## 4 Application to Simulation Signal  
In order to verify the effectiveness of the RS-LOD method and compare it with other methods, the same simulation signal is analyzed using RS-LOD, EMD, and LOD methods, respectively. In a comparison of the analysis processing, the end effects are not solved. It is to say that end effects  
Figure 5: A flowchart of RS-LOD method.  
may appear in all methods. Therefore, the severity of end effects can be used as an assessing index to compare the three methods. Additionally, the standard deviation (SD) criterion [7] is adopted as the iterative stopping condition, that is, by calculating the standard deviation of the results of two consecutive iterations, and comparing with the predetermined threshold to determine the iterative termination. The predetermined threshold is 0.5 in this paper.  
Here we consider a multi-component AM-FM simulation signal \(x(t)\):  
\[x_{1}(t)=10\sin(50\pi t), \tag{17}\]  
\[x_{2}(t)=5\sin(10\pi t), \tag{18}\]  
\[x_{3}(t)=5\exp(-0.5t)\sin(3\pi t), \tag{19}\]  
\[x(t)=x_{1}(t)+x_{2}(t)+x_{3}(t), \tag{20}\]  
where \(t\in[0,\,6]\), the setting sampling frequency 1000 Hz, and the simulation signal consists of a modulated signal and two sine signals. The time-domain waveforms of \(x(t)\) and its three components \(x_{1}(t)\), \(x_{2}(t)\), and \(x_{3}(t)\) are shown in Figure 6.  
The simulation signal \(x(t)\) is decomposed by using the LOD, EMD, and RS-LOD method and the decomposition results are presented in Figures 7-9, respectively. It can be seen from Figures 7-9 that all three methods can decompose the original signal from high frequency to low frequency in turn, and the decomposed three components correspond to the \(x_{1}(t)\), \(x_{2}(t)\), and \(x_{3}(t)\) of the original signal, respectively. However, it is also obvious found that there exist some differences in the results of their decomposition. In Figure 7, the decomposition result of the LOD is serrated, and the smoothness is poor (marked by a red circle). The RS-LOD decomposition result shown in Figure 9 (set \(p=5\)) does not show this phenomenon, and because of the good smoothness, the cumulative error is small; thus the amplitude of the residue is obviously smaller than that of LOD. In Figure 8, due to the cubic spline function used by EMD, the end effects are obvious [20], and the amplitude of residue is also larger than that of RS-LOD because of the error of the end effects. In summary, we can see that the decomposition result of the RS-LOD is better than that of LOD and EMD as a whole.  
Figure 6: Time-domain waveform of the simulation signal \(x(t)\) and its components.  
Figure 8: Empirical mode decomposition (EMD) results of the simulation signal \(x(t)\).  
Figure 7: LOD decomposition results of the simulation signal \(x(t)\). (Red circle: exhibit the poor smoothness of the decomposition results).  
In addition, in order to quantitatively compare the decomposition results of the three methods, the assessing indicators, including the index of orthogonality (IO) [7], the number of sifting times, and the consuming time are used to evaluate the results. The three algorithms have been run 30 times using MATLAB R2017b to above simulation signal \(x(t)\) on the same computer, respectively, and all using SD \(<\) 0.5 as iterative stopping condition. The average values of three assessing indicators are calculated and counted, and the results are shown in Table 1.  
It can be seen from Table 1, the IO value, which reflects the decomposition effects, the RS-LOD is the best of the three. Comparing the LOD method which uses linear transformation completely, the consuming time of the RS-LOD is a little longer, but they are both obviously shorter than that of the EMD method. Therefore, through a quantitative comparison, the RS-LOD method also has some advantages in terms of decomposition effects and computational efficiency.