## 4 Results
### Systematic Assessment of Comparative Methods with Test Sets and Stability Discussion  
On the basis of systematic assessment with the whole test sets in Case 2, the SISNRs for a group of 100 randomly selected test signals are first displayed in Figure 12a,b. The similarity coefficients for the same test group are displayed in Figure 12c,d. In addition, the averaged values and standard deviations of the group SISNRs and group similarity are listed in Table 3.  
Figure 11: Zoomed-in sections of the separated signals in Figure 10 using (**a**) ICA, (**b**) SNN, (**c**) TFBSS, and (**d**) the proposed method. Figure 10: Separated signal comparison with the actual railway noise between separated sources using (**a**) ICA, (**b**) SNN, (**c**) TFBSS, and (**d**) the proposed method. As stated earlier, the similarity investigates the degree of resemblance between the separated signal and the source signal, reflecting how well the separation retains the details of the source signal. The SISNR assesses the extent of noise suppression during separation based on Equation (11). From Figure 12c, in the separation of crack signals, most of the similarity coefficients can reach above 0.95. The stability of the separation exhibited varying degrees in different methods, and this point is extremely obvious in several cases when the similarity coefficient abruptly decreased. In contrast, the separation of noise signals produced far more chaotic results. The variance of ICA was the highest, indicating the lowest averaged metrics and a minimal separation stability, while the averaged similarity level and variance of SNN and TFBSS were quite close, and both were inferior to the proposed method. It demonstrated that the proposed method could efficiently recover more time-frequency details of the sources in most separation processes. The overall and averaged SISNRs for the crack and noise signals are shown in Figure 12a,b. In terms of SISNR, the four algorithms show significant differences in noise suppression effects. All the compared methods can provide roughly satisfactory separation results, with 15 dB for defect signals. While for railway noise signal separation, the ICA and SNN algorithms were significantly weaker (about 5 dB lower) than the TFA-based algorithms, and the separated signals were still interfered with by the residual abnormal noises. Moreover, the dynamic robustness of the compared methods against different Gaussian noise levels was evaluated, as shown in Figure 13. As the noise amplitude decreased from \(-10\) dB to \(-60\) dB, the compared methods gradually converged and reached their maximal SISNRs, where TFBNN obtained an approximately 20 dB higher SISNR than the  
Figure 12: SISNRs for the separated sources in a test group of 100 noisy signals: (**a**) separated rail crack signal and (**b**) separated railway noise signal. Similarity coefficients for the separated sources in a test group of 100 noisy signals: (**c**) separated rail crack signal and (**d**) separated railway noise signal. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**Averaged SISNR**} & \multicolumn{2}{c}{**Averaged Similarity**} \\ \cline{2-5}  & **Defect** & **Railway Noise** & **Defect** & **Railway Noise** \\ \hline ICA & 18.0983 \(\pm\) 2.8019 & 18.2298 \(\pm\) 5.0321 & 0.9657 \(\pm\) 0.048 & 0.9509 \(\pm\) 0.0678 \\ SNN & 17.8321 \(\pm\) 2.4989 & 19.5157 \(\pm\) 4.302 & 0.9663 \(\pm\) 0.0419 & 0.9673 \(\pm\) 0.0513 \\ TFBSS & 18.9886 \(\pm\) 1.8957 & 23.7881 \(\pm\) 1.0311 & 0.9784 \(\pm\) 0.008 & 0.9832 \(\pm\) 0.0017 \\ TFSNN & 19.1102 \(\pm\) 1.8263 & 23.9424 \(\pm\) 0.9326 & 0.9796 \(\pm\) 0.007 & 0.9935 \(\pm\) 0.0014 \\ \hline \end{tabular}
\end{table} 
Table 3: Averaged SISNR and similarity for the separated sources in a test group of 100 signals. time domain methods and 5 dB higher than TFBSS. This indicated the separation accuracy or convergence limits of the time-frequency methods were much higher than the time domain methods.