EEMD-Based cICA method for single-channel signal separation and fault feature extraction of gearbox

1. Introduction

In general, the goal of independent component analysis (ICA) [1-4] 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 single-channel 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 single-channel 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 time-frequency 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 (ICA-R) 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. Zhi-Lin Zhang [8] developed a morphological cICA algorithm to extract weak temporally correlated signals from a pregnant woman ECG data, this method used second-order statistics based approach to design the suitable reference signal. Zhan-Li 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 ICA-R algorithm for the non-invasive extraction of the fetal ECG (FECG), which alternately maximizes the negentropy contrast function for FastICA and the closeness measure function in ICA-R. Xiang Wang et al. [11] extended the conventional cICA framework to the case of complex-valued mixing model and presented different prior information, the method is named as ICA with cyclostationary constraint (ICA-CC) and ICA with spatial constraint (ICA-SC). Zhiyang Wang et al. [12, 13] introduced cICA into the machine fault diagnosis, and attained some successful applications.

In practice, for most of the ICA-based 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, single-channel observation signal separation, the number of sensor is only one. This is a very undesirable requirement for real-world applications because the number of active source signals is unknown in advance in most practical situations. In this case, single-channel 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 single-channel observation signal. Therefore, single-channel 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 PCA-based 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 non-self-adaptive signal processing method in nature. Empirical mode decomposition (EMD) algorithm [16, 17] can self-adaptively 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 single-channel signal based on EMD and ICA. Q. Miao et al. [23] used EMD-based 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 non-linear 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 EEMD-based 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 so-called EEMD-based cICA method is proposed and applied to the BSE of single-channel 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 single-channel observation signal with source noise and measured noise. The single-channel signal separation and fault feature extraction method of EEMD-based 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 single-channel measured signal based on ICA

3. Single-channel signal separation and fault feature extraction

4. Simulation analysis

The aim of the simulation is to extract the desired low-frequency 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.

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 single-channel 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 single-channel simulated signal $x\left(t\right)$, whose time-domain 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 low-frequency 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.

Fig. 4Suitable reference signal rt and its extracted desired signal y*t using EEMD-based cICA

a) Constructed suitable reference signal $r\left(t\right)$

b) Extracted signal $y_{*}\left(t\right)$ with EEMD-based 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 low-frequency $f_r$ (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 low-frequency 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 EMD-based 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, EMD-based 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 EEMD-based cICA method. The simulated results show that the proposed method can effectively extract the low-frequency weak gear fault signals from the single-channel observation signal.

Fig. 6Extracted signal y*t using EMD-based cICA and its FFT spectrum and envelope spectrum

a) Extracted signal $y\left(t\right)$ with EMD-based 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 real-world signal from a multi-stage gearbox to verify the effectiveness of our approach, the single-channel vibration signals with a missing tooth and a chipped tooth localized on the gear $Z_3$ (= 36) of the two-stage fixed-shaft 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

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 fixed-shaft 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. 9Single-channel signal x1t with a missing tooth and its FFT spectrum & envelope spectrum

a) Single-channel 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 single-channel 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 fixed-shaft 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 EEMD-based 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 EMD-based cICA method is shown in Fig. 13, the result is not as good as the EEMD-based cICA method.

Fig. 13Extracted results using EMD-based 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. 14Single-channel signal x2t with a chipped tooth and its FFT spectrum and envelope spectrum

a) Single-channel 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 single-channel 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 fixed-shaft gearbox.

Similarly, if we use the EMD-based cICA method to analysis the signal $x_2\left(t\right)$, the effective low-frequency 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 low-frequency fault feature extraction, especially for the weak fault feature extraction of the gearbox single-channel observation signal.

Fig. 16Extracted results using EEMD-based 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 EMD-based 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 single-channel 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 low-frequency fault feature extraction for the single-channel observation signal, the results verify the effectiveness of this proposed method, which is suitable for the gearbox fault diagnosis, especially for the low-frequency 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.