## 3 UBSS Algorithm Based on TF Soft Mask
### Source Signal Recovery by TF Soft Mask  
Ideally, the reconstruction of the source signal is achieved by inverse short-time Fourier transform (ISTFT) in the frequency band \([f_{i}-\Delta,f_{i}+\Delta]\) 10. However, the traditional clustering method is only a kind of hard segmentation of power points to obtain the TF binary mask. When the frequency spectrum is overlapped, the short-term spectral power  
Figure 4: The angular probability density distribution. Figure 3: The statistical histogram of angular distribution. of the observed signals is the superposition of the short-term spectral power of the source signals. The integrity of the source signal reconstruction cannot be guaranteed through the method of hard segmentation. Thus a matching soft mask matrix \(M(t,f)\) is required to fully recover the source signal. \[Y_{i}(t,f)=X(t,f)\odot M_{i}(t,f) \tag{10}\]  
where \(\odot\) represents the Hadamard product; \(Y_{i}(t,f)\) denotes the \(i\)th signal to be reconstructed, and \(M_{i}(t,f)\) is the corresponding TF soft mask matrix that needs to be constructed. In this paper, the construction of the TF soft mask matrix is performed by designing a contribution degree function based on the cosine distance. It can be seen from Figure 2 that among all the power points inside the \(i\)th linear cluster, the endpoint is subject to the least interference. That is, when \(f_{u}=f_{i}\), the contribution degree \(\eta_{u}^{i}\) of the \(i\)th source signal to the mixed signal can be taken as 1, and the cosine distance between the endpoint and itself is 1 as well (\(f_{i}\) is the frequency corresponding to the endpoint of the \(i\)th linear clustering cluster). When \(f_{u}\) is further away from \(f_{i}\), the contribution degree \(\eta_{u}^{i}\) will be decreased. The corresponding cosine distance \(d_{u}^{i}\) between the power point and the \(i\)th endpoint is decreased as well. Extremely, \(\eta_{u}^{i}\) is taken as 0 when \(f_{u}\) is the frequency of the nearest adjacent endpoint. Hence the contribution degree \(\eta_{u}^{i}\) can be characterized by the cosine distance between all power points and the endpoint of the \(i\)th linear cluster. A monotonically increasing contribution function is designed to characterize the positive correlation between contribution degree \(\eta_{u}^{i}\) and cosine distance \(d_{u}^{i}\):  
\[\eta_{u}^{i}= \begin{cases}1,&d_{u}^{i}=1\\ \left(\frac{d_{u}^{i}-d_{u}^{i}}{1-d_{u}^{i}}\right)^{\gamma},&0<d_{u}^{i}<d_ {u}^{i}\\ 0,&otherwise\end{cases}  \tag{11}\]  
where \(d_{i}^{i}\) denotes the maximum of the cosine distance between the endpoint of the \(i\)th linear cluster and the endpoints of the remaining linear clusters; \(\gamma\) denotes the scaling factor of the contribution function, whose different values will affect the separation performance of the algorithm. In Section 4.1, we will discuss the impact of \(f\) on separation performance. The cosine distance of the \(u\)th power point from the endpoint of the \(i\)th linear cluster \(d_{u}^{i}\) is obtained as  
\[d_{u}^{i}=CD\big{\{}\big{[}P_{\alpha}(f_{i}),\ \ P_{\beta}(f_{i})\big{]},\ \ \ \big{[}P_{\alpha}(f_{u}),\ \ P_{\beta}(f_{u})\big{]}\big{\}} \tag{12}\]  
where \(CD\) denotes the cosine distance between two points. The frequencies corresponding to the endpoints of the clustering clusters can be detected by seeking the first \(\hat{n}\) peak points in the sum of the power data marked in red Figure 5. The points marked in red are the endpoints of the detected linear clusters to be determined in Figure 6. Therefore, the contribution degree of the \(u\)th point to the \(i\)th linear cluster can be calculated by the contribution degree function given above. Based on the contribution degree \(\eta_{u}^{i}\) obtained from Equation (11), the \(i\)th TF soft mask matrix \(\mathbf{M}_{i}\) is constructed by the following formula:  
\[\mathbf{M}_{i}=