# Study on Denoising Method of Weld Defect Signal Based on SSA-VMD-WPD
## 3 Results and Discussion
### Evaluation of Denoising Effect
#### 3.3.2 Signal Noise Reduction Studies  
This study utilized a Gaussian model to establish an ultrasonic echo signal model.  
\[X(t)=\beta\text{e}^{-\alpha\left(t-\tau\right)^{2}}\cos[2\pi f_{c}(t-\tau)+\phi] \tag{28}\]  
Set up a defect echo with 512 sampling points, where the amplitude coefficient \(\beta=0.8\), the bandwidth factor \(\alpha=50\) (MHz)\({}^{2}\), the center frequency \(fc=5.0\) MHz, the echo arrival time \(\tau=2.5\) us, and the phase \(\phi=8.89\). Gaussian white noise with signal-to-noise ratios of 5 dB, 10 dB, and 15 dB was added separately. This results in noisy signals with signal-to-noise ratios of 5.2540 dB, 10.0703 dB, and 15.1890 dB. The waveform plots of the original signal and noisy signals can be seen in Figure 12.  
Using envelope entropy as the fitness function, parameter optimization was conducted using the SSA algorithm. Initialization of the parameters begins by setting the minimum values for the decomposition mode number \(K\) and the penalty factor \(\alpha\) to [200,2] and the maximum values to [3000,10]. The number of iterations was set to 20, and the population size was set to 30. Taking the noise-containing signal with a signal-to-noise ratio of 15.1890 dB as an example, the diagram of the iterative optimization process can be seen in Figure 13.  
Figure 12: Simulated signal waveform.  
From the graph, it is evident that during the iterative process, the fitness value decreased sharply, indicating a good optimization effect. With the optimized parameters \(K=4\) and \(\alpha=498\), the signal was decomposed using VMD, and the variance contribution rate (\(VCR\)) of each mode component, along with its correlation coefficient R with the noisy simulated signal, was computed. The results are shown in Table 5.  
The time-frequency plots and spectrograms of each modal component of the VMD decomposition are shown in Figure 14. According to the selection criteria, modal components with correlation coefficient \(R<0.3\) or variance contribution rate \(VCR\leq 0.01\) were identified as noise components. Hence, IMF3 and IMF4 were determined to be the noise modal components. Analysis from Figure 14 reveals that the time-frequency plots of IMF1 and IMF2 resemble the noisy simulated signal, with their energy concentrated around 5 MHz, consistent with the central frequency set for the simulated signal. Therefore, IMF1 and IMF2 were considered noise-free modal components, while IMF3 and IMF4 were deemed noise modal components. Further wavelet packet denoising was applied to IMF3 and IMF4 using T as the indicator for selecting the denoising level and optimal wavelet basis function. Finally, the deno  
Figure 13: Diagram of the iterative optimization process.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Modal Components** & **IMF1** & **IMF2** & **IMF3** & **IMF4** \\ \hline Correlation coefficient \(R\) & 0.7629 & 0.7632 & 0.0975 & 0.0839 \\ Variance contribution ratio \(VCR\) & 0.4829 & 0.5076 & 0.0047 & 0.0049 \\ \hline \end{tabular}
\end{table}
Table 5: Calculation of metrics for each modal component of the VMD decomposition.  
By analyzing Table 6 and Figure 15, the following results can be derived:  
1. The denoising performance of wavelet packet analysis surpasses that of wavelet decomposition denoising. Additionally, when combined with parameter-optimized variational mode decomposition for joint denoising, wavelet packet analysis still outperforms wavelet decomposition. Hence, compared to wavelet decomposition, wavelet packet analysis is more suitable for denoising ultrasound phased array A-scan signals.
2. After processing with SSA-VMD-WPD, the maximum signal-to-noise ratio (\(SNR\)) obtained was 21.3816, with a minimum root mean square error (\(RMSE\)) of 0.0090. Compared to SSA-VMD, WPD, SSA-VMD-wavelet decomposition, and wavelet decomposition, the denoising effect of the proposed method in this paper is the best.