## 3 The Construction of Wavelet Packet Signature Analysis  
[PERSON] in 1989 introduced a recursive algorithm known as the pyramid algorithm [36]. This algorithm, which is deemed to be an important algorithm for computing the DWT coefficients, consists of a conjugate pair of low-(H) and high-(G) pass filters followed by down sampling giving two coefficients, namely details and approximations. In the next decomposition levels, the filters are only applied to the approximations. Compared with DWT, only the approximate coefficients will be further decomposed at each level, both the approximate coefficients and the detail coefficients will be further decomposed at each level in WPD that offers a richer signal analysis [37; 38]. Wavelet packet atoms are waveforms with three naturally interpreted indices. \[\psi_{j,k}^{i}(t)=2^{\frac{i}{2}}\psi^{i}\Big{(}2^{j}t-k\Big{)}\ i=0,\ 1,\ 2,\ 3\ldots \tag{2}\]  
where integer \(j\) determines the dilation, and \(k\) and \(i\) are called the time-localization and modulation parameters of the wavelet packet, respectively. Wavelet packet atoms are defined by the following sequence of functions:  
\[\psi_{0,0}^{2i}(t)=\sqrt{2}\sum_{k}h(k)\psi_{1,k}^{i}(2t-k) \tag{3}\]  
\[\psi_{0,0}^{2i+1}(t)=\sqrt{2}\sum_{k}g(k)\psi_{1,k}^{i}(2t-k) \tag{4}\]  
The conjugate mirror filters \(h(k)\) and \(g(k)\) with finite impulse responses (FIR) of size \(k\) can define the fast binary WPD algorithm of the signal. The first two WP functions \(\psi_{0,0}^{0}(t)=\varphi(t)\) and \(\psi_{0,0}^{1}(t)=\psi(t)\) are also called the scaling function and wavelet function, respectively [36; 39; 40; 41]. Each output of the filter consists of \(N/2\) wavelet coefficients. The wavelet packet coefficients (WPC) of signal \(f(t)\) are calculated by taking the inner product of the signal and basic function:  
\[C_{j}^{i}(k)=\int_{-\infty}^{\infty}f(t)\psi_{j,k}^{i}(t)dt \tag{5}\]  
Since the wavelet coefficients will highpoint the changes in signals, the wavelet coefficients-based features are relatively suitable for early and high sensitivity fault detection. The wavelet packet coefficient features have been broadly used for characterizing machine faults. Although these coefficients are associated with frequency components, they are modified in the time domain (each coefficient, \(C^{i}\), corresponds to a time range). Each \(C_{j}^{i}(k)\) coefficients measure a specific sub-band frequency content. For a discrete signal,  
\[C_{j+1}^{2i}(\tau)=\sum_{k}h(k-2\tau)C_{j}^{i}(k) \tag{6}\]  
\[C_{j+1}^{2i+1}(\tau)=\sum_{k}g(k-2\tau)C_{j}^{i}(k) \tag{7}\]  
The frequency bandwidth of WPT coefficient can be calculated by [42; 43]. \[\left[\frac{if_{s}}{2^{j+1}}.\frac{(i+1)f_{s}}{2^{j+1}}\right] \tag{8}\]  
The wavelet transform is commonly used in the time domain, which is why the frequency localization of wavelet packets is more complicated to analyze. The popularity of wavelets is due to its dilation and translation properties. Dilation property is used to adjust the width of the frequency band along with the location of its center frequency, and the translation property can be used to automatically zoom in and out in order to locate the positions of high-frequency and low-frequency changes. As mentioned above, the parameter \(i\) is the interpretation of oscillation. The basic idea of the wavelet packets is that for fixed values of \(j\) and \(k\), \(\psi_{j,k}^{i}(t)\) analyzes the fluctuations of the signal roughly around the position \(2^{j}\ k\), at the scale \(2^{j}\), and at various frequencies for the different admissible values of the last parameter \(i\). Note that the method of decomposition described above does not result in a WPT tree displayed in increasing frequency order. In fact, the delicate point is to realize that the naturally order for \(i\) = 0, 1, 2, 3, 4, 5, 6, 7 does not matched exactly the order defined by the number of oscillationsin wavelet packet decomposition. The frequency order is \(i\) = 0, 1, 3, 2, 6, 7, 5, 4 as highlighted in Figure 1 by dashed lines.