## 3 Numerical Simulation Signal Analysis and Discussion  
A numerical simulation is done to verify the proposed method. Without loss of generality, the modulated sine burst, which is a narrow-band signal, is employed in the simulation. From [36], the simulation signal can be expressed as follows:  
Figure 4: The detected boundaries which are denoted by the blue dot-dashed lines, the four LWM and the minimum distance of two adjacent LWM. Figure 3: The detection process of the first LWM. (**a**) Detect the four local maxima and determine the window. (Red circles are the local maxima; blue dashed lines are the boundaries of the window. ); (**b**) zero the amplitude in the window, obtain the first LWM which is denoted by pentagram, and redetect the other three local maxima which are denoted by the red circles. \[x(t)=A[H(t)-H(t-\frac{N}{f_{\rm C}})][1-\cos(\frac{2\pi f_{\rm c}t}{N})]\sin(2\pi f _{\rm c}t), \tag{12}\]  
where \(A\) is the signal amplitude; \(f_{\rm c}\) is the central frequency; \(t\) is the time; \(H(t)\) is the Heaviside step response; \(N\) is the wave numbers. For example, when \(A=5\), \(N=20\), \(t=1\) ms, \(f_{\rm C}=40\,kHz\), and the sample rate is 1 MHz, the simulation signal is shown in Figure 5.  
During the simulation process, the first step is to generate the mixed signal according to the Equation (12), if it is composed of \(n\) different modes, then it can be expressed as:  
\[s(t)=\sum_{i=1}^{n}x_{i}(t) \tag{13}\]  
When \(n=9\), and all the simulation parameters are as given in Equation (14), the simulated sensor signal is shown in Figure 6.  
\[ \begin{array}{c}A=\left[\begin{array}{ccccccccc}4&3&2&3&2&1&2&1&0.5\end{array} \right]\\ N=\left[\begin{array}{ccccccccc}30&30&30&40&40&40&50&50&50\\ 40&35&45&70&65&75&110&105&115\end{array}\right]\end{array}  \tag{14}\]  
It is difficult to separate the simulated sensor signal into different components only from the waveform information in Figure 6a. But there are nine different frequency peaks in the spectrum (Figure 6b). Here the parameters of window value and segmentation numbers will be discussed. The window value relies on the practical requirement and experience, for example if the three components of 35, 40 and 45 kHz can be regarded as only one component, the window will be 10 kHz, but the segmentation numbers can be obtained by the energy of the component, if the energy proportion of one component is larger than a specify value, then the component can be considered as a segmentation. When the window value is 10 kHz and the segmentation numbers are three, the detected local maxima of the original EWT and the LWM method, which are the red circles and red stars, respectively, are illustrated in Figure 7. As shown in Figure 7a, the first two local maxima are very close, which may generate two narrow bands in the decomposition results, and this is not applicable for analyzing the non-stationary AE signals which have the dispersion phenomenon when they propagate on the composite structure. Therefore, compared with the original EWT method, the proposed LWM approach gives better results for the local maxima, which are shown in Figure 7b. Based on the local maxima, segmentation boundaries are obtained which are the red dashed lines in Figure 7b, and three spectrum segmentations are shown, denoted by the gray, cyan and green colors. This indicates that the proposed LWM algorithm performs well in searching for the local maxima and segmenting the spectrum. Figure 5: The (**a**) waveform and (**b**) spectrum of the simulation signal. The decomposition results of the LWM-EWT method are depicted in Figures 8 and 9, in order to have a more obvious contrast, the original signals with offset of 20 V (each of which is the mixed signal of three nearest frequency simulation signal), the separated signals and error with offset of \(-20\) V between them are depicted in one figure.