## 2 Background and Methodology
The qualitative pattern of LSH in the high decomposition level of the signals can be a good indicator for the fault detection, as shown in \[\omega_{bMMF}=-s\omega_{syn}+\omega_{m}=-s\omega_{syn}+(1-s)\omega_{syn}=(1-2 s)\omega_{syn} \tag{1}\]1 Hence, the reconstructed signal that comes from the detail and approximation of level seven, and also the approximation of level six, is used as an input signal for calculating the fault-related feature. The relevant frequency bands of D7 (Detail of level seven), A7 (approximation of level seven) and A6 (approximation of level six) are shown in  
\[\omega_{bMMF}=-s\omega_{syn}+\omega_{m}=-s\omega_{syn}+(1-s)\omega_{syn}=(1-2 s)\omega_{syn} \tag{1}\]2 Dyadic wavelet decomposition algorithm. Entropy in information theory describes the amount of information provided by a signal or event. It relates the uncertainty of the signal or event associated with a given probability distribution. The concept of entropy has found broad applications in engineering, including, for instance, fault diagnoses [28; 29; 30; 31]. A survey of recent methods for the fault diagnosis of rotating machinery using entropy techniques was discussed in [32]. According to these studies, entropy has been used a feature for the diagnosis of a failure in electrical machines. The features used in this research are log-energy entropy and Shannon entropy. These two features are common concepts in many fields, mainly in signal processing, and are listed as bellow:  
\[\omega_{bMMF}=-s\omega_{syn}+\omega_{m}=-s\omega_{syn}+(1-s)\omega_{syn}=(1-2 s)\omega_{syn} \tag{1}\]3  
with the convention \(0\text{log}(0)=0\). \[\omega_{bMMF}=-s\omega_{syn}+\omega_{m}=-s\omega_{syn}+(1-s)\omega_{syn}=(1-2 s)\omega_{syn} \tag{1}\]4  
with the convention \(\log(0)=0\). where, \(X_{n}\) is the reconstructed signal from level \(n\). Once the features are determined, statistical techniques should be used in exploratory data investigation. The movement of features is surveyed to discover the overlapping of healthy and faulty conditions at different levels of load. A number of methods have been implemented for visualizing and checking normality in multi-level experiments to provide a good understanding of the overall characteristics of the data. Thus, by listing all data with a multiplier, a boxplot can be used to check the \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Decomposition Level**} & \multicolumn{2}{c}{**Frequency Ranges (Hz)**} \\ \cline{2-3}  & **Detail (D)** & **Approximation (A)** \\ \hline level 1 & 2500–1250 & 1250–0 \\ level 2 & 1250–625 & 625–0 \\ level 3 & 625–312.5 & 312.5–0 \\ level 4 & 312.5–156.2 & 156.2–0 \\ level 5 & 156.25–78.12 & 78.12–0 \\ level 6 & 78.12–39.06 & 39.06–0 \\ level 7 & 39.06–19.53 & 19.53–0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Frequency ranges for the wavelet decomposition of the signal. Figure 5: Theoretical behavior of \(f_{LSH}\) as a function of slip. assumption of normality. Boxplots, in this research, are considered to analyse and compare various features of parameter. The parameters investigated are two types of motor conditions, healthy and faulty, and four different stages of starting load are considered for each condition. The estimation of the statistical significance of differences was verified with a two-way Analysis of variance (ANOVA) based on the motor condition, the starting load, and their interaction. [PERSON]'s honest significant differences technique was also utilized for multiple comparisons after ANOVA. There are three sets of hypotheses should be tested with the two-way ANOVA in this research. The null hypotheses for each of the sets are given below:  
1. The population means of the first factor (motor condition) are equal;
2. The population means of the second factor (load) are equal;
3. There is no interaction between the two factors (motor condition and load). Accordingly, the most important point of current research is to identify the features related to BRB fault detection in LS-PMSM.