## 3 Redundant Symmetric Lifting Schemes and the Improved Neighboring Coefficients
### The Proposed Method Using Customized Multiwavelets and the Improved Neighboring Coefficients  
In this paper, redundant symmetric lifting scheme is applied to construct customized multiwavelets with specified properties for specific signals. Then the lifting coefficients of Equation (13) must be underdetermined to ensure there are free parameters in these equations. Generally for a fixed vanishing moment \(p\), the more functions are used for the lifting, the longer support of the obtained multiwavelet is. And more functions mean a greater freedom in lifting. However, longer support will reduce the localization of multiwavelet. Therefore, a tradeoff has to be chosen between vanishing moment and numbers of functions used to do lifting. We construct new multiwavelet starting from Hermite splines [19] as the original scaling function. In dynamic signals, impulses are always the fault symptoms of defected components and Hermite spline multiwavelet is very similar to impulse components. Multiscaling functions and multiwavelet functions of Hermite spline multiwavelet are shown in Figure 5. There is more freedom and flexibility to construct new multiwavelets with prescribed properties because of the simple waveform of Hermite splines. The proposed methods can act as an effective and promising tool for fault detections of planetary gears. Figure 4: The algorithm of improved neighboring coefficients. There are \(N_{f}=(p^{\cdot}-p)-Rank(M_{B})\) free parameters, which are vital for the redundant symmetric lifting scheme. The customized multiwavelet construction is performed by the optimization of the free parameters. Kurtosis is widely used for fault feature detections because it is sensitive to sharp variant structures, such as impulses. The bigger the impulses in signals, the larger the kurtosis [35]. Furthermore, it is a dimensionless parameter, which is independent of the amplitudes of the signal. The definition of kurtosis is shown in Equation (20):  
\[K_{p}=\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})^{4}}{n\sigma^{4}} \tag{20}\]  
where \(x_{i}\) is the \(ith\) point of the signal \(x\), \(\overline{x}\) is the mean value of \(x\), \(n\) is the signal length of \(x\), \(\sigma\) is the standard deviation of \(x\). The forth moment of \(x\) enables the numerator of \(K_{p}\) increases quickly while the denominator increases slowly when incipient faults occurred. Therefore, kurtosis is sensitive for incipient faults with less impulses but descending when impulses are more because of the fault exacerbation. More impulses mean a distinct periodicity. Therefore, kurtosis cannot accurately depict the real signal trend in the detection of periodic shock impulses. In dynamic response signals, the mechanical fault often expresses as periodic impact features which can be detected through the envelope spectrum of the vibration signal. Hence, the envelope spectrum entropy is selected as an evaluation indicator to obtain the customized multiwavelets. Figure 5: Hermite spline multiscaling functions and multiwavelet functions. (**a**) Multiscaling functions. (**b**) Multiwavelet functions. Assume \(\{c_{i}\}_{i=1,\ldots,M}\) is the class of the normalized coefficients. When it is divided by \(\sum_{i=1}^{M}c_{i}\), \(\{d_{i}\}_{i=1,\ldots,M}\) is obtained:  
\[d_{j}=c_{j}\big{/}{\sum_{i=1}^{M}c_{i}} \tag{21}\]  
The multiwavelet entropy \(En_{mwt}\) is calculated by:  
\[En_{mwt}=-\sum_{i=1}^{M}d_{i}\ln d_{i} \tag{22}\]  
According to the information theory, the most uncertain probability distribution has the maximum entropy value, and the entropy value reflects the uniformity of the probability distribution. So \(En_{mwt}\) provides the information about the definite degree of the envelope spectrum. The smaller the value of the envelope spectrum entropy \(En_{mwt}\) is, the more distinct periodic impact features will be. Our objective is to find the optimal multiwavelets by finding the minimum value of envelope spectrum entropy \(En_{mwt}\).