## 3 The Proposed IAVMD Method  
### Waveform Matching Extension  
In the waveform matching extension, we first look for all available similar waveforms, then pick out the best matching waveform within them. Last, we connect the waveband before/after the best matching waveform to both left and right ends of the raw waveform to finish the task of boundary processing. Figure 1 shows the flowchart of the waveform matching extension. For one compound waveform \(x(t)\), the waveform matching extension procedure is described below [56].  
(1) Find all extreme points and boundary points of the original signal \(x(t)\) through the peak searching method, and determine the characteristic waveform. Take Figure 2 for an example, \(M_{i}(i=1,2,3,\cdots)\) is the maximum value of \(x(t)\), corresponding to the time \(tm_{i}\), \(N_{i}\) is the minimum value of \(x(t)\), corresponding to the time \(tn_{i}\), and \(S_{1}\) is the left boundary point of the signal \(x(t)\). We regard the triangular waveform \(S_{1}-M_{1}-N_{1}\) of the original signal \(x(t)\) as the characteristic waveform to search the optimum matching waveform.  
(2) Seek for all matching waveforms resembling the characteristic waveform \(S_{1}-M_{1}-N_{1}\) on the basis of the time \(ts_{i}\) corresponding to the starting value \(S_{i}\) of the matching waveforms, where the time \(ts_{i}\) of the horizontal axis is achieved by the linear interpolation, which is expressed as:  
\[ts_{i}=\frac{tm_{1}\times tn_{i}-tn_{1}\times tm_{i}}{tm_{1}-tn_{1}} \tag{6}\]  
(3) Estimate the matching error generated in searching all similar waveforms according to the following Equation (7):  
\[E_{i}=\left|S_{i}-S_{1}\right|+\left|N_{i}-N_{1}\right|+\left|M_{i}-M_{1} \right|+\left|M_{i+1}-M_{2}\right| \tag{7}\]  
where \(\left|M_{i+1}-M_{2}\right|\) represents the trend term of the similar waveforms, which can reveal the relative extremum position of all similar waveforms.  
(4) Compare the matching error, determine the smallest error, and consider the similar waveform \(S_{1}-M_{1}-N_{1}\) containing the smallest error as the best matching waveform.  
(5) Use the data before the best matching waveform to extend the left ends of the raw waveform \(x(t)\).  
(6) Apply the same way to extend the right ends of the raw waveform \(x(t)\), and finally obtain the extended waveform. This indicates that the waveform matching extension procedure of the original signal \(x(t)\) is finished in this stage. That is, the left and right boundaries of the original signal \(x(t)\) have been processed thoroughly.  
Here, two representative examples for a signal with an abrupt change are utilized to describe intuitively the procedure of waveform matching extension. Without loss of generality, the number of data points extended in both left and right ends are selected as 100 points. Figure 3a,b shows the processing results achieved when applying the waveform matching extension for a cycle shock pulse signal and a modulated compound signal, respectively. Obviously, the waveform extended in both the left and right ends is in line with the variation trend of the raw signal, which indicates that the waveform matching extension is feasible in analyzing the modulated compound signal.  
Figure 1: Flowchart of the waveform matching extension.  
Figure 3: Intuitive examples of the waveform matching extension: (**a**) a cycle shock pulse signal and (**b**) a modulated compound signal.  
Figure 2: Sketch map of the waveform matching extension.