## 3 Experimental Results and Analysis
### Simulation Experiment  
The simulation experiments were designed to provide a controlled environment, allowing us to systematically investigate the behavior of the proposed gradual de-noising method under certain parameters and conditions. As previously stated, the closer the diagonal elements' values are to \(1\) and the closer the off-diagonal elements' values are to \(0\), the better the independence between the signal sources. If the diagonal elements' values deviate from \(1\) and the off-diagonal elements have larger values, this may indicate that there is a certain correlation between the original signals. For the dynamic-deflection time-series data collected from three adjacent points on the bridge, their dynamic trends are similar, showing strong correlation. Considering that our goal is to extract the similar variation trends of the three sets of time-series data and reduce the interference of random noises in the monitoring data of each point, this study constructed the following simulation signals. Signal \(g_{1}(t)\) is the pure value without any noise, treated as true value. Signals \(g_{2}(t)\) and \(g_{3}(t)\) are simulated observed signals that are superimposed with random noise \(r(t)\), respectively. The construction of the simulated signals is expressed as shown in Equation (6). These three sets of simulation signals have a sampling frequency of 200 Hz, a sampling time of 5 s, and 1000 sample points per set. \[\left\{ \begin{array}{l}g_{1}(t)=\sin(2\pi\cdot 5t)\\ g_{2}(t)=\sin(2\pi\cdot 5t)+0.1\times r(t)\\ g_{3}(t)=\sin(2\pi\cdot 5t)+0.1\times r(t)\end{array} \right. \tag{6}\]  
The corresponding three sets of simulation signals are shown in Figure 2. The three sets of signals were then subjected to SOBI processing to obtain three sets of separated signals. The separated-signal diagrams and their corresponding frequency diagrams are shown in Figure 3. And the transformation matrix \(\mathbf{A}\) was estimated as shown in Equation (7). It should be noted here that it is normal for the elements in matrix \(\mathbf{A}\) to have negative values. The presence of negative values can be attributed to various factors, including the phase discrepancies among the signal sources, the second-order statistical characteristics of the signals, the effects of noise present in the system, etc. [54]. As can be seen from Figure 3a, the trend evident in separated signal I is consistent with the waveform of the original noise-free signal \(g_{1}(t)\), while the waveforms of separated signals II and III are chaotic. Additionally, examining the frequency domain distribution of the three separated signals reveals that separated signal I exhibits a primary frequency of 5 Hz, which matches the principal frequency of the original noise-free signal \(g_{1}(t)\). In contrast, the primary frequency information for separated signals II and III is indistinct, further indicating that separated signal I contains primary useful information. To verify the effectiveness of the method selected in this paper, we performed EWT decomposition on the simulation signals \(g_{1}(t)\) and \(g_{2}(t)\); the separated-signal diagrams and their corresponding frequency diagrams are shown in Figure 4.  
After performing EWT and SOBI decomposition, we directly reconstructed the signals without any further processing of the decomposition results. The EWT reconstruction results were compared with the noise-free pure signal \(g_{1}(t)\), as shown in Figure 5. The SOBI reconstruction results were compared with the noise-free pure signal \(g_{1}(t)\), as shown in Figure 6. The EWT method exhibits significant limitations in signal recovery, with the reconstructed signal losing a substantial amount of useful information, and the amplitude of the valid signal decreasing noticeably. In contrast, the SOBI method, even without the application of noise control techniques, is able to reconstruct the separated signal with a  
Figure 2: Three sets of simulated signals (\(g_{1}(t)\), \(g_{2}(t)\), and \(g_{3}(t)\)). \[\mathbf{A}=\left[