### 3.1 Principles of AMULW

The schematic of the proposed AMULW in one stage is illustrated in Figure 2. In AMULW, the update lifting is applied to modify the approximation signal, while the detail signal remains unchanged in the decomposition process, namely \(y^{\prime}_{1} = y_{1}\).

The AMULW comprises three main steps. First, the raw signal *x*
_{0} is split into two parts, producing an approximation signal *x*
_{1} and a detail signal *y*
_{1}. This partition can be fulfilled by some special wavelet transform. The simplest one is to directly split *x*
_{0} into odd and even samples [19]:

$$\begin{aligned} &x_{1} = \psi^{ \uparrow } (x_{0} )(n) = x_{0} (2n) \hfill \\ &y_{1} = \omega^{ \uparrow } (x_{0} )(n) = x_{0} (2n + 1) \hfill \\ \end{aligned}$$

(5)

Then, a two-valued decision map *D* is used to control the choice of the update filter, which is expressed as follows:

$$D(n) = \left\{ \begin{aligned} 1\quad \quad g(n) > T \hfill \\ 0\quad \quad g(n) \le T \hfill \\ \end{aligned} \right.$$

(6)

where *T* is the threshold value; *g*(*n*) denotes the local gradient information of adjacent three samples in a signal, which is defined as:

$$g(n) = \left| {x_{1} (n) - y_{1} (n - 1)} \right| + \left| {y_{1} (n) - x_{1} (n)} \right|$$

(7)

Subsequently, the approximation signal *x*
_{1} is updated using the information hidden in the detail signal *y*
_{1}, yielding a new approximation signal \(x^{\prime}_{1}\). If *D*(*n*) = 1, the morphological dilation-erosion filter is used as the update operator *U*
_{1}:

$$U_{1} (n) = y_{1} (n - 1) \wedge y_{1} (n) - y_{1} (n - 1) \vee y_{1} (n)$$

(8)

$$\begin{aligned} &x^{\prime}_{1} (n) = x_{1} (n) + U_{1} (n) = x_{1} (n) + \hfill \\ &y_{1} (n - 1) \wedge y_{1} (n) - y_{1} (n - 1) \vee y_{1} (n) \hfill \\ \end{aligned}$$

(9)

where \(\vee\) represents morphological dilation operation and \(\wedge\) represents morphological erosion operation.

If *D*(*n*) = 0, the average filter is utilized as the update operator *U*
_{0}:

$$U_{0} (n) = y_{1} (n - 1) + y_{1} (n)$$

(10)

$$\begin{aligned} x^{\prime}_{1} (n) &= \frac{1}{3}(x_{1} (n) + U_{0} (n)) \hfill \\ &= \frac{1}{3}(x_{1} (n) + y_{1} (n - 1) + y_{1} (n)) \hfill \\ \end{aligned}$$

(11)

Correspondingly, *x*
_{1} can be easily reconstructed from \(x^{\prime}_{1}\) and \(y^{\prime}_{1}\) as Eq. (12) or (13). Once *x*
_{1} is obtained, the synthesis signal *x*
_{0} is also easy to get.

$$x_{1} (n) = x^{\prime}_{1} (n) + y_{1} (n - 1) \vee y_{1} (n) - y_{1} (n - 1) \wedge y_{1} (n)$$

(12)

$$x_{1} (n) = 3x^{\prime}_{1} (n) - y_{1} (n - 1) - y_{1} (n)$$

(13)

The above analysis process is limited to the scope of one stage decomposition. A raw signal *x*
_{0} can be decomposed into multiple levels as a multiple stage decomposition involves:

$$\begin{aligned} x_{0} \to \left\{ {x^{\prime}_{1} ,y^{\prime}_{1} } \right\} \to \left\{ {x^{\prime}_{2} ,y^{\prime}_{2} ,y^{\prime}_{1} } \right\} \to \cdots \hfill \\ \to \left\{ {x^{\prime}_{n} ,y^{\prime}_{n} ,y^{\prime}_{n - 1} ,y^{\prime}_{n - 2} , \cdots ,y^{\prime}_{2} ,y^{\prime}_{1} } \right\} \hfill \\ \end{aligned}$$

(14)

On account of the presence of the decision map *D*, the update lifting operator *U* can be adaptively selected to modify the approximation signal. The morphological dilation-erosion filter is adopted to strengthen the impulsive features when there is an impulse in a signal, while the average filter is employed to smooth a signal at the time when the signal amplitudes vary weakly.

### 3.2 The Selection of the Threshold *T*

The function of the threshold *T* is to distinguish the stationary data and the impulses in a signal. If *T* is too large, some useful impulsive information might be smoothed by using the average filter; on the contrary, if *T* is too small, some tiny fluctuations might be treated as impulses. In the present paper, *T* is defined as:

$$T = k \cdot \hbox{max} (g(n))\quad \quad k \in [0,1]$$

(15)

In order to reduce the adverse effects of the threshold *T*, we perform the AMULW multiple times with *k* increasing from 0 to 1, with a step size of 0.1. As a result, 11 analysis results would be obtained. Then, characteristic frequency intensity coefficient (CFIC) [20] of the above 11 signals are computed. The CFIC can be expressed as:

$$\text{CFIC} = \frac{{\sum\limits_{i = 1}^{N} {A_{{if_{c} }} } }}{{\sum\limits_{j = 1}^{M} {A_{{f_{j} }} } }}$$

(16)

where \(A_{{if_{c} }}\) is the amplitude of the *i*th harmonic of the fault characteristic frequency, *N* is the number of the harmonics of the fault characteristic frequency, \(A_{{f_{j} }}\) is the amplitude of the frequencies analyzed, and *M* is the number of all frequency components.

The CFIC represents the portion of the fault frequency amplitude to the overall frequency amplitude. A larger value of CFIC implies a better fault detection performance. Other statistic criterion, such as kurtosis [21], smoothness index [22], crest factor [23], peak energy [24] and fusion criterion of kurtosis, smoothness index and crest factor [25], can also be used as a guide to select the fault relevant information. In this study, CFIC is employed. The signal with the largest CFIC value is picked out from the whole 11 candidates and only this signal will be utilized in the fault detection.

### 3.3 Perfect Reconstruction

Assume that *α*, *β* and *γ* are lifting wavelet coefficients:

$$x^{\prime}_{1} (n) = \alpha x_{1} (n) + \beta y_{1} (n - 1) + \gamma y_{1} (n)$$

(17)

Ref. [17] has proved in theory that in order to fulfill perfect reconstruction, it requires:

$$\alpha + \beta + \gamma = M$$

(18)

where *M* is a constant and usually set as 1.

In Eq. (11), \(\alpha = \beta = \gamma = {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}\), which meets the requirement of Eq. (18). For Eq. (9), we discuss the influence of the value of *y*
_{1}(*n*−1) and *y*
_{1}(*n*) on perfect reconstruction. In summary, there are three situations:

Case 1: *y*
_{1}(*n* − 1) > *y*
_{1}(*n*), at this moment Eq. (9) can be simplified as

$$x^{\prime}_{1} (n) = x_{1} (n) + y_{1} (n) - y_{1} (n - 1)$$

(19)

Case 2: *y*
_{1}(*n* − 1) < *y*
_{1}(*n*), at this moment Eq. (9) can be simplified as

$$x^{\prime}_{1} (n) = x_{1} (n) + y_{1} (n - 1) - y_{1} (n)$$

(20)

Case 3: *y*
_{1}(*n* − 1) = *y*
_{1}(*n*), at this moment Eq. (9) can be simplified as

$$x^{\prime}_{1} (n) = x_{1} (n)$$

(21)

Regardless the value of *y*
_{1}(*n* − 1) and *y*
_{1}(*n*), the situations in Eqs. (19)–(21) accords with the requirement of Eq. (18). Therefore, the proposed AMULW fits the perfect reconstruction condition.