## 3 Methodology
### Wavelet Transform  
Though frequency information in Fourier transform is extracted for a whole duration of the signal, it is normally determined by calculating the average over the complete length of the signal, which is a major drawback of Fourier transform [56]. To overcome this problem, many time-frequency domain approaches have been used, such as Gabor transform, short-time Fourier transform (STFT), Wigner-Ville transform, and wavelet transform. Among them, wavelet transform is based on Fourier transform and STFT to allow transformation of the time domain signal into time-frequency domain. In wavelet transform, small signals are mathematically integrated into one complete signal, and the small signals are known as wavelets. The wavelet is a short duration oscillation starting and ending at zero. There are several orthogonal basis functions which are known as the mother wavelets. Number of other wavelets are created from wavelet by scaling (stretching or shrinking) with the mother wavelet itself. The shifting of scaled wavelets along the time axis provides information about localization of different frequency contents of the corresponding signal. There are two variants of wavelets namely continuous wavelet transform (CWT) and discrete wavelet transform (DWT).  
The CWT is defined in [57] as  
\[CWT_{(a,b)}(t)=\frac{1}{\sqrt{a}}\int\limits_{-\infty}^{+\infty}x(t)\psi^{*}( \frac{t-\tau}{a})dt \tag{7}\]  
where \(a\), \(\tau\), and \(\psi\) represent the scale parameters, translation parameter, and mother wavelet, respectively. \(\psi^{*}\) is the complex conjugate of \(\psi\).  
The DWT is derived from discretization of _CWT_ (\(a\),\(b\)) as  
\[DWT_{(jk)}(t)=\frac{1}{\sqrt{2j}}\int\limits_{-\infty}^{+\infty}x(t)\psi^{*}( \frac{t-2 jk}{2^{j}})dt\text{ where, }a=2^{j}\text{ and }\tau=2^{j}k \tag{8}\]  
Generally, a multiresolution analysis decomposes the signal into a smoother version of the original signal (approximations) and a set of detailed information at various scales. Here, \(j\) and \(k\) can be any positive integer values such as 1,2, 3,... The high frequency and low frequency components are known as detail coefficients (cD) and approximate coefficients (cA), respectively. This process is performed using a series of high and low pass filters and can be expressed as  
\[x(t)=A_{j}+\sum_{j\leq j}D_{j} \tag{9}\]  
where \(A_{j}\) and \(D_{j}\) represent the low frequency bands (approximations) and high frequency bands (details) of the signal, respectively. At the transient state, the high frequency components will be evaluated to analyze the signal. In addition, the DWT exposes aspects of data such as discontinuities of higher derivatives, breakdown point, and self-similarity in a more efficient manner than any other signal processing technique.  
A number of mother wavelets exists for both CWT and DWT all of them are not equally applicable for any signal. It is the nature of signal (e.g., image, time-series data) and field of application which influences which wavelet to be used. Researchers need to look for the wavelet function that best correlates with the function or signal being analyzed to extract the most effective information. As found in the literature, for time-series signals, Haar, db4, and sym4 wavelets are widely used [23; 58], as is represented in Figure 6. Therefore, we explored the effectiveness of these three wavelets in decomposing motor current signal-based fault analysis.  
Among them, the Daubechies (db) and Symlets (Sym) show asymmetric basic function, where the Haar wavelet possesses absolute symmetry in nature.