BroadBand-Adaptive VMD with Flattest Response
Abstract
:1. Introduction
2. Review of VMD
2.1. Mode Definition
2.2. VMD Model
2.3. Wiener Filtering of VMD
3. Ideas for Improving VMD
3.1. The Flattest Response
1 | 1 | |||||
2 | 1 | |||||
3 | 2 | 2 | 1 | |||
4 | 2.61312593 | 3.41421356 | 2.61312593 | 1 | ||
5 | 3.23606798 | 5.23606798 | 5.23606798 | 3.23606798 | ||
6 | 3.86370331 | 7.46410162 | 9.14162017 | 7.46410162 | 3.86370331 | 1 |
3.2. To Set the Bandwidth
3.3. Harmonics
4. Improved VMD
4.1. Improved Optimal Problem
4.2. Solution to the Problem
4.3. Minimization w.r.t
4.4. Minimization w.r.t
4.5. Minimization w.r.t
4.6. Complete Algorithm
Algorithm 1: Complete optimization of iVMD |
Initialize , , , , |
Repeat |
For do |
Update for all : |
Update : |
in Table 2 |
End for |
Update Lagrangian multiplier for all : |
Until convergence |
4.7. Reconstruction versus Denoising
5. Experiments and Results
5.1. Example 1 with Linear Trend
5.2. Example 2 with a Piecewise Signal
5.3. Example 3: Intrawave Frequency Modulation
5.4. Example 4: Sawtooth Signal
5.5. Example 5: An Electrocardiogram
6. Conclusions and Outlook
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Shen, X.; Li, R. BroadBand-Adaptive VMD with Flattest Response. Mathematics 2023, 11, 1858. https://doi.org/10.3390/math11081858
Shen X, Li R. BroadBand-Adaptive VMD with Flattest Response. Mathematics. 2023; 11(8):1858. https://doi.org/10.3390/math11081858
Chicago/Turabian StyleShen, Xizhong, and Ran Li. 2023. "BroadBand-Adaptive VMD with Flattest Response" Mathematics 11, no. 8: 1858. https://doi.org/10.3390/math11081858
APA StyleShen, X., & Li, R. (2023). BroadBand-Adaptive VMD with Flattest Response. Mathematics, 11(8), 1858. https://doi.org/10.3390/math11081858