## 3 Experimental Results and Analysis
### Simulation Experiment
\begin{array}{ccc}0.7071&0.007&-0.0043\\ 0.7121&-0.0947&0.0756\\ 0.7086&0.0466&0.0859\end{array} \right] \tag{7}\]  
Further analysis of the estimated transformation matrix \(\mathbf{A}\) reveals additional insights into the contribution of the separated components to the reconstructed signals. The values in the first column of the estimated transformation matrix \(\mathbf{A}\) are significantly larger than those in the last two columns, demonstrating that separated signal I contributes the most to the reconstruction of the three sets of signals. Figure 7 presents the reconstructed signals for the three sets. However, the non-zero values in the last two columns of matrix \(\mathbf{A}\) still represent some contributions (the non-zero values in the last two columns are not zeros) from the separated components II and III to the three reconstructed signals. Thus, it can be seen from Figure 7 that there are certain differences between the reconstructed signals and the true values. In order to more clearly express this phenomenon, we added more complicated noises, including colored noise \(e(t)\), Gaussian white noise \(\partial(t)\), and random noise \(r(t)\), to the original pure data to construct a new set of simulated signals, as shown in Equation (8). The corresponding simulated signals are shown in Figure 8. The signals after SOBI separation and their corresponding frequencies are depicted in Figure 9. The transformation matrix \(\mathbf{A}\)  
Figure 6: The results of direct reconstruction of the three simulation signals. Figure 7: Comparison of the three sets of reconstructed signals under traditional SOBI with the true values. was estimated as indicated in Equation (9). Figure 10 presents the reconstructed signals for the three sets. \[\left\{ \begin{array}{l}g_{4}(t)=\sin(2\pi\cdot 5t)+0.5\times r(t)+r(t:t+200)\\ g_{5}(t)=\sin(2\pi\cdot 5t)+0.05\times\vartheta(t)\\ g_{6}(t)=\sin(2\pi\cdot 5t)+0.03\times e(t)+r(t:t+200)\\ \end{array} \right. \tag{8}\]  
Figure 8: Three sets of simulated signals (\(g_{4}(t)\), \(g_{5}(t)\), and \(g_{6}(t)\)). Figure 10: Comparison of the three sets of reconstructed signals under traditional SOBI with the true values. Figure 9: (**a**) Separated-signals diagrams acquired from the SOBI method and (**b**) corresponding frequency diagrams. \[\mathbf{A}=\left[ \begin{array}{cccc}0.7208&0.3820&0.1617\\ 0.7074&-0.0079&-0.0529\\ 0.7171&-0.2023&0.1964\end{array} \right] \tag{9}\]  
As can be seen from Figure 9 and matrix \(\mathbf{A}\) in Equation (9), separated signal I is strongly correlated with the three observed signals. However, the values in the last two columns of matrix \(\mathbf{A}\) are significantly larger, even reaching 0.3820, which is close to half the contribution value of separated signal I. This indicates that separated signals II and III contain useful information. As shown in Figure 10, if the traditional SOBI algorithm is still used for de-noising, it will lead to a worse noise reduction effect and even introduce extra noise, as shown through the red line. Therefore, the gradual de-noising method proposed is used for processing. First, the I-SOBI proposed in this study is used for the first stage de-noising. Taking the noise reduction of observed signal \(g_{5}(t)\) as an example, it can be found through matrix \(\mathbf{A}\) that separated signal III is more strongly correlated than separated signal II. Therefore, \(-0.0079\) in matrix \(\mathbf{A}\) is set to zero. Additionally, according to the frequency spectrum diagram in Figure 9b, the main frequency of separated signal I is 5 Hz. Thus, the information associated with separated signal III below 5.2 Hz is retained (a buffer bandwidth is added to the main frequency to retain the frequency component of the main frequency and to prevent spectral leakage) and replaces the original separated signal III. And then the reconstruction, that is, the de-noised data, of observed signal \(g_{5}(t)\) can be obtained, as shown in Figure 11a.