## 2 Theory
### Numerical Simulation  
Numerical simulations were performed for the case where noise sources with similar frequency bands overlap, or are located nearby, showing a high correlation with each other. The assumption was made that there are three sources, two of which have similar frequency bands and physical locations, resulting in a high correlation. Thus, it was assumed that the two sources are mixed with each other during measurement and are indistinguishable (Figure 3). A simulation was performed based on these numerical assumptions, as shown in the  
Figure 4 shows the results of the frequency analysis performed for each measured signal \(x\).  
The frequencies of \(s_{1}\) and \(s_{2}\) are the same at 9 Hz, and \(s_{1}\) and \(s_{2}\) are mixed and measured through matrix \(A\). In other words, because these two signals are physically overlapped or are located nearby, the measured signals \(x_{1}\) and \(x_{2}\) have similar characteristics to those of the frequency band. For this numerical model, OCF was performed for the measured signals \(x_{1}\), \(x_{2}\), and \(x_{3}\) and the output \(y\) (Figure 5).  
The measured \(x_{1}\) and \(x_{2}\) signals, which are highly correlated with each other, show the same contribution to the output (Figure 5a,b). Signal \(x_{3}\) contributes to the output at 14 and 17 Hz (Figure 5c). Thus, PCF was performed to identify noise sources at 9, 11, and 23 Hz, and the results are shown in Figure 6.  
Figure 4: Frequency response of \(x_{1}\), \(x_{2}\), and \(x_{3}\).  
Figure 3: Numerical simulation model.  
The results of the PCF showed that the measured signal \(x_{2}\) had no contribution to the output (Figure 6b). This is because the ordering of \(x_{1}\) is high in the process of calculating the mutual contribution of \(x_{1}\) and \(x_{2}\). Thus, it is difficult to accurately evaluate the contribution of coherence functions because the signals of the noise source are mixed, showing similar contributions, and it is challenging to show the contribution of the actual noise source to the output because the PCF calculates the mutual contribution based on this situation.  
Thus, to solve this problem, it is necessary to restore the signal using ICA and evaluate the contribution through OCF based on the obtained signals. Because the signals are independent of each other when the signals are restored using ICA, there is no need to remove the correlation. Thus, it is possible to evaluate the contribution through OCF alone without using PCF.  
First, a frequency analysis was performed by restoring the noise measured for three input sources using ICA (Figure 7).  
Figure 7: Frequency response of \(s_{1}\), \(s_{2}\), and \(s_{3}\).  
It can be seen that the restored signal showed its major frequency characteristics to be nearly identical to those of the original signal. The contribution of the restored signal to the output signal was evaluated using OCF (Figure 8).  
The evaluation results were found to be consistent with those set in the simulation. \(s_{1}\) and \(s_{2}\) showed the contribution to the output at 9 Hz. Furthermore, the results showed that \(s_{1}\) made a contribution to the output at 23 Hz, \(s_{2}\) contributed at 11 Hz, and \(s_{3}\) made a contribution at 14 and 17 Hz. The results showed the same results as the previous set in the simulation. Thus, it would be useful to utilize the contribution evaluation using ICA under the condition where signals that are physically overlapped or located nearby with overlapping frequency bands show a high correlation with each other.