Fig. 1. Flow chart of VMD denoising based on parameter optimization
Fig. 2. Noise-free ultrasonic echo signal and frequency spectrum. (a) Echo signal; (b) frequency spectrum
Fig. 3. EMD modes and frequency spectra under noise-free signal. (a) EMD modes; (b) frequency spectra
Fig. 4. VMD modes and frequency spectra under noise-free signal. (a) VMD modes; (b) frequency spectra
Fig. 5. VMD modes and frequency spectra under noise-free signal when [k, α]=[4, 2500]. (a) VMD modes; (b) frequency spectra
Fig. 6. VMD modes and frequency spectra under noise-free signal when [k, α]=[5, 2500]. (a) VMD modes; (b) frequency spectra
Fig. 7. VMD modes and frequency spectra under noise-free signal when [k, α]=[6, 2500]. (a) VMD modes; (b) frequency spectra
Fig. 8. Ultrasonic echo signal and frequency spectrum after adding white Gaussian noise with SNR of 20 dB. (a) Echo signal; (b) frequency spectrum
Fig. 9. VMD modes and frequency spectra under white Gaussian noise. (a) VMD modes; (b) frequency spectra
Fig. 10. EMD modes and frequency spectra under white Gaussian noise. (a) EMD modes; (b) frequency spectra
Fig. 11. Ultrasonic echo signal and frequency spectrum after adding FFN1 with SNR of 15 dB. (a) Echo signal; (b) frequency spectrum
Fig. 12. VMD modes and frequency spectra under FFN1. (a) VMD modes; (b) frequency spectra
Fig. 13. EMD modes and frequency spectra under FFN1. (a) EMD modes; (b) frequency spectra
Fig. 14. Ultrasonic echo signal and frequency spectrum after adding FFN2 with SNR of 17 dB. (a) Echo signal; (b) frequency spectrum
Fig. 15. VMD modes and frequency spectra under FFN2. (a) VMD modes; (b) frequency spectra
Fig. 16. EMD modes and frequency spectra under FFN2. (a) EMD modes; (b) frequency spectra
Fig. 17. Denoised reconstructed signal and frequency spectrum when white Gaussian noise is added. (a) Reconstructed signal; (b) frequency spectrum
Fig. 18. Denoised reconstructed signal and frequency spectrum when FFN1 is added. (a) Reconstructed signal; (b) frequency spectrum
Fig. 19. Denoised reconstructed signal and frequency spectrum when FFN2 is added. (a) Reconstructed signal; (b) frequency spectrum
Fig. 20. Noisy signal and frequency spectrum in experiment. (a) Noisy signal; (b) frequency spectrum
Fig. 21. VMD modes and frequency spectra of noisy signal. (a) VMD modes; (b) frequency spectra
Fig. 22. EMD modes and frequency spectra of noisy signal. (a) EMD modes; (b) frequency spectra
IMF component | Correlation coefficient |
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IMF1 | 0.9443 | IMF2 | 0.0076 | IMF3 | 0.0124 |
|
Table 1. Correlations between IMF components and original signal
Denoising method | Wavelet threshold | EMD+wavelet threshold | EEMD+wavelet threshold | VMD+wavelet threshold | Proposed method |
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SNR | 10.0754 | 12.8064 | 13.2024 | 13.3214 | 14.2134 | MSE | 0.1123 | 0.0863 | 0.0639 | 0.0596 | 0.0396 |
|
Table 2. Performance comparisons of 5 denoising methods