## 2 Background and Methodology
Wavelet transform has been shown to be one of the best tools for non-stationary signal analysis [20]. The applicability of using wavelet transform for decomposing a signal and extracting its sensitive bands is considered to be good for the health monitoring of an electrical motor and fault diagnosis [21]. Discrete wavelet transform (DWT) is proposed to be used because it provides a dyadic band-pass filtering of the signal as well as the advantages of its simplicity, low computational requirements and the easy interpretation of the obtained results [18]. The DWT is computed through the Mallat algorithm, in which low-pass and high-pass filtering of a time-domain signal is performed while the sampling rate changes [22]. DWT allows high frequency components to be analyzed with short time intervals, and low frequency components to be analyzed with long time intervals [23]. The decomposition of a sampled signal \(i(t)\). using DWT results \(n+1\) wavelet signals, which are an approximation signal, \(A_{n}(t)\), and \(n\) detail signals \(D_{j}(t)\) with \(j\) varying from 1 to \(n\)[24]. \[i(t)=(A_{n}(t)+D_{n}(t)+\cdots+D_{1}(t)) \tag{3}\]  
The parameter \(n\) is an integer known as the number of decomposition levels and it is set based on the sampling rate (\(f_{s}\)), which is used for capturing \(i(t)\). The n also can be computed based on the frequency interval covered by the fault-related component. Taking into account whether the second rule is considered, \(n\) should be less than predicted by the sampling rate. The approximation signal \(A_{n}(t)\) is obtained using the scaling function, deduced by the father wavelet (\(\varphi_{j}^{n}\)) and scaling coefficients \(a_{j}^{n}\):  
\[A_{n}(t)=\sum_{i}a_{i}^{n}\cdot\varphi_{j}^{n}(t) \tag{4}\]  
The detail signal \(D_{n}(t)\) is calculated using \(d_{i}^{j}\) as a base; the wavelet coefficients with level \(j\), which is a scaled and time-expanded version of the wavelet functions, are deduced by the mother wavelet (\(\psi_{i}^{j}\)):  
\[D_{n}(t)=\sum_{i}d_{i}^{j}\cdot\psi_{i}^{j}(t) \tag{5}\]  
Figure 3: Behavior \(f_{LSH}\) during the start-up for LS-PMSM. Figure 4 illustrates the dyadic wavelet decomposition algorithm regarding the coefficients of the transform at the different levels according to the description by [25]. In this figure, the length of those coefficients and the frequency content at each level is shown, considering an original signal with a sampling rate of \(f_{s}\) samples/s. The analysis of these signals reveals the time evolution of the components of the signal \(i(t)\) which are contained within its corresponding frequency band. The approximation signal includes the low-frequency components of the signal, belonging to the interval:  
\[f(A_{n})\in[0.2^{-(n+1)}f_{s}]\mathrm{Hz} \tag{6}\]  
More concretely, the detail contains the information concerning the signal components with frequencies included in the interval:  
\[\omega_{bMMF}=-s\omega_{syn}+\omega_{m}=-s\omega_{syn}+(1-s)\omega_{syn}=(1-2 s)\omega_{syn} \tag{1}\]0  
There are many different mother wavelet functions that are categorized based on their real type, such as the Coiflet wavelet, and their complex type, such as the Morlet wavelet. In fault detection algorithms, different types of mother wavelets have been used with satisfactory results [26; 27]. In this work, the DWT of the motor start-up current has been performed using a fourth-order biorthogonal Coiflet wavelet. The sampling frequency used to capture the data in this research is 5000 samples/s. To define the suitable level of decomposition, the main signal of the motor current was initially decomposed into seven levels, where each level has its own range of frequencies. Table 1 presents the frequency bands equivalent to each level. Clearly, the fundamental frequency is located in the frequency range corresponding to the detail of level six.