# A Novel Rolling Bearing Fault Diagnosis and Severity Analysis Method
## 3 Wavelet Packet Energy Entropy  
The WT is increasingly used in engineering applications, especially in signal feature extraction. Unlike the FFT, which is frequency-localized, the WT is a suitable tool for detecting and characterizing specific phenomena in both the time and frequency spaces, but it does not split the high-frequency bands. The wavelet packet transform (WPT) is a further generalization of the WT and is an effective method for vibration signal processing because it can process the entire set of frequency bands, including both the high- and low-frequency bands [13]. Notably, this coverage includes the high-frequency bands where the information of a rolling bearing fault always exists. A signal can be decomposed into a set of wavelet packet nodes with the form of a full binary tree by the WPT. The structure diagram of the wavelet packet three-layer is shown in Figure 1, which can also be called the wavelet packet decomposition tree. In the binary tree structure, each wavelet packet node is indexed by \((j,n)\) and the corresponding wavelet packet coefficient is defined as \(d_{j}^{n}\), where \(j\) represents the decomposition levels and \(n\) indicates the node index in level \(j\). \((j,n)=(0,0)\) is the root of the decomposition tree corresponding to the original signal \(S_{0,0}\). It is clear that the original signal \(S_{0,0}\) splits into two parts in the first level during the WPT. The left branch undergoes low-pass filtering and a vector of approximation coefficients \(d_{1}^{0}\) can be obtained. The right branch undergoes high-pass filtering and a vector of detail coefficients \(d_{1}^{1}\) can be acquired. Using the same scheme, \(d_{1}^{0}\) and \(d_{1}^{1}\) are split into two parts, obtaining \(d_{2}^{0}\), \(d_{2}^{1}\), \(d_{2}^{2}\), and \(d_{2}^{3}\). If the decomposition level is \(j\), the number of nodes after decomposition is \(2^{j}\). That is, the frequency band of the sampling signals is subdivided into \(2^{j}\) small segments. The wavelet packet algorithm consists of two parts: The decomposition algorithm and the reconstruction algorithm. When the orthogonal scaling function \(\phi(t)\) and the wavelet function \(\psi(t)\) are set, the relationships between them are as follows:  
\[\left\{ \begin{array}{l}\phi(t)=\sqrt{2}\underset{k}{\sum}h_{k}\phi(2t-k)\\ \psi(t)=\sqrt{2}\underset{k}{\sum}g_{k}\phi(2t-k)\end{array} \right., \tag{11}\]  
where \(k\) is a transformation parameter, \(h_{k}=1/\sqrt{2}\big{(}\phi(t),\phi(2t-k)\big{)}\) is the low-pass filter coefficient, and \(g_{k}=1/\sqrt{2}\big{(}\psi(t),\psi(2t-k)\big{)}\) is the high-pass filter coefficient. Here, \(\langle\cdot,\cdot\rangle\) represents the inner product operator. The decomposition algorithm for the coefficients is defined by the following recursive relationships:  
\[\left\{ \begin{array}{l}d_{j+1}^{2n}[k]=\sqrt{2}\underset{l}{\sum}h_{l-2k}d _{j}^{n}[k]\\ d_{j+1}^{2n+1}[k]=\sqrt{2}\underset{l}{\sum}g_{l-2k}d_{j}^{n}[k]\end{array} \right., \tag{12}\]  
where \(d_{j}^{n}[k]\) are the wavelet packet coefficients, \(d_{j+1}^{2n}[k]\) are the approximation coefficients, \(d_{j+1}^{2n+1}[k]\) are the detailed coefficients, and \(h_{l-2k}\) and \(g_{l-2k}\) are the low-pass and high-pass filter coefficients for decomposition, respectively. Figure 1: Structure diagram of wavelet packet three-layer decomposition. After decomposition by the wavelet packet, the reconstruction algorithm for the wavelet packet coefficients is deduced as:  
\[d_{j}^{n}[k]=\sum_{l}h_{k-2l}d_{j+1}^{2n}[k]+\sum_{l}g_{k-2l}d_{j+1}^{2n+1}[k]. \tag{13}\]  
The corresponding reconstructed signal of wavelet packet coefficients \(d_{j}^{n}\) is defined as S\({}_{j,n}\). Assuming that the decomposition level \(j=2\), the original signal can be represented as:  
\[S_{0,0}=d_{0}^{0}=S_{2,0}+S_{2,1}+S_{2,2}+S_{2,3} \tag{14}\]  
The reconfiguration signal of each node after wavelet packet decomposition reflects the corresponding frequency component in the distribution of the original signal.