## 3 PNCC Method Based on Wavelet Packet Transform
### Improved PNCC Method Based on Wavelet Package
In PNCC, the background noise processing of audio signal based on mid-time frame analysis. The mid-time power is calculated according to the following Equation  
\[\tilde{Q}[m,l]=\frac{1}{2M+1}\sum_{m^{\prime}=m-M}^{m+M}P\left[m^{\prime},l\right] \tag{2}\]  
where \(\tilde{Q}[m,l]\) represents mid-time power. In this study, the time integration factor \(M\) is 2, because it can better suppress background noise such as white noise. The asymmetric nonlinear noise suppression filter includes time mask, asymmetric, low-pass filter and other modules. Its complete characteristics can be described by the following Equation  
\[\tilde{Q}_{\text{out}}\left[m,l\right]=\left\{ \begin{array}{cc}\lambda_{a} \tilde{Q}_{\text{out}}\left[m-1,l\right]+(1-\lambda_{a})\tilde{Q}_{\text{in}} \left[m,l\right]&\\ \text{if}&\tilde{Q}_{\text{in}}\left[m,l\right]\geq\tilde{Q}_{\text{out}} \left[m-1,l\right]\\ \lambda_{b}\tilde{Q}_{\text{out}}\left[m-1,l\right]+(1-\lambda_{b})\tilde{Q}_{ \text{in}}\left[m,l\right]&\\ \text{if}&\tilde{Q}_{\text{in}}\left[m,l\right]<\tilde{Q}_{\text{out}} \left[m-1,l\right]\end{array} \right. \tag{3}\]  
where \(\tilde{Q}_{\text{in}}\left[m,l\right]\) and \(\tilde{Q}_{\text{out}}\left[m,l\right]\) represent input and output, respectively. Specifically, mid-time power is used as input for filtering. \(\lambda_{a}\) and \(\lambda_{b}\) are constants between 0 and 1. The selection of \(\lambda_{a}\) and \(\lambda_{b}\) values affects the recognition accuracy to some extent. In this study, the values are 0.999 and 0.5. These two values are selected to maximize the recognition accuracy of undisturbed audio and the performance of this method in the case of noise. Concretely, the lower envelope of the average noise power \(\tilde{Q}_{le}[m,l]\) is calculated firstly by  
\[\tilde{Q}_{le}[m,l]=\mathcal{AF}_{0.999,0.5}[\tilde{Q}[m,l]] \tag{4}\]  
where \(\tilde{Q}_{le}[m,l]\) is processed by nonlinear low-pass filter\(\tilde{Q}[m,l]\), and its initial value \(\tilde{Q}_{le}[0,l]\) is \(0.9\tilde{Q}[m,l]\). \(\tilde{Q}_{le}[m,l]\)generates the rectified output \(\tilde{Q}_{0}[m,l]\) via an ideal half-wave linear rectifier, and then calculates the lower envelope of \(\tilde{Q}_{0}[m,l]\) again  
\[\tilde{Q}_{f}[m,l]=\mathcal{AF}_{0.999,0.5}\left[\tilde{Q}_{0}[m,l]\right] \tag{5}\]  
Then, take the larger value in \(\tilde{Q}_{f}[m,l]\) and \(\tilde{Q}_{tm}[m,l]\) based on \(\tilde{Q}_{0}[m,l]\) to obtain \(\tilde{Q}_{1}[m,l]\)  
\[\tilde{Q}_{1}[m,l]=\max\left(\tilde{Q}_{tm}[m,l],\tilde{Q}_{f}[m,l]\right) \tag{6}\]  
Finally, the final output is determined according to Equation (7)  
\[ \begin{array}{cc}\tilde{R}[m,l]=\tilde{Q}_{1}[m,l]\text{ if }\tilde{Q}[m,l]\geq c \tilde{Q}_{le}[m,l]\\ \tilde{R}[m,l]=\tilde{Q}_{f}[m,l]\text{ if }\tilde{Q}[m,l]<c\tilde{Q}_{le}[m,l] \end{array}  \tag{7}\]  
where \(c\) is a fixed constant with a value of 2. It is considered that if the value is less than several times its lower envelope, then it is not a large enough incentive, and vice versa. Time masking processing is based on the fact that the human auditory system pays more attention to the beginning of the input power. To put this simply, the moving peak of each frequency channel is obtained, and the instantaneous power is suppressed when the instantaneous power is lower than the envelope. Specifically, the following Equation is used to obtain the peak power of each frequency channel  
\[\tilde{Q}_{p}[m,l]=\max\left(\lambda_{t}\tilde{Q}_{p}[m-1,l],\tilde{Q}_{0}[m, l]\right) \tag{8}\]where \(\lambda_{t}\) is the forgetting factor to obtain peak power. The time mask for audio is implemented using the following Equation  
\[\tilde{Q}_{tm}[m,l]=\left\{ \begin{array}{cc}\tilde{Q}_{0}[m,l],&\tilde{Q}_{0}[ m,l]\geq\lambda_{t}\tilde{Q}_{p}[m-1,l]\\ \mu_{t}\tilde{Q}_{p}[m-1,l],&\tilde{Q}_{0}[m,l]<\lambda_{t}\tilde{Q}_{p}[m-1,l] \end{array} \right. \tag{9}\]  
The best value of \(\lambda_{t}\) and \(\mu_{t}\) is 0.85 and 0.2.