## 4 Results  
### Defect Separation Performance with Simulated Test Signals  
The defect separation performance in Case 1 is first analyzed with the simulated mixture signals. To validate the superiority of the TFSNN, in this section it was compared with the representative methods of ICA [7], a fully-connected shallow neural network (SNN) [14], and TFBSS [22]. The separated signals and original source signals are shown in Figure 9. To visualize them on the same scale, the separated signals were Z-score standardized to ensure they have the same mean and variance as the corresponding sources. From the visual results, it can be intuitively seen that the proposed method significantly outperformed the other three algorithms in separating the crack signal, the ramp, and the square wave signals. Particularly, the recovered details of the ramp and defect signals were difficult to distinguish in ICA and SNN. Due to a lack of TF exploitation, both ICA and SNN seemed to suffer more seriously from the interference non-stationary crack signal. Next, we evaluate the separation effects of the above algorithms horizontally using two evaluation criteria, a scale-invariant signal-to-noise ratio and a similarity coefficient. Figure 8: Typical two-channel observations for the noisy mixture under actual railway noise: (**a**) Channel 1 and (**b**) Channel 2. Figure 7: STFDs for (**a1,a2**) a typical rail crack AE signal from the tensile dataset; (**b1,b2**) an exemplified section of stationary railway background noise; and (**c1,c2**) an exemplified section of non-stationary railway abnormal noise. Note that subfigures in the first rows and second rows were obtained from SPWVD and CWD, respectively. Considering the scale ambiguity for BSS, the classical signal-to-noise ratio (SNR), referring to the ratio of power between the target source signal and the residual error, is recently proved inaccurate or misleading in BSS tasks [32]. An improved version of SNR, scale-invariant signal-to-noise ratio (SISNR), measures the level of distortion or noise interference in a processed signal by comparing it to a reference signal in a way that is invariant to the scaling of the signals [33]. This metric is useful for evaluating speech enhancement and source separation by eliminating the scale ambiguity and ensuring that the residual errors are indeed orthogonal or uncorrelated to the target. Therefore, SISNR is selected as the first evaluation metric in this paper. SISNR is calculated according to the following formula, where \(\mathbf{s}\) is the reference signal and \(\mathbf{\hat{s}}\) is the separated or processed signal:  
\[SISNR_{i}=10\text{lg}\frac{\left\|\mathbf{a}\mathbf{s}_{i}^{2}(t)\right\|^{2} }{\left\|\mathbf{a}\mathbf{s}_{i}^{2}(t)-\mathbf{s}_{i}^{2}(t)\right\|} \tag{11}\]  
where \(\mathbf{\alpha}=\mathbf{\hat{s}}^{\mathcal{\Lambda}}\mathbf{s}/\left\| \mathbf{\hat{s}}\right\|^{2}\). A higher SISNR value indicates a lower level of the noise signal, implying a better separation effect of the blind signal separation algorithm. The SISNRs for different separated signals in Figure 8 are listed in Table 1.  
On the one hand, the SISNRs clearly indicated that both of the TFA-based methods could suppress the residual noise to a much lower level than the pure time-domain methods, hence providing an over 10 dB increment in averaged SISNR. In particular, the proposed method further exhibited a 2.87 dB higher SISNR than the classical TFBSS, which verified \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Crack** & **Square Pulse** & **Ramp** \\ \hline ICA & 10.6537 & 12.5581 & 20.1588 \\ SNN & 15.3378 & 13.4418 & 20.7815 \\ TFBSS & 24.0757 & 29.3030 & 25.0376 \\ TFSNN & 26.5158 & 32.2905 & 28.2357 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: SISNRs (dB) of blind separated signals using different methods. Figure 9: Performance comparison with the simulated mixtures between separated sources using (a) ICA, (b) SNN, (c) TFBSS, and (d) the proposed method. the necessity and efficiency of applying network optimization.