In this paper, three sets of signal simulation tests are designed. Gaussian white noise with different signaltonoise ratios is added for three different original signals (heavy sine signal, bumps signal, and doppler signal). The noisy signals are denoised using the superposition denoising with 20 signals (SD20), the superposition method with 5 signals (SD5), the wavelet threshold denoising (WTD) and the wavelet threshold denoising after superposition with 5 signals (SD5WTD). The four denoising methods are compared according to the SNR and MSE, and the optimal denoising results are selected.
The steps of the simulation tests are as follows:
a) Randomly generate 20 sets of noiseadded signals.
b) Denoising by the superposition method using the 20 sets of signals and calculating the SNR and MSE.
c) Five groups of signals from the 20 groups were randomly selected for denoising by the superposition method and calculating the SNR and MSE.
d) Wavelet threshold denoising was performed by randomly selecting 1 group from 20 groups of signals, and the SNR and MSE were calculated.
e) Performing wavelet threshold denoising based on the results of step b) and calculating the SNR and MSE.
f) Select the denoising method with the largest SNR and smallest MSE.
Heavy Sine signal test
Add Gaussian white noise with SNR = 20 for the Heavy Sine signal. The decomposition level of wavelet threshold denoising is chosen as 6 layers, and the wavelet base is "db4".
Figure 2 shows the original Heavy Sine signal, the noisy Heavy Sine signal, and the Heavy Sine signal denoised by each of the four methods. The SD5WTD can reveal a smoother Heavy Sine signal and retain many signal details. Other methods reduce the noise to varying degrees, but the SD5WTD is closest to the original signal.
Table 1 shows the SNR and MSE of each method for the denoising of Heavy Sine signals. The results show that the SD5WTD has the highest SNR and the smallest MSE, which is the most effective method among the four denoising methods.
Table 1
Results of SNR and MSE for Heavy Sine signal denoising.
Denoising Method

SNR

MSE

SD20

32.9464

0.0048

SD5

26.8945

0.0195

WTD

29.3057

0.0112

SD5WTD

33.7294

0.0040

Bumps signal test
Add Gaussian white noise with SNR = 12 for the Bumps signal. The decomposition level of wavelet threshold denoising is chosen as 6 layers, and the wavelet base is "sym8".
Figure 3 shows the original Bumps signal, the noisy Bumps signal, and the Bumps signal denoised by each of the four methods. Due to the excessive noise added, the fluctuation of the noisy signal is obvious. However, SD5WTD achieved the best performance as well. The characteristics of the original signal are restored more realistically. Although SD20 restores the overall trend of the signal, there is still too much high frequency noise.
Table 2 shows the SNR and MSE of each method for the denoising of Bumps signals. The results show that the SD5WTD has the highest SNR and the smallest MSE, which is the most effective method among the four denoising methods. Combined with Fig. 3, the SD5WTD superimposed has the most obvious denoising effect.
Table 2
Results of SNR and MSE for Bumps signal denoising.
Denoising Method

SNR

MSE

SD20

25.2472

0.0097

SD5

18.9110

0.0416

WTD

19.9496

0.0328

SD5WTD

26.0987

0.0080

Doppler signal test
Add Gaussian white noise with SNR = 15 for the Doppler signal. The decomposition level of wavelet threshold denoising is chosen as 6 layers, and the wavelet base is "sym8".
Figure 4 shows the original Doppler signal, the noisy Doppler signal, and the Doppler signal denoised by each of the four methods. The Doppler signal has much highfrequency information, which overlaps with the noise after adding noise, and the superposition denoising retains this highfrequency information to a certain extent. As the number of superimposed signals increases, the denoising effect becomes more obvious, but this is not compatible with the actual application process. The signal obtained by denoising with the SD5WTD is relatively smooth but missing in the high frequency part.
Table 3 shows the SNR and MSE of each method for the denoising of Doppler signals. The results show that the SD5WTD has the highest SNR and the smallest MSE. Combined with Fig. 4, the SD5WTD has significant advantages in denoising lowfrequency signals, and the overall denoising effect is better than superposition denoising although it is slightly inadequate for highfrequency signals.
Table 3
Results of SNR and MSE for Doppler signal denoising.
Denoising Method

SNR

MSE

SD20

28.1470

0.00013

SD5

21.9504

0.00055

WTD

23.0395

0.00043

SD5WTD

29.0310

0.00011

3.4 Elastic wave signal test
To test the practical application of the SD5WTD for elastic wave signal, a bending element type piezoelectric transducer is used to receive 5 elastic wave signals generated from a stable source. Figure 5 shows the elastic wave signal and the denoised elastic wave signal.
The results show that the SD5WTD can obtain clear waveforms and enhance the accuracy of further analysis of the elastic wave signals.