## 4 Analysis of Marginal Spectrum Change of an Active Lamb Wave
### Application Analysis of Marginal Spectrum Change in Plate Structural Health Monitoring  
In the calculation of the marginal spectrum of the signal, the first step is to decompose the signal into multiple order IMFs by using the EMD. The EMD is an adaptive signal decomposition method based on the extreme signal points [20]. It is impossible to forecast the influence of minor changes in the extreme points on the IMFs, because the EMD process is complicated and includes interpolation, end-point processing, and IMF judgment. Namely, the signals are highly consistent from the sequence perspective, but they have randomly scattered extreme points, which may result in poor consistency in the IMFs obtained through the decomposition. The inconsistency degree can be so high that it can be impossible to analyze the marginal spectrum change of a signal before and after damage. Therefore, to guarantee the consistency in the IMFs acquired through the EMD process, the randomly scattered extreme points must be filtered out. Notably, in this study, the phase shift information of  
Figure 5: Instantaneous frequency and marginal spectrum of the Lamb wave packet with a center frequency of 150 kHz. (**a**) Sequence diagram and instantaneous frequency of the wave packet, and subfigure (**I**) shows the wave packet before and after damage; subfigure (**II**) shows the instantaneous frequency before and after damage. (**b**) The intrinsic mode function (IMF) and marginal spectrum of the wave packet. Subfigure (**I**) shows the IMFs after empirical mode decomposition (EMD) decomposition of the wave packet, and subfigure (**II**) shows the marginal spectrum of the wave packet before and after damage. the instantaneous frequency of the signals that cannot be generated by a filter is analyzed, and thus the zero-phase filtering method must be used [31; 32]. Besides, Wave packet 1 of the health sensing signal with a center frequency of 100 kHz, shown in Figure 4 bI, is used as an example. The first-order IMFs obtained by the EMD before filtering out randomly scattered extreme points are presented in Figure 6a, wherein it can be seen that there was a poor consistency between the first-order IMFs of the ten same-condition signals. The first-order IMFs obtained by the EMD after filtering out randomly scattered extreme points are presented in Figure 6b, wherein it can be seen that the consistency between the IMFs of the ten same-condition signals could be achieved. Under the premise of ensuring consistency in signal decomposition, the Lamb wave packet presented in Figure 4b was decomposed into multiple IMFs. As shown in Figures 5b and 6b, the signal energy was mainly concentrated in the first several order IMFs. According to the description in Section 4.1, only the first three orders of IMFs were needed to calculate the marginal spectrum, which can be further used to study the changes in the Lamb wave marginal spectrum before and after damage. Since the energy was mainly concentrated near the center frequency, the marginal spectrum corresponding to the frequency band near the center frequency was used. To compare the distribution of the marginal spectrum in the frequency domain before and after the damage more easily, the frequency band was divided into several continuous bands, and then the marginal spectrum amplitudes corresponding to all the frequencies in each frequency band were considered to be the marginal spectrum amplitudes of that band. To make the experimental data analysis conclusion reflect the marginal spectrum variation characteristics of the active Lamb wave signal after damage as closely as possible, the number of damage and health state signals used in the analysis was 10. The 10 marginal spectral amplitudes before and after the damage of the same frequency band are displayed in the same subfigure. The marginal spectrum of the Lamb wave with a center frequency of 100 kHz is presented in Figure 7a, where the abscissa corresponds to the signal number and the ordinate corresponds to the amplitude of the marginal spectrum.