## 2 Methods
### Scale Adaptive Wavelet Packet Transform
#### 2.3.2 Selection of Wavelet Base and Scale  
According to the wavelet packet decomposition principle, it can decompose the high and low frequency signals into corresponding components to perform local time-frequency analysis. In order to ensure the non-redundancy of signal decomposition and reduce the loss of PD signal, it is necessary to select an appropriate decomposition scale. In order to better preserve the characteristics of signal, it is necessary to select the best wavelet base. The threshold function and the compactness are used as the preferred indicators to find the best wavelet base. Through sufficient simulations, the minimum energy loss is achieved by the db4 wavelet base due to the maximum similarity between the db4 wavelet base and the PD signal acquired by HFCT. So, the db4 wavelet base is used for separation of signal and noise in this paper. As mentioned above, the decomposition scale J in wavelet packet decomposition will affect the decomposition depth and the decomposition result largely. In order to illustrate this point, the single-exponential attenuation signal as described in (17) with white noise is used to simulate the cable partial discharge signal [13] to evaluate the effect of decomposition scale. \[f(t)=Ae^{-(t-t_{0})/\tau} \tag{17}\]  
where \(A\) is the amplitude of PD signal, \(t_{0}\) is the time of PD occurred, \(\tau\) is the attenuation coefficient. When the SNR is \(-13\) dB, different decomposition scales J are set respectively (J = 1,2,3,4) for wavelet packet decomposition. In order to quantitatively analyze the influence of the scale on de-noising effect, the normalized correlation coefficient (NCC) of the first six components and the input noisy PD signal, \(\rho_{k}\), the NCC of the first component and the simulated real PD signal, the SNR of de-noised PD signal, and the relative amplitude error, are calculated synchronously. The results are shown in Table 1.  
It can be seen from Table 1 that as the decomposition scale increases, the correlation coefficient between each component and the noisy signal decreases, especially when the decomposition scale is 4, the correlation coefficient decreases sharply, which is called over decomposition. It is obvious that, when the decomposition scale is 1, the relative amplitude error is small, but the similarity with the simulated signal is low, which means the relative poor ability of noise suppression. When the decomposition scale is 4, the noise suppression ability is better, but the relative error is large. When the decomposition scale is 3, a better balance between noise suppression and similarity is achieved. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline J & \(\rho_{1}\) & \(\rho_{2}\) & \(\rho_{3}\) & \(\rho_{4}\) & \(\rho_{5}\) & \(\rho_{6}\) & NCC & SNR & D (\%) \\ \hline J = 1 & 0.794 & 0.608 & & & & & 0.6744 & \(-0.1\) dB & 10.0 \\ J = 2 & 0.672 & 0.424 & 0.414 & 0.446 & & & 0.7741 & 2.6 dB & 16.7 \\ J = 3 & 0.446 & 0.251 & 0.251 & 0.230 & 0.210 & 0.238 & 0.8294 & 4.6 dB & 23.3 \\ J = 4 & \(-2.5\times 10^{-4}\) & \(-0.006\) & \(1.8\times 10^{-4}\) & \(-6.2\times 10^{-4}\) & 0.002 & 0.066 & 0.8229 & 4.8 dB & 43.3 \\ \hline \end{tabular}
\end{table} 
Table 1: Results at different decomposition scales. Figure 2: Schematic diagram of three-scale wavelet packets decomposition. F denotes an original signal, D1 and A1 denote the high-frequency and low-frequency component of the first scale decomposition respectively; DA2 and AA2 denote the high-frequency and low-frequency component of the second-scale decomposition, respectively. Therefore, the three-scale decomposition of the wavelet packet is optimal in terms of the noisy PD signal with the SNR of \(-\)13 dB. Based on the above analysis, a scale adaptive wavelet packet decomposition algorithm for noisy PD signal is employed in this paper. Firstly, according to the signal characteristics, the wavelet base is determined as db4. Perform the decomposition from the minimum scale.