## 3 Ensemble Multiwavelet Analysis Method  
### Multiwavelet Lifting Scheme  
[PERSON] developed alifting scheme that made use of an existing wavelet and scaling functions to generate a new wavelet with prescribed or required properties via transformations in the time domain, which makes it possible to produce an adaptive wavelet. Assuming an initial set of biorthogonal filter operators \(\left\{H_{j}^{\mathit{old}},\widetilde{H}_{j}^{\mathit{old}},G_{j}^{\mathit{old}},\widetilde{G}_{j}^{\mathit{old}}\right\}\), then a new set of biorthogonal filter operators \(\left\{H_{j},\widetilde{H}_{j},G_{j},\widetilde{G}_{j}\right\}\) can be generated by conducting the lifting scheme as follows [16]:  
\[H_{j} =H_{j}^{\mathit{old}}\] \[G_{j} =G_{j}^{\mathit{old}}+S_{j}H_{j}^{\mathit{old}}\] \[\widetilde{H}_{j} =\widetilde{H}_{j}^{\mathit{old}}-S_{j}^{\ast}\widetilde{G}_{j}^ {\mathit{old}} \tag{8}\] \[\widetilde{G}_{j} =\widetilde{G}_{j}^{\mathit{old}}\]  
The forward and corresponding inverse operations of the wavelet lifting scheme are shown in Figure 1. The noteworthy advantage of producing a new wavelet basis function based on the lifting scheme is that the abundant choices of \(S\) can be realized. Once \(S\) is selected, the lifting scheme ensures and reserves that all filters have the property of biorthogonality. Thus, as can be seen, the foremost advantage of the lifting scheme is that it supplies the possibility of generating a customized wavelet. Even now, there are remarkable advantages for multiwavelet over scalar wavelet, but the fixed multiwavelet basis function, independent of the special vibration data, is still not the optimal selection for a specified engineering application. Thus, the lifting scheme is used to generate customized multiwavelet basis function or ensemble multiwavelet basis function. Based on the multiwavelet lifting scheme, a changeable set of biorthogonal filter operators can be obtained as follows [18]:  
\[H_{\mathit{new}}(z) =H(z)\] \[G_{\mathit{new}}(z) =T(z^{2})(G(z)+S(z^{2})H(z))\] \[\widetilde{H}_{\mathit{new}}(z) =\tilde{H}(z)-S^{\ast}(z^{2})\widetilde{G}(z) \tag{9}\] \[\widetilde{G}_{\mathit{new}}(z) =(T^{\ast}(z^{2}))^{-1}\widetilde{G}(z)\]  
where the determinant of \(T(z)\) is a monomial and \(S(z)\) and \(T(z)\) are finite-degree. One of the most important properties of the multi-scaling function, which has noteworthy significance in engineering applications, is the approximation order. Based on the wavelet theory, we know that if a multi-scaling function owns an approximation order \(m\), this indicates that the corresponding multiwavelet function owns \(m\) vanishing moments. In the following, the procedure of generating a new multiwavelet, based on the multiwavelet lifting scheme, and by use of an original multiwavelet with required numbers of vanishing moments will be explained. Firstly, select the original multiwavelet \(\omega_{0}(x)\) (\(\omega_{0}(x)=\Psi_{1}\) or \(\Psi_{2}\) ) from the basis function library and a set of translation  
Figure 1: The forward and inverse transform of the lifting scheme. quantity \(k\) of scaling functions as well as wavelet functions \(\omega_{{}_{1}}(x),\ldots,\omega_{{}_{k}}(x)\). Next, generate the new multiwavelet by use of the \"lifting coefficients equation\" as follows:  
\[G_{j} =G_{j}^{\mathit{old}}+S_{j}H_{j}^{\mathit{old}}\]0  
If the vanishing moment of a multiwavelet required to be lifted from \(p\) to _p'_, both sides of \"lifting coefficients equation\" are integrated. Then, a set of linear equations in the matrix form is obtained and displayed as follows:  
\[