## 2 PQD Feature Detection System Model Using WPTM-Based Metrics
### PQD Feature Detection Using WPTM-Based Metrics
#### 2.3.2 WPTM and WPTM-Based Metrics  
WTM using WCs has been widely applied to locate (or detect) the discontinuities of transient signals, such as images [2]. This paper similarly defines WPTM using WPCs and suggests several WPTM-based metrics for reliable PQD feature detection under severe power signal conditions. In a binary tree, as shown in Figure 3, the top node (i.e., top WPC) is \(\textbf{W}_{0}^{0}=V_{0}=\{x[i]|i=0,1,\ldots,l-1\}\). A WP space \(\textbf{W}_{l}^{p}\) (where the level index \(l=0,\ldots,L-1\) and the packet index \(p=0,\ldots,2^{l}-1\)) is defined with its corresponding orthonormal basis functions \(\psi_{l}^{p}\). By using the dyadic WPD, a WP space \(\textbf{W}_{l}^{p}\) is decomposed into two WP subspaces (producing the corresponding two WPCs)--\(\textbf{W}_{l}^{p}=\textbf{W}_{l+1}^{2p+1}\oplus\textbf{W}_{l+1}^{2p+1}\)--which are, respectively, defined with their corresponding basis functions \(\psi_{l}^{p}\) as follows:  
\[\psi_{l+1}^{2p}[i]=\sum_{n=-\infty}^{\infty}h[n]\psi_{l}^{p}[i-r],\]  
\[\psi_{l+1}^{2p+1}[i]=\sum_{n=-\infty}^{\infty}g[n]\psi_{l}^{p}[i-r],\]  
\[h[n]=\langle\psi_{l+1}^{2p}[u],\psi_{l}^{p}[u-r]\rangle,\]  
\[g[n]=\langle\psi_{l+1}^{2p+1}[u],\psi_{l}^{p}[u-r]\rangle\]  
where \(r=2^{l}n\) denotes the translation index and \(\psi_{l}^{p}\) can be represented with \(2^{l}\) elementary basis functions \(\psi^{p}[\cdot]\): \(\psi_{l}^{p}[i-r]=\frac{1}{\sqrt{2}}\psi^{p}\left[\frac{i-r}{2^{l}}\right]\), \(p=0,\ldots,2^{l}-1\). Let us assume that we construct an admissible tree for the input signal \(x[i]\) with all \(l\)th-level wavelet basis functions \(\psi_{l}^{p}\). Then, the corresponding WPCs of \(x[i]\) become  
\[\textbf{W}_{l}^{p}=\Big{\{}W_{l}^{p}[n]\Big{|}n=1,\ldots,N\Big{\}},\]  
Figure 3: WD and WPD. where \(W_{l}^{p}[n]=\langle x[i],\psi_{l}^{p}[i-r]\rangle=\sum_{i=0}^{l-1}x[i],\psi_{l}^{p} [i-r]\) and \(x[i]=\sum_{p=0}^{2^{l}-1}\sum_{i=0}^{l-1}\langle x[i],\psi_{l}^{p}[i-r]\rangle \Psi_{l}^{p}[i-r]\). Considering the transient (high-frequency) component of the PQD signal \(x[i]\), we can approximate the component with all WPCs besides approximation (\(\textbf{W}_{l}^{0}\)). Hence, for level \(l\), the WPC matrix **W** of the signal \(x[i]\) can be approximately expressed with \(P=2^{l}-1\) WPCs (except approximation) as follows:  
\[\textbf{W}=\left[ \begin{array}{c}\textbf{W}_{l}^{1}\\ \vdots\\ \textbf{W}_{l}^{p}\end{array} \right]=\left[ \begin{array}{ccc}W_{l}^{1}[1] &\cdots&W_{l}^{1}[N]\\ \vdots&\ddots&\vdots\\ W_{l}^{p}[1]&\cdots&W_{l}^{p}[N]\end{array} \right]. \tag{11}\]  
For level \(l\), WPTM exhibits an exponentially increasing degrees of freedom, that is, \(DoF=P=2^{l}-1\), whereas WTM exhibits a linearly increasing degrees of freedom, that is, \(DoF=2\) or 3 (except approximation) [1]. Furthermore, while WTM has a limited spectrum support (only low-frequency part), due to the nature of WD, WPTM maintains the entire spectrum support, due to the nature of WPD (Figure 3). Hence, WPTM exhibits better correlation properties than WTM while appropriately retaining the high-frequency components of the PQD signal. We can also specify the basis functions \(\psi_{l}^{p}\) with \(\theta_{l}(\textbf{\emph{u}})=2^{-l}\theta\Big{(}2^{-l}\textbf{\emph{u}} \Big{)}\) (where \(\textbf{\emph{u}}=\big{\{}u_{p}|p=1,\ldots,P\big{\}}\) indicates the orthonormal vectors defining the wavelet packet signal space and \(\theta(\cdot)\) is the convolution kernel), where conventional WTM-based schemes have been used for the edge detection of nonstationary signals, such as images [2]. In this study, for PQD detection, we use scaled wavelets, denoted by  
\[\overline{\psi}_{l}^{p}=2^{l}\frac{\partial\overline{\theta}_{l}}{\partial u _{p}}, \tag{12}\]  
which can smoothen the PQD signal space surface. In (12), \(\overline{\psi}_{l}^{p}(\textbf{\emph{u}})=\psi_{l}^{p}(-\textbf{\emph{u}})\) and \(\overline{\theta}_{l}(\textbf{\emph{u}})=\theta_{l}(-\textbf{\emph{u}})\).