## 3 Results  
### Simulated Signal Analysis  
To verify the effectiveness of the DIMIRLMD algorithm, a strong nonstationary simulation signal is defined by the superposition of a periodic shock wave, a sinusoidal wave, and noise:  
\[\begin{array}{c}y(t)=y_{1}(t)+y_{2}(t)+y_{3}(t)\\ y_{1}(t)=y_{n}e^{-\xi\omega_{n}t}sin\omega_{n}\sqrt{1-\xi^{2}t}\\ y_{2}(t)=y_{m}sin\omega_{m}t\\ y_{3}(t)=NOISE\end{array} \tag{20}\]  
The four methods LMD, CSILMD, MPCHILMD, and DIMIRLMD are each employed to separately break down the signal. The decomposition results of the first three PF components are shown in Figures 9-12. In theory, the decomposition results should be decomposed into more than three components from high frequency to low frequency. The noise component should be represented by the first three components, and the impulse component and the sinusoidal component should be contained in the simulation signal.  
Figure 8: Flowchart of DIMIRLMD.  
Through the comparison between decomposition results, significant differences are presented in the four LMD methods. We can see that the second and the third component in Figure 9 for the original LMD have deviated from simulation components, indicating that obvious modal aliasing occurs. The first three components of CSILMD and MPCHILMD are shown in the left panel of Figure 10. The second component is the third component in Figure 11. The second component is the third component in Figure 11. The third component is the third component in Figure 11.  
are generally similar to the simulation components. Due to the influence of the shock signal, the third sinusoidal PF component of these two methods is mixed with obvious shock disturbance. In the decomposition result of the DIMIRLMD method, although the third component contains a slight shock disturbance, its effect is significantly improved compared with the other three methods.  
The most useful assessment metric for different LMD decomposition outputs is the mean square error (MSE) between the PF component and the associated signal. To further compare the analysis results, the MSE of the last two PF components for four LMD methods is shown in Table 2. Usually, the more precise the envelope function is, the fewer iterations are needed to complete the decomposition; the iteration number of each PF component is also presented in Table 2.  
Corresponding to the conclusions drawn from Figures 9-12, Table 2 further proves this fact based on MSE and iterations from a quantitative point of view. Due to serious modal aliasing, the second and third PF components of the original LMD appear to have an excessive MSE and too much iteration compared to the other three methods. CSILMD and MPCHILMD both show a clear improvement in MSE and iterations for two PF components. CSILMD is more excellent in the sinusoidal PF component for its smooth envelope, and MPCHILMD is better in the impulsive PF component for its flexible envelope. The DIMIRLMD presents an outstanding advancement in MSE and iterations for both PF component methods. This is because the optimal double interpolation scheme adaptively matches the best interpolation method for PF components with different characteristics, and the interval reconstruction envelope scheme directly suppresses the demodulation mutation phenomenon. The combination of these two schemes effectively ensures the accuracy and stability of the decomposition.