## 4 Simulation and Experimentation  
### Separation of Independent Source Signals  
This simulation experiment is intended to demonstrate the usability of the proposed algorithm for independent source signals. Three independent source signals are generated in Python: s1, s2, and s3. s1 is a sinusoidal signal \(s1=sin(20~{}\pi t)\) with a frequency of 10 Hz; s2 is a sinusoidal signal \(s2=sin(100~{}\pi t)\) with a frequency of 50 Hz; s3 is a random signal \(s3=rand(1,L)-0.5\); L is the length of the signal L = 1000. The number of sample points Ne is also 1000. Their waveforms and spectra are shown in Figure 4.  
The linear mixing matrix A = {0.3763 2.0243 \(-1.3216\); \(-0.2270\)\(-2.3595\)\(-0.6361\); \(-1.1489\)\(-0.5100\)\(0.3179\)} is a third-order square matrix, and the source signal is linearly mixed to obtain the observed signals x1, x2, x3, whose waveforms and spectrograms are shown in Figure 5.  
The waveforms and spectra of the separated signals are shown in Figure 6 after separating the observed signals with the BAS-BCA algorithm.  
Figure 4: The source signals. (**a**) Waveforms. (**b**) Spectra.  
Figure 5: The observed signals. (**a**) Waveforms. (**b**) Spectra.  
And obtain the separation matrix \(W=\{-0.7862\)\(0.6179\)\(-0.0033\); \(-0.6177\)\(-0.7858\)\(0.02762\); \(-0.0144\)\(-0.0237\)\(-0.9996\). In order to see the separation effect more objectively and accurately, the correlation coefficient \(\rho\) between the separated signal and the source signal is chosen as the evaluation criterion in this paper [31], as shown in Equation (21), and the results are shown in Table 3 as follows:  
\[\rho\ =\ \frac{cov(s,y)}{\sqrt{cov(s,s)cov(y,y)}} \tag{21}\]  
where \(cov(\cdot)\) denotes the variance, \(s\) denotes the source signal, and \(y\) denotes the separated signal. From the knowledge of probability theory, it is known that the similarity coefficient has the following properties:  
1. \(|\rho|\leq 1\), represents the similarity of s and \(y\), and the closer to 1, the more similar,
2. \(|\rho|=1\), which means that \(s\) and \(y\) are completely similar,
3. \(|\rho|=0\), which means \(s\) and \(y\) are not similar at all.  
From Table 3, it can be seen that the correlation coefficients of s1 and y1, s2 and y2, and s3 and y3 are all close to 1, which corresponds to Figures 4 and 6. From Figures 4 and 6, it can be seen that three independent source signals are successfully separated based on the BAS-BCA algorithm.  
As a comparison, the signal interference ratio (SIR) [32] is also selected to evaluate the separation results by using FastICA, SCA, and the BCA algorithm from the literature [31]. The equation of SIR is shown in Equation (22) as follows:  
\[SIR=-10 lg\frac{\left\|y_{i}-s_{i}\right\|^{2}}{\left\|s_{i}\right\|^{2}} \tag{22}\]  
where \(y_{i}\) is the i-th signal in the separated signal \(y\) and \(s_{i}\) is the i-th signal in the source signal \(s\). Signal processing-related knowledge has shown that the separation is improved  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **y1** & **y2** & **y3** \\ \hline
**S1** & 0.97990255 & 0.01395991 & 0.00630897 \\
**S2** & 0.00404252 & 0.96984297 & 0.00778653 \\
**S3** & 0.00741474 & 0.00730825 & 0.94597251 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Correlation coefficient.  
Figure 6: The separated signals. (**a**) Waveforms. (**b**) Spectra.  
by increasing the SIR value. We made 10 sets of experiments with different e signal-to-noise ratios (SNRs), and the results are shown in Figure 7.  
As can be seen from Figure 7, all algorithms' separation accuracy increases with increasing SNR. The average NSIR of the BAS-BCA algorithm, the BCA algorithm in [4], SCA, and the FastICA algorithm are 21.95, 20.89, 19.96, and 21.78, respectively. The BAS-BCA algorithm has the best separation accuracy, followed by the FastICA algorithm. Therefore, the superiority of the proposed algorithm in independent signal separation is proved.