## 3 UBSS Algorithm Based on TF Soft Mask
### Source Number Estimation  
In the problem of UBSS, the source number estimation is an indispensable step. The accurate estimation of the number of sources is the prerequisite for precise source signal recovery [20; 21]. In order to estimate the number of sources accurately, the KDE algorithm is used to calculate the angular probability density distribution of the mixed signal in the sparse domain, followed by detecting the number of peak points of the distribution. In order to eliminate the interference of noise on the signal, a threshold \(\varepsilon\) needs to be set to screen out the low power points. The set of frequencies \(F\) after removing the low power points can be obtained by  
\[F=\left\{f\bigg{|}\sum_{k=1}^{m}P_{k}(f)>\varepsilon\underset{f}{\text{max}}[ \sum_{k=1}^{m}P_{k}(f)]\right\} \tag{7}\]  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Variable** & \multicolumn{1}{c}{**Meaning**} \\ \hline \(F\) & The set of frequencies after removing the low power points. \\ \hline \(\theta\) & The angular distribution. \\ \hline \(G_{\text{rx}}(\theta)\) & The angular probability density distribution of \(\theta\). \\ \hline \(Y_{i}(t,f)\) & The STFT of the \(i\)th signal to be reconstructed. \\ \hline \(M_{i}(t,f)\) & The corresponding \(i\)th TF soft mask matrix that needs to be constructed. \\ \hline \(\eta_{u}^{i}\) & The contribution degree of the \(i\)th source signal to the mixed signal. \\ \hline \(d_{u}^{i}\) & The corresponding cosine distance between the power point and the \(i\)th endpoint. \\ \hline \(y_{i}(t)\) & The \(i\)th source signal to be reconstructed. \\ \hline \hline \end{tabular}
\end{table}
Table 2: Variable definitions in this section.  
Figure 2: The power scatters of mixed signals with TF aliasing.  
where \(\varepsilon\) is the empirical parameter, which is taken as 0.1 in this paper. In the later expressions, \(f_{u}\) is used to denote the \(u\)th element of \(F\). Afterward, convert the filtered power points into angular distribution \(\theta\):  
\[\theta_{u}=\arctan\bigl{[}P_{\kappa}(f_{u})/P_{\beta}(f_{u})\bigr{]} \tag{8}\]  
The statistical histogram after converting to an angular distribution is shown in Figure 3. As seen from the figure, the angular distribution is more densely distributed around the intervals 0.07, 0.52, 1.15, and 1.48 than the other intervals. Thus the angular distribution \(\theta\) is modeled based on the KDE algorithm [22]:  
\[G_{\kappa}(\theta)=\frac{1}{l}\sum_{u=1}^{l}K_{\kappa}(\theta-\theta_{u}) \tag{9}\]  
where \(G_{\kappa}(\theta)\) denotes the angular probability density distribution of \(\theta\); \(l\) is the length of the vector \(\theta\); \(K_{\kappa}(\cdot)\) represents the kernel function with a smoothing factor \(\kappa\) of the bandwidth. In this paper, we choose the Gaussian kernel as the kernel function and take \(\kappa\) as 0.25. The angular probability density distribution obtained by the KDE algorithm is shown in Figure 4. The number of peak points of the angular probability density distribution is the number of sources to be estimated. As we can see from Figure 4, there are four peak points marked by red circles, which means that the number of estimated source signals \(\hat{n}\) is four.