**1. Introduction**

The hydraulic pump is a power source and supplies pressure energy to hydraulic systems. The pump has been applied in many important industrial fields such as aeronautics, astronautics, metallurgy, petrochemical engineering, and engineering machinery. The hydraulic system in aforementioned fields possess the characteristics such as large-scale continuation, integration and automation, and thus the working condition of pump facing some challenges as high temperature, high pressure and high speed [1,2]. Unfortunately, these challenges accelerate the deterioration of the health status of pumps, so it is very important to diagnose the faults for hydraulic pump [3,4]. The mechanical and fluid impact can cause the vibration of the pump, and vibration severity increases once the pump is broken [5,6]. A lot of fault feature information is contained in the vibration signal, and the signal is also contaminated by many interferences [7,8]. Recently, many scholars have applied vibration signal to diagnose the faults in domestic and foreign. Lan et al. applied wavelet packet transform (WPT), local tangent space alignment (LTSA), empirical mode decomposition (EMD) and local mean decomposition (LMD) to process the fault signal of the pump to extract the eigenvectors, and then the faults could be diagnosed by an extreme learning machine (ELM) [9]. Sun et al. calculated cyclic autocorrelation functions (CAFs) of a fault signal of the pump, and the corresponding slices were extracted from the CAFs and processed by fast Fourier transform (FFT), and then indicators were extracted from the FFT spectra to diagnose the faults [10]. Considering that the early fault signal of the pump is a periodic weak signal, Zhao et al. proposed an intermittent chaos, sliding window symbol

sequence statistics-based method, and it was used to detect the early faults [11]. Du et al. presented a new method based on the layered clustering algorithm to diagnose multiple faults of an aircraft pump, and thus the faults could be diagnosed based on risk priority number and their severity layer by layer [12]. Lu et al. applied ensemble empirical mode decomposition (EEMD) to decompose the fault signal of the pump, and eigenvectors were extracted in time, frequency and time-frequency domains as the input, and then the optimized support vector regression (SVR) model was used to diagnose the faults [13].

Wavelet transform (WT) is an effective tool to process a nonlinear and non-stationary signal [14,15]. The time-domain signal can also be decomposed by WT into different frequency bandwidth groups [16,17]. Compared with short-time Fourier transform (STFT), the basic difference between WT and STFT is the basis function. The STFT only uses sine and cosine as basis functions. WT also adopts many kinds of wavelet functions specifying a certain mathematical property. Furthermore, WT fuses the principles of Fourier transform basis function and STFT window function to develop a new oscillation and attenuation wavelet basis function. Thus, WT overcomes the shortcoming of the fixed window, and the new function can change the resolution of time and frequency information with the change of scale factor. Owing to the width adjustment of WT, the signal details can be analyzed [18,19]. Currently, WT has been widely used to diagnose some faults at home and abroad. Aiming at the typical faults of bearing, some researches proposed the method based on WT and utilized it to diagnose the faults successfully [18,19]. Moreover, the method based on WT was used by Kordestani et al. to diagnose the faults of spoiler system effectively [20]. Pointing at the faults of planetary gearboxes, Zhao et al. also proposed a method based on WT, and proved the faults could be diagnosed effectively [21].

Although WT has widely used in some fields, it has some imperfect aspects. Firstly, in matching wavelet basis function with morphological features of a signal, there is no accuracy matching principle and criterion to follow. Secondly, the signal possesses many kinds of features, but the wavelet basis function is not changed when it works, and thus WT is not adaptive. Thirdly, if the basis function and scale factor are selected, the resolution is fixed, and thus it does not have the adaptability. Thirdly, although WT has the capability of multi-scale and multi-resolution, some parts of the signal contained in the window must be approximate stationary (pseudo stationary) [22].

EMD was proposed by Huang in 1998, and it is good at processing the nonlinear and non-stationary signal, and the signal is made up of some multi-component modes [23]. One single-component mode can be separated adaptively from the signal based on its morphological feature information (local maxima points), and the single-component mode is called intrinsic mode function (IMF) [24,25]. Thus far, a large number of studies about EMD have been made by scholars on a global scale. EMD was effectively applied to monitor the healthy condition of wind turbine [19]. Moreover, EMD was also employed to process some fault signals such as gearbox [25], bearings [26–28], and so on.

Although EMD is widely applied, the cubic splines are employed in EMD to fit the lower and upper envelopes in each sift, which can lead to two obvious defects [29]. One is mode-mixing, and another is endpoint effect. In mode mixing, an IMF may be consisted of two or more component modes, and thus it may lead to wrong decomposition. In the endpoint effect, the endpoints of an IMF are divergent, and the divergence is gradually into the inside with the iteration of the decomposition. If the iterative number becomes larger, IMF will be more distorted, then mode mixing and illusive components happen.

Aiming to improve the shortcomings of WT and EMD, a method for processing the nonlinear and non-stationary signal is proposed by Gilles, and the method is called empirical wavelet transform (EWT) [30]. In EWT, Fourier amplitude spectrum of a multi-component mode signal is segmented, and the wavelet orthogonal basis function is established based on each segment. Thus, the signal can be decomposed by EWT into several AM-FM single-component modes, and each mode is compact, supported, and centered on a specific frequency. If the signal is contaminated by interference components, there are also interference components in Fourier amplitude spectrum of the signal, and the segment of EWT is determined by the spectrums of the signal and the interference components together. Thus, the spectrum increase of interference component leads to many segments, which leads to mode mixing and over-decomposition [31]. Dong et al. applied sparsity to improve EWT, and adopted it to diagnose the faults of rolling element bearings [31]. Jiang et al. proposed a novel method based on EWT and ambiguity correlation classifiers, and EWT was used to decompose the fault signal, and ambiguity correlation classifiers was adopted to diagnose the faults of rolling element bearings [32]. Cao et al. used the EWT to decompose the signal of wheel-bearing, and the mode which was rich of fault feature information was selected to analyze the spectrum, and then the faults were diagnosed [33].

In order to resolve the above problems, a new method named improved empirical wavelet transform (IEWT) is proposed. Firstly, the power spectrum of the loose slipper fault signal is obtained. Secondly, different threshold values are adopted to eliminate the power spectrum of the interference components, and the bad influence of interference component on segment acquirement is largely reduced. Then, the best segment number is obtained based on the feature energy ratio (*FER*). Thus, the fault signal of hydraulic pump can be best decomposed by IEWT. The acquired results provide an important basis for the application extension to faults diagnosis study of other rotating machinery.

The rest of the paper is organized as follows. In Section 2, the algorithms of *FER* and IEWT are introduced, and then the flowchart of IEWT is presented. Section 3 depicts the EWT application in a simulated signal and a hydraulic pump fault signal in detail. In Section 4, some conclusions of this investigation are summarized.

#### **2. Methodology**

#### *2.1. Feature Energy Ratio*

Impact vibration is often caused by the faults of rotating machinery, and the impact vibration energy is generated, and thus the energy is rich in fault feature information. The big value of feature energy ratio (*FER*) signifies that the amount of fault feature information is large.

For the sake of measuring amount of fault features information in a signal, the *FER* is proposed in reference [34]. It can be rewritten as

$$FER = (E\_1 + E\_2 + \dots \ + E\_n)/E \tag{1}$$

where *E*1, *E*2, ... , *En* are respectively the energy, and they are respectively presented in fault feature frequency and its harmonics, and *E* is the total energy of the signal.

#### *2.2. Improved Empirical Wavelet Transform*

The algorithm of IEWT is described as follows:

(1) Calculating power spectrum of signals

Different from the Fourier amplitude spectrum of EWT, IEWT is based on power spectrum.

Let *x*(*t*) = (*x*1, *x*2, ... , *xn*) be a signal, and *X*(ω) is its Fourier amplitude spectrum in the frequency domain, and then the power spectrum of the signal is denoted as *P*, and it is defined as

$$P = \lim\_{T \to \infty} \frac{1}{2T} \int\_{-T}^{T} x^2(t)dt = \frac{1}{2\pi} \int\_{-\infty}^{\infty} \lim\_{T \to \infty} \frac{1}{2T} \left| X\_T(\omega) \right|^2 d\omega \tag{2}$$

#### (2) Applying the threshold processing

The step is also different from EWT, and the threshold processing is introduced into IEWT. The threshold values are defined as

$$THVA = coefficient \times \text{mean}(P) \tag{3}$$

where *coe*ffi*cient* is an integer and equals to 1, 2, ... , *L*.

Figure 1 demonstrates how the threshold processing works.

**Figure 1.** Demonstration of threshold processing.

In Figure 1, the threshold value *THVA* is applied to eliminate the spectrums whose values are smaller than the values (the eliminated spectrum values are set to 0), and thus a new spectrum distribution *Pcoe*ffi*cient* can be obtained. Thus, the bad influence of interference components on adaptive segment acquirement is much reduced.

(3) Decomposing the Signals

Periodicity of a normalized Fourier axis is 2 π, and the discussion is restricted to ω ∈ [0, π] based on *Pcoe*ffi*cient*. It is supposed that the Fourier support [0, π] is segmented into *N* contiguous segments, which means that there are *N* + 1 boundaries. However, 0 and π are always used in definition and it is need to be found *N* − 1 extra boundaries. To find the boundaries, local maxima value of power spectrum are selected, and the values are sorted in decreasing order (0 and π are excluded). It is assumed that the algorithm found *M* maxima, and two cases can appear:

(1) *M* ≥ *N*: enough maxima are selected to define the wanted segment number, and then the first *N* − 1 maxima are adopted.

(2) *M* ≤ *N*: the component mode number of a signal is smaller than expected, and all the selected maxima are kept, and rest *N* to the appropriate value.

In Figure 2, each segments is defined as <sup>Λ</sup>*<sup>n</sup>* <sup>=</sup> [ω*n*−1, <sup>ω</sup>*n*], and it can be found that **<sup>U</sup>***<sup>N</sup> <sup>n</sup>*=**<sup>1</sup>** Λ*<sup>n</sup>* = [0, π]. Centered around each ω*n*, a transition phase (the gray hatched areas on Figure 2) *Tn* of width 2τ*<sup>n</sup>* is defined.

**Figure 2.** Segments of the Fourier axis.

*Processes* **2019**, *7*, 824

Based on construction of Littlewood-Paley and Meyer's wavelets, some empirical wavelets of EWT are actually band pass filters on each Λ*n*. Then ∀*n* > 0, it can be defined that the empirical scaling function and the empirical wavelets by expressions of Equations (4) and (5) respectively.

$$\mathcal{O}\_{n}(\omega) \begin{cases} 1 & \text{if } |\omega| \le \omega\_{n} - \tau\_{n} \\ \cos\left[\frac{\pi}{2}\beta \left(\frac{1}{2\tau\_{n}}(|\omega| - \omega\_{n} + \tau\_{n})\right)\right] & \text{if } \omega\_{n} - \tau\_{n} \le |\omega| \le \omega\_{n} + \tau\_{n} \\ 0 & \text{otherwise} \end{cases} \tag{4}$$

and

$$
\hat{\beta}\_{n}(\omega) \begin{cases}
1 & \text{if } \omega\_{n} + \tau\_{n} \le |\omega| \le \omega\_{n+1} - \tau\_{n+1} \\
\cos\left[\frac{\pi}{2}\beta \left(\frac{1}{2\tau\_{n+1}}(|\omega| - \omega\_{n+1} + \tau\_{n+1})\right)\right] & \text{if } \omega\_{n+1} - \tau\_{n+1} \le |\omega| \le \omega\_{n+1} + \tau\_{n+1} \\
\sin\left[\frac{\pi}{2}\beta \left(\frac{1}{2\tau\_{n}}(|\omega| - \omega\_{n} + \tau\_{n})\right)\right] & \text{if } \omega\_{n} - \tau\_{n} \le |\omega| \le \omega\_{n} + \tau\_{n} \\
0 & \text{otherwise}
\end{cases}
(5)$$

where <sup>γ</sup><*minn* [ω*n*+<sup>1</sup> <sup>−</sup> <sup>ω</sup>*n*/ω*n*+<sup>1</sup> <sup>+</sup> <sup>ω</sup>*n*], <sup>β</sup>(*x*) <sup>=</sup> <sup>35</sup>*x*<sup>4</sup> <sup>−</sup> <sup>8</sup>*x*<sup>5</sup> <sup>+</sup> <sup>70</sup>*x*<sup>6</sup> <sup>−</sup> <sup>20</sup>*x*7.

The definition of IEWT is the same as EWT, and IEWT is defined in the same way as for the classic wavelet transform. Detail coefficient *W*<sup>ε</sup> *f* (*n*, *t*) is given by the inner products with the empirical wavelets in Equation (6). Approximation coefficient *W*<sup>ε</sup> *f* (0, *t*) is given by the inner products with the scaling function in Equation (7).

$$\mathcal{W}\_f^\varepsilon(n, t) = \langle f, \Psi\_n \rangle = (f(\omega) \overline{\hat{\Psi}\_n(\omega)})^V \tag{6}$$

$$\mathcal{W}\_f^{\boldsymbol{\omega}}(0, t) = \langle f, \mathcal{Q}\_1 \rangle = \left( f(\boldsymbol{\omega}) \overline{\mathcal{Q}\_{1n}(\boldsymbol{\omega})} \right)^{\boldsymbol{V}} \tag{7}$$

The signal is reconstructed, and it is defined in Equation (8).

$$f(t) = \mathcal{W}\_f^\varepsilon(0, t)\mathcal{Q}\_1(\omega) + \sum\_{n=1}^N \mathcal{W}\_f^\varepsilon(n, t)\mathcal{V}\_n(t) = \left(\hat{\mathcal{W}}\_f^\varepsilon(0, \omega)\hat{\mathcal{Q}}\_1(\omega) + \sum\_{n=1}^N \hat{\mathcal{W}}\_f^\varepsilon(n, \omega)\hat{\mathcal{V}}\_n(\omega)\right) \tag{8}$$

Thus the empirical modes are given in Equations (9) and (10)

$$F\_0(t) = \mathcal{W}\_f^\varepsilon(0, t) \mathcal{Q}\_1(t) \tag{9}$$

$$F\_k(t) = \mathcal{W}\_f^\times(k, t)\Psi\_k(t) \tag{10}$$

#### (4) Selecting the best decomposition result based on *FER*

The step is very important and also different from EWT, and the purpose of the step is to find the best decomposition.

Because *coe*ffi*cient* = 1, 2, ... , *L*, there are *P*1, *P*2, ... , *PL*, and thus *L* decomposition results can be got. Each result is comprised of some component modes, and the mode corresponding to the biggest *FER* value *FERcoe*ffi*cient, max* is compared with that corresponding to the second biggest *FER* value *FERcoe*ffi*cient, max*, and the comparison result is denoted in Equation (11).

$$A\_{\text{coffcient}} = (FER\_{\text{coffcient}, \text{max}} - FER\_{\text{coffcient}, \text{sccondmax}}) / FER\_{\text{coffcient}, \text{sccondmax}} \tag{11}$$

Maximum value of *Acoe*ffi*cient* is denoted as *Amax*, and thus *Amax* corresponds to the best decomposition result, and the corresponding threshold value is the best decomposition value. Therefore, the mode to *FERcoe*ffi*cient, max* contains the richest fault feature information.
