## 4 HRF Detection  
Based on the above, time-frequency domain detection method was used to extract the characteristics of arc HRFs. Orthogonal wavelet transformation and singular value decomposition (SVD) extract information in the time and frequency domains, respectively. After orthogonal wavelet transformation, the original signal is decomposed into various components at different scales when \(h\) is a wavelet-based low-pass filter, \(g\) is a wavelet based high-pass filter, \(a_{j}(n)\) is the low-frequency part of the signal, and \(d_{j}(n)\) is the high-frequency part of the signal. As the decomposition time increases, \(a_{j}(n)\) and \(d_{j}(n)\) are given by Equations (19) and (20), respectively:  
\[a_{  
After wavelet transformation, SVD is used to distinguish signals of different frequency bands; this can be applied to decompose non-square matrices, and it is assumed that matrix \(A\) is an \(m^{*}n\)-order matrix. There then exists an \(m^{*}n\)-order orthogonal matrix \(U\) and \(m^{*}n\)-order orthogonal matrix \(V\), so that Equation (21) holds:  
\[A=UDV^{T} \tag{21}\]  
where \(D\) is an \(m\times n\)-order diagonal matrix and \(D=(diag(\sigma_{1},\sigma_{2},\ldots,\sigma_{q}),O)\). The diagonal element \(\sigma_{1},\sigma_{2},\ldots,\sigma_{q}\) is the singular value of matrix \(A\). Common forms of matrix \(A\) include the Toeplitz, cycle, and Hankel matrices; the latter was used. During SVD, the phase of the decomposed semaphore in the original signal remains unchanged; in other words, the decomposition simply subtracts the decomposed signal from the original signal. The Hankel matrix of the original signal \(y(n)\) is given by Equation (22):  
\[A=\begin{bmatrix}y(1)&y(2)&\cdots&y(n)\\ y(2)&y(3)&\cdots&y(n+1)\\ \vdots&\vdots&\vdots&\vdots\\ y(m)&y(m+1)&\cdots&y(N)\end{bmatrix} \tag{22}\]  
where \(u_{1},u_{2},\ldots,u_{q}\) are the elements in matrix \(u\) of order \(m^{*}1\), and \(v_{1},v_{2},\ldots,v_{q}\) are the elements in matrix \(v\) of order \(n^{*}1\). Orthogonal wavelet transformation is combined with SVD for time-frequency domain decomposition. First, the current signal (collected on the line) is decomposed at various scales to obtain coefficients that are used to construct a Hankel matrix; SVD then yields singular value spectra at different wavelet decomposition scales. Among the frequency domain ranges represented by each scale after wavelet decomposition, \(a_{j}[n]\) decomposed to layer \(j\) is the lowest-frequency region.  
If the DC arc current is small, most current energy is of low frequency. After an HRF fault, an obvious gain change will be seen at low frequency. The amplitude of random line noise is usually small and can thus be ignored. The arc frequency band is predominantly low-frequency, and the line high-frequency components do not interact.  
Then sum the singular value amplitudes of all points of the low-frequency singular value sequence of the \(j\)th layer decomposed by the wavelet; thus, sum the singular value of the \(a_{j}[n]\) layer of the fault line and singular value sequence of the \(a_{j}[n]\) layer of the non-fault line at least k times. The fault detection criterion is represented by Equation (23):  
\[a_{j\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_