## 3 Proposed Methodology
### Characterization of the Root Cause of Fault
#### 3.1.3 Time-Frequency Domain Analysis Using Discrete Wavelet Transform (DWT)  
This subsection describes the time-frequency domain analysis of voltage signal using DWT. The DWT which is time-frequency analysis has been deployed as it was capable to extract features in time resolution and frequency resolution respectively [27; 28; 29; 30]. The detail wavelet transform with mathematical algorithm is discussed in the subsequent section. (a) Discrete Wavelet Transform Algorithm  
Wavelet transform is a powerful signal processing tool used in recognizing power disturbance pattern based on its features extraction [31]. It has the capability to analyze the signal in multi resolution either localized in time or space. Previously, Fourier Transform was used to analyze the stationary signal with limited capability for non-stationary analysis as the time information was lost [31]. Fourier Transform equation based on frequency domain can be defined as:  
\[F(\omega)=\int_{-\infty}^{+\infty}f(t)e^{-j\omega t}dt \tag{18}\]  
From the equation, \(-\infty\) to \(+\infty\) indicated that time information will be lost. Therefore, the wavelet transform is an effective tool to analyze non stationary signal because of the mother wavelet function used as the basis function. Mother wavelet function \(\psi\) (t) equation is defined as:  
\[\psi\ (t)_{(a,b)}=\frac{1}{\sqrt{a}}\psi\left[\frac{t-b}{a}\right] \tag{19}\]  
where \(1/a\) is frequency and \(1\sqrt{a}\) is the normalizing constant of each scale parameter. Meanwhile, \(b\) is parallel translation of time axis. The continuous wavelet transform (CWT) is defined as:  
\[CWT_{\psi}x(a,b)=\frac{1}{\sqrt{|a|}}\int_{-\infty}^{+\infty}x(t)\psi^{*} \left(\frac{t-b}{a}\right)dt \tag{20}\]  
Terms \(a\) and \(b\) indicate the dilation and translation which determined the frequency length of wavelet and shifting position respectively. The \(\psi\) is the mother wavelet, the \(*\) indicates that the complex conjugate was used in the case of a complex wavelet. The extension of CWT, known as the discrete wavelet transform (DWT) is introduced to overcome the computational derived from CWT. The DWT is defined as:  
\[DWT(m,n)=\int_{-\infty}^{+\infty}x(t)\psi_{m,n}^{*}(t)dt \tag{21}\]  
where; \(\psi_{m,n}(t)=a_{0}^{\frac{-m}{2}}\left(\frac{(t-na_{0}^{n}b_{0})}{a_{0}^{n}}\right)\)  
These parameters are a = \(a_{0}^{m}\), \(b=nb_{0}a_{0}^{m}\)  
Where \(m,\ n\ Z;\ m,\ n\) represent the frequency localization and time localization respectively. The \(x(t)\) denoted as signal in time domain. In this paper, db4 mother wavelet is used to detect the disturbance signal and obtain time detection information and detail frequency. 2. Decomposition  
Discrete wavelet transform is a very useful technique to analyze the transient phenomenon. Multiresolution analysis (MRA) is one of the tools of DWT, which decomposes a non-stationary signal into low frequency signal known as approximation and high frequency signal called details. In this stage, the original disturbance waveforms are decomposed using DWT at the desired level \(j\) with \"Daubechies\" wavelet function of order \(n\). The decomposition of PQ waveform into various frequency bands is achieved by applying high pass filter and low pass filter to the time domain signals. The flow of division filters is further explained in the next subsection. Figure 9 (D4)-(D1) illustrates four level decomposition coefficient of voltage dip fault utilised in this paper. 3. Sub-Band Filters  
In DWT, the signal is analyzed at different frequency bands with different resolutions using the digital filtering techniques. This is significant to divide the signal into approximation and detail signals. The signal will comply into high pass and low pass filters. At the first stage, an original signal is slashed into two halves of bandwidth and shipped to both the filters. Next, the output of the low pass filter is further divided into half of the frequency bandwidth, and further shipped for the next stage.