $  0.685  0.552  0.296  0.298  0.216  0.192  0.055  0.031  0.014  0.008 
Fig. 4Suitable reference signal rt and its extracted desired signal y*t using EEMDbased cICA
a) Constructed suitable reference signal $r\left(t\right)$
b) Extracted signal ${y}_{*}\left(t\right)$ with EEMDbased cICA method
The FFT spectrum and envelope spectrum of the extracted signal ${y}_{*}\left(t\right)$ are shown in Fig. 5. Apparently, the modulated sidebands of lowfrequency ${f}_{r}$ (1.5 Hz) around the center frequency 2${f}_{m}$_{}(93 Hz) is very evident. Of course, the original source signal ${s}_{2}$ is not completely recovered for the strong source noise, but the lowfrequency weak feature has been extracted from the mixed signal $x\left(t\right)$ with other strong signals and noise influence.
Fig. 5FFT spectrum and envelope spectrum of the extracted signal y*t
a) FFT spectrum of the extracted signal ${y}_{*}\left(t\right)$
b) Envelope spectrum of the extracted signal ${y}_{*}\left(t\right)$
Fig. 6 shows the decomposition results and its FFT spectra with EMDbased cICA method for the mixed signal $x\left(t\right)$. The SIR value of the extracted signal $y\left(t\right)$ is 1.58 dB. Obviously, EMDbased cICA method can also expresses the feature frequency ${f}_{r}$ (1.5 Hz) of signal $x\left(t\right)$, but its effect is a bit worse than the EEMDbased cICA method. The simulated results show that the proposed method can effectively extract the lowfrequency weak gear fault signals from the singlechannel observation signal.
Fig. 6Extracted signal y*t using EMDbased cICA and its FFT spectrum and envelope spectrum
a) Extracted signal $y\left(t\right)$ with EMDbased cICA method
b) FFT spectrum of the extracted signal $y\left(t\right)$
c) Envelope spectrum of the extracted signal $y\left(t\right)$
5. Experimental signals analysis
Next, we use the realworld signal from a multistage gearbox to verify the effectiveness of our approach, the singlechannel vibration signals with a missing tooth and a chipped tooth localized on the gear ${Z}_{3}$ (= 36) of the twostage fixedshaft gearbox in this experiment are studied, respectively. The schematic diagram of gearbox test rig is shown in Fig. 7.
Fig. 7Schematic diagram of gearbox test rig
Table 3Characteristic frequencies of gearbox
Items  Singlestage planetary gearbox  Twostage fixedshaft gearbox  
Tooth number $Z$  ${Z}_{S}=$ 28  ${Z}_{P}=$ 36  ${Z}_{R}=$ 100  ${Z}_{1}=$ 29  ${Z}_{2}=$ 100  ${Z}_{3}=$ 36  ${Z}_{4}=$ 90 
Shaft rotating frequency $f$/ Hz  ${f}_{S}={f}_{r}=$ 24  ${f}_{C}=$ 5.25  ${f}_{r1}=$ 5.25  ${f}_{r2}=$ 1.52  ${f}_{r3}=$ 0.6  
Meshing frequency $f$_{}/ Hz  ${f}_{pm}=$ 525.0  ${f}_{m1}=$ 152.3  ${f}_{m2}=$ 54.8 
Experimental fault gear photos are shown in Fig. 8. The rotating frequency ${f}_{r}$ of motor is 24.0 Hz, sampling frequency is 5120 Hz and data length is 15 kB samples. Characteristic frequencies are listed in Table 3, where ${f}_{r}$. ${f}_{S}$ and ${f}_{C}$ denote the rotating frequency of motor, sun gear, planet carrier, respectively, ${f}_{m}$ and ${f}_{pm}$ denote the meshing frequency of fixedshaft gearbox and planetary gearbox, respectively. The gear fault characteristic frequency is ${f}_{r2}$ (1.52 Hz), and its corresponding meshing frequency is ${f}_{m1}$ (152.3 Hz).
Fig. 8Photos of faulty gear Z3: a) missing a tooth; b) a chipped tooth
a)
b)
5.1. A missing tooth signal analysis
Fig. 9 illustrates the FFT spectrum and envelope spectrum of the gear vibration signal ${x}_{1}\left(t\right)$ with a missing tooth. The main frequency components are the meshing frequency ${f}_{pm}$ (525 Hz) of planetary gearbox and its harmonics, the modulated frequency is the planet carrier rotating frequency ${f}_{c}$ (5.25 Hz), which does not mean that the planetary gearbox has any fault according to the reference [30]. However, it is difficult to distinguish any obvious fault feature frequency ${f}_{r2}$ (1.52 Hz) because the fault feature with a missing tooth is not apparent.
Fig. 9Singlechannel signal x1t with a missing tooth and its FFT spectrum & envelope spectrum
a) Singlechannel observation signal ${x}_{1}\left(t\right)$ with a missing toth
b) FFT spectrum of ${x}_{1}\left(t\right)$
c) Envelope spectrum of ${x}_{1}\left(t\right)$
Table 4Kurtosis and correlation coefficients of IMFs by EEMD with a missing tooth
IMFs  ${c}_{1}$  ${c}_{2}$  ${c}_{3}$  ${c}_{4}$  ${c}_{5}$  ${c}_{6}$  ${c}_{7}$  ${c}_{8}$  ${c}_{9}$  ${c}_{10}$ 
$K$  3.29  4.15  3.79  10.09  3.32  2.95  1.91  2.96  2.40  2.64 
$\rho $  0.693  0.795  0.530  0.155  0.139  0.078  0.048  0.017  0.007  0.0005 
Fig. 10 depicts the decomposition results of the singlechannel observation signal ${x}_{1}\left(t\right)$ in Fig. 9(a) by using EEMD method. The kurtosis of each IMF (${c}_{1}$${c}_{10}$) and the correlation coefficient between each IMF (${c}_{1}$${c}_{10}$) and the fault signal ${x}_{1}\left(t\right)$ with a missing tooth are listed in Table 4. Based on the criterions of kurtosis and correlation coefficient, we select the IMFs ${c}_{1}$${c}_{5}$ ($K>$ 3.2 and $\rho >$ 0.1) combined with the original signal ${x}_{1}\left(t\right)$ to construct a new observation vector. Through generating a proper reference signal $r\left(t\right)$ (shown in Fig. 11(a)) with the meshing frequency of ${f}_{m2}$ (54.8 Hz), we successfully extract the desired fault signal ${y}_{1*}\left(t\right)$ (shown in Fig. 11(b)) with cICA method. Obviously, the periodical impacts at $T=$ 0.67 s $(\approx 1/{f}_{r2}=$ 1/1.52) in time domain are very evident. The corresponding FFT spectrum and envelope spectrum of the extracted signal ${y}_{1*}\left(t\right)$ are shown in Fig. 12(a) and (b), respectively. As shown in Fig. 12, it can be clearly distinguished that there are plentifully modulated sidebands around the right side of frequency 2${f}_{m2}$ (109.6 Hz). The obvious fault feature frequency is ${f}_{r2}$ (1.52Hz), which is corresponding to the shaft 2 rotating frequency ${f}_{r2}$_{}of the fault gear ${Z}_{3}$ (= 36) with a missing tooth on the fixedshaft gearbox.
Fig. 10EEMD decomposition results of the original signal x1t with a missing tooth
a)
b)
Fig. 11Proper reference signal rt and its extracted desired fault signal y1*t using EEMDbased cICA method with a missing tooth
a) Proper reference signal $r\left(t\right)$
b) Extracted fault signal ${y}_{1*}\left(t\right)$
Fig. 12FFT spectrum and envelope spectrum of the extracted desired fault signal y1*
FFT spectrum ${y}_{1*}\left(t\right)$
b) Envelope spectrum ${y}_{1*}\left(t\right)$
To compare the effect, the extracted result of EMDbased cICA method is shown in Fig. 13, the result is not as good as the EEMDbased cICA method.
Fig. 13Extracted results using EMDbased cICA method with a missing tooth
a) Extracted signal ${y}_{1}\left(t\right)$
b) FFT spectrum of ${y}_{1}\left(t\right)$
c) Envelope spectrum of ${y}_{1}\left(t\right)$
5.2. A chipped tooth signal analysis
Fig. 14 demonstrates the FFT spectrum and envelope spectrum of the gear fault vibration signal ${x}_{2}\left(t\right)$ with a chipped tooth. The main frequency components are also the meshing frequency ${f}_{pm}$ (525 Hz) of planetary gearbox and its high order harmonics, and the fault modulated frequency is still the planet carrier rotating frequency ${f}_{c}$ (5.25 Hz), which is uninterested for us. However, it is much difficult to identify the fault feature frequency ${f}_{r2}$ (1.52 Hz) because the fault signal with a chipped tooth is much fainter.
Fig. 14Singlechannel signal x2t with a chipped tooth and its FFT spectrum and envelope spectrum
a) Singlechannel observation signal ${x}_{2}\left(t\right)$ with a chipped tooth
b) FFT spectrum ${x}_{2}\left(t\right)$
c) Envelope spectrum ${x}_{2}\left(t\right)$
Fig. 15 shows the decomposition results of the singlechannel observation signal ${x}_{2}\left(t\right)$ in Fig. 14(a) using EEMD method. The kurtosis of each IMF (${c}_{1}$${c}_{10}$) and the correlation coefficient between each IMF (${c}_{1}$${c}_{10}$) and the fault signal ${x}_{1}\left(t\right)$ with a chipped tooth are listed in Table 5. Based on the criterions of kurtosis and correlation coefficient, we select the IMF components ${c}_{1}$${c}_{5}$ ($K>$ 3.2 and $\rho >$ 0.1) combined with the original signal ${x}_{2}\left(t\right)$ to construct a new observation vector. The unchanged reference signal $r\left(t\right)$ is shown in Fig. 11(a), then we utilize cICA method to successfully extract the desired fault signal ${y}_{2*}\left(t\right)$, whose time domain waveform, FFT spectrum and envelop spectrum are shown in Fig. 16. From Fig. 16(a), the periodical impacts at $T=$ 0.67 s ($\approx 1/{f}_{r2}=$ 1/1.52) in time domain is evident, but it is not as clear as that shown as in Fig. 11 (b). In Fig. 16 (b), we can clearly distinguish that there are some modulated sidebands around the right side of frequency 2${f}_{m2}$ (109.6 Hz). The fault feature frequency is 1.52 Hz from Fig. 16(c), which is also corresponding to the shaft 2 rotating frequency ${f}_{r2}$ of the faulty gear ${Z}_{3}$ (= 36) with a chipped tooth on the fixedshaft gearbox.
Similarly, if we use the EMDbased cICA method to analysis the signal ${x}_{2}\left(t\right)$, the effective lowfrequency fault feature ${f}_{r2}$ (1.52 Hz) will not be extracted, as shown in Fig. 17.
Table 5Kurtosis and correlation coefficients of IMFs by EEMD with a localized chipped tooth
IMFs  ${c}_{1}$  ${c}_{2}$  ${c}_{3}$  ${c}_{4}$  ${c}_{5}$  ${c}_{6}$  ${c}_{7}$  ${c}_{8}$  ${c}_{9}$  ${c}_{10}$ 
$K$  4.14  3.44  4.22  3.63  3.29  2.71  2.59  2.74  2.53  2.48 
$\rho $  0.650  0.696  0.168  0.161  0.107  0.085  0.065  0.005  0.003  0.001 
Fig. 15EEMD decomposition results of the original signal x2t with a chipped tooth
a)
b)
The experimental results indicate that the proposed method is effective and available for lowfrequency fault feature extraction, especially for the weak fault feature extraction of the gearbox singlechannel observation signal.
Fig. 16Extracted results using EEMDbased cICA method with a chipped tooth
a) Extracted signal ${y}_{2*}\left(t\right)$
b) FFT spectrum of ${y}_{2*}\left(y\right)$
c) Envelope spectrum ${y}_{2*}\left(t\right)$
Fig. 17Extracted results using EMDbased cICA method with a chipped tooth
a) Extracted signal ${y}_{2}\left(t\right)$
b) FFT spectrum of ${y}_{2}\left(t\right)$
c) Envelope spectrum of ${y}_{2}\left(t\right)$
6. Conclusions
Aiming at the shortcomings of traditional ICA method and trying to solve the key problem of the extremely underdetermined singlechannel blind source separation and fault feature extraction with source noise and measured noise, we proposed an approach combining the advantages of EEMD and cICA. Through simulation and experiments of gear lowfrequency fault feature extraction for the singlechannel observation signal, the results verify the effectiveness of this proposed method, which is suitable for the gearbox fault diagnosis, especially for the lowfrequency and weak fault diagnosis of gearbox. Further study is yet required to introduce the additional denoising processes to enhance this proposed method performance in the low SNR case. Notably, this proposed method is also suitable for other signals feature extraction that show periodicity characteristics, such as the bearing fault signal, the internal combustion engine fault signal.
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About this article
This work was supported by the Project of China National Coal Association (Grant No. MTKJ2015261), Doctoral Fund of Henan Polytechnic University (Grant No. B201728) and Foundation of innovative research team of Henan Polytechnic University (Grant No. T20173). The authors would like to thank the reviewers for many valuable comments and suggestions.