## 3 Wavelet Packet Transform-Based Feature Extraction
### Signal Features Extraction  
Feature extraction is an important aspect of the systems involving machine-learning algorithms [30]. In the proposed technique, the wavelet packet entropy features were computed using the wavelet packet transform coefficients. The extracted features were used to classify and recognize the target signal or frequency sweeping jamming signals. The coefficients of each node at each layer of the three-layer wavelet packet transform are preferred as the object of feature extraction. As shown in Figure 3, the three-layer wavelet packet decomposition has a total of 15 nodes, which are nodes (0,0), (1,0), (1,1), (2,0), (2,1), (2,2), (2,3), (3,0), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), and (3,7). The description of the features is as follows. In the three-layer wavelet packet transform, the wavelet packet coefficients matrix of the \(n\)-th node in the \(i\)-th layer is \(c_{i}^{n}\). The length of the wavelet packet coefficient matrix of the \(n\)-th node in the \(i\)-th layer is expressed as follows:  
\[len\_c_{i}^{n}=length(c_{i}^{n}) \tag{5}\]  
The Shannon entropy feature of the \(n\)-th node in the \(i\)-th layer is defined as \(En_{i}^{n}\). \[En_{i}^{n}=\sum_{k=1}^{len\_c_{i}^{n}}\left[p_{i}^{n}(k)\right]^{2}*\left[ \log_{2}(p_{i}^{n}(k))\right]^{2} \tag{6}\]  
where  
\[En_{i}^{n}=\sum_{k=1}^{len\_c_{i}^{n}}\left[p_{i}^{n}(k)\right]^{2}*\left[ \log_{2}(p_{i}^{n}(k))\right]^{2} \tag{7}\]  
Figure 3: Schematic diagram of 3-layer wavelet packet decomposition. In this paper, the number of the wavelet decomposition layers is three, so \(i=0,1,2,3\), \(n=1,2,\ldots,15\). \(k\) denotes the index of the wavelet packet coefficient. Therefore, for each signal, 15 wavelet packet node Shannon entropy features can be extracted, i.e., \(En_{0}^{0}\), \(En_{1}^{0}\), \(En_{1}^{1}\), \(En_{2}^{0}\), \(En_{2}^{1}\), \(En_{2}^{2}\), \(En_{2}^{2}\), \(En_{3}^{2}\), \(En_{3}^{1}\), \(En_{3}^{2}\), \(En_{3}^{3}\), \(En_{3}^{4}\), \(En_{3}^{5}\), \(En_{3}^{6}\), \(En_{3}^{7}\). Daubechies series wavelet is orthogonal and sensitive to irregular signals, so the 'dbl' wavelet is selected for wavelet transformation in this paper. For the target signal and amplitude modulation frequency sweeping jamming signals, the wavelet packet node Shannon entropy features \(En_{0}^{0}\sim En_{3}^{7}\) are extracted, respectively, and their probability density distribution functions are shown below. The probability density distribution of the wavelet packet node entropy features in the figure below shows that the target signal and the frequency sweeping jamming signals have different differential significance in the features of different wavelet packet nodes. Due to the real-time performance requirement of radio fuze, it is necessary to select the wavelet packet node features with the greatest significance of differences, which not only reduces the dimensionality of the features to improve the computational speed, but also reduces redundant features and improves the classification accuracy. In order to quantify the significance of the differences in the entropy characteristics of the wavelet packet node coefficients for the different signals shown in Figure 4, the Kruskal-Wallis test method was selected for the significance of the differences in the characteristics of the different nodes. The Kruskal-Wallis test returns a value of \(p\) to indicate the size of the difference in the data distribution. A smaller \(p\)-value indicates a greater difference in the features and the more favorable the features are for classification and identification. Table 2 shows the \(p\)-values of the significant differences between the different signal characteristics of the different nodes.