## 3 PNCC Method Based on Wavelet Packet Transform
### Wavelet Packet Transform  
As described in the previous paragraph, the original PNCC method uses short-time Fourier transform as the front-end processing of audio signal. The audio signal of the trunk borer in a severe noise environment is a typical non-stationary signal. At present, typical non-stationary signal processing methods include short-time Fourier transform (FFT), wavelet transform (WT), empirical wavelet transform (EWT), wavelet packet transform (WPT), empirical mode decomposition (EMD), and variational mode decomposition (VMD). Based on Hilbert transform, EMD and VMD transform the original signal into a complex analytical signal domain for analysis and processing. Short-time Fourier transform, wavelet transform and wavelet packet transform carry out signal processing in the time-frequency domain. Therefore, using wavelet packet transform to improve the original PNCC method can retain the noise processing process of the original PNCC method as completely as possible. At the same time, considering that the wavelet packet transform has a higher high-frequency resolution, it can better listen to the information of the high-frequency components of trunk borer. In fact, we think that, in theory, we can use the VMD or EMD method to replace the preprocessing process of the original PNCC method. However, the subsequent processing process needs to be changed accordingly, and a complete theoretical derivation and practical effect verification are needed at the same time. Considering that the work is too large, and less related to the theme of this study, and the possibility of failure is high, we abandoned this scheme. However, this would be a good direction for future research.  
WPT is a universalization of discrete wavelet transform (DWT), and the necessary frequency resolution is able to be realized [26]. Wavelet packet transform achieves more high-frequency resolution at the cost of abandoning time resolution. Figure 5 shows an example of the WPT decomposition tree structure (three levels) of a set of clear audio signals of trunk borers \(f(t)\).  
The wavelet packet tree is regarded as a filter bank, which is generated by two sets of orthogonal wavelet base filter coefficients  
\[\varphi^{2i}=\sqrt{2}\sum_{k}h(k)\varphi^{i}(2t-k) \tag{10}\]\[\varphi^{2i+1}=\sqrt{2}\sum_{k}g(k)\varphi^{i}(2t-k) \tag{11}\]  
where \(g(k)\) and \(h(k)\), denoted as group-conjugated orthogonal filters, are quadrature filters related to the mother wavelet function and scaling function, respectively. \(\varphi\) stands for wavelet. Each element in the wavelet tree is considered to be a separate filter. The overall response of the filter bank can be described by the following Equation  
\[\varphi^{i}_{j,k}(t)=2^{j/2}\varphi^{i}\Big{(}2^{j}t-k\Big{)} \tag{12}\]  
where \(j\),\(i\) and \(k\) represent scale, modulation and translation parameters, respectively. WP coefficients of the time domain \(f(t)\) are calculated using the following Equation  
\[C^{i}_{j,k}(t)=\int_{-\infty}^{\infty}f(t)\varphi^{i}_{j,k}(t)dt \tag{13}\]  
where \(C^{i}_{j,k}\) wavelet packet coefficient and meets the orthogonality condition:  
\[\varphi^{m}_{j,k}(t)\varphi^{n}_{j,k}(t)=0,\quad(m\
eq n) \tag{14}\]  
The WP component of a signal at the specific node of the decomposition tree is  
\[f^{i}_{j}(t)=\sum_{k=-\infty}^{\infty}c^{i}_{j,k}(t)\varphi^{i}_{j,k}(t) \tag{15}\]  
After the decomposition of the \(j\)th level, the raw signal is built up via the summation of \(2^{j}\) elements, shown as  
\[f(t)=\sum_{i=1}^{2^{j}}f^{i}_{j}(t) \tag{16}\]