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