## 3 Proposed Method
### Separation of Non-Sparse Components  
Based on the statistical results in Section 3.2, we have validated the half-K assumption that the active source number in a certain TF bin does not exceed half the total number of simultaneously occurring sources. It means that for \(K\) sources, there are no more than \(K/2\) sources active at a certain TF bin. We set \(K\leq 6\) to ensure the localization accuracy in this work. Based on the proposed assumption, we can conclude that the number of active sources at a certain TF bin (\(K_{a}\)) is no more than three. It should be noted that \(K_{a}=1\) means that there is only one source active at a certain TF bin, i.e., the sparse bin. The set of these sparse bins are the above-mentioned sparse components. Aiming to separate the corresponding non-sparse components of each source, we have to know all the active sources of the non-sparse components. Here, the problem is solved by first dividing the TF band into several zones with same width, and then calculating the similarity between the separated sparse components and the mixture signal in this TF zone. In detail, if the similarity exceeds a certain threshold, the frequency components of the regions are mostly derived from this source signal and the signal set that satisfies the threshold is named the active source in the current TF zone of non-sparse components. The normalized cross-correlation function is utilized for similarity calculation, and the cross-correlation coefficient between the mixture signal \(\mathbf{S}_{W}\) and a sparse component signal \(\mathbf{S}_{i}^{C}\) in a TF zone \(\mathcal{Z}\) is calculated as follows:  
\[R_{\mathbf{S}_{W},\mathbf{S}_{i}^{C}}(\mathcal{Z})=\frac{\sum\limits_{(n,l) \in\mathcal{Z}}|S_{W}(n,l)|\cdot|S_{i}^{C}(n,l)|}{\sqrt{\sum\limits_{(n,l)\in \mathcal{Z}}|S_{W}(n,l)|^{2}\cdot\sum\limits_{(n,l)\in\mathcal{Z}}|S_{i}^{C}( n,l)|^{2}}} \tag{13}\]  
where \(\mathbf{S}_{i}^{C}\) represents the separated sparse component signal and \(i=1,2,...,K\). To obtain all the active sources of non-sparse components in a TF-analyzed one \(\mathcal{Z}\), we define a active detecting vector \(\mathbf{\mathfrak{D}}=[D_{1},D_{2},...,D_{K}]\); the \(i\)th element \(D_{i}\) can be obtained by:  
\[D_{i}=\left\{ \begin{array}{l}1,\text{if }R_{\mathbf{S}_{W},\mathbf{S}_{C_{i} }}(\mathcal{Z})>\epsilon\\ 0,else\end{array} \right. \tag{14}\]  
where \(\epsilon\) is an experimental threshold, in order to ensure that the number detected active sound source is not larger than the real one. We set \(\epsilon=0.8\) in this paper (informal testing found this value generally led to satisfactory results but future work can explore the optimization of this value). The index value of the detected active source is recorded in a vector \(\mathbf{\mathcal{I}}\), which can be obtained by a \(INDEX^{0}\) function as:  
Figure 4: Average probability of active source number (PASN) among six simultaneously occurring sources. \[\mathbf{\mathcal{I}}=INDEX^{0}(\mathbf{\mathfrak{D}}) \tag{15}\]  
where \(INDEX^{0}(\cdot)\) is a non-zero index searching function; the output parameter is a vector that contains all the indexes of non-zero elements of the input vector. \(||\mathbf{\mathfrak{D}}||_{0}=3\), i.e., \(\mathbf{\mathcal{I}}=[\mathcal{I}_{1},\mathcal{I}_{2},\mathcal{I}_{3}]\), where \(||\cdot||_{0}\) counts the number of non-zero components in its argument. This means there are three active sources at the current zone. Hence, the corresponding TF coefficient of these active sources is rewritten to find a solution to the linear equations:  
\[\left[ \begin{array}{ccc}\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}&\frac{\sqrt{2 }}{2}\\ \cos\mu_{\mathcal{I}_{1}}&\cos\mu_{\mathcal{I}_{2}}&\cos\mu_{\mathcal{I}_{3}} \\ \sin\mu_{\mathcal{I}_{1}}&\sin\mu_{\mathcal{I}_{2}}&\sin\mu_{\mathcal{I}_{3}} \end{array} \right]\left[ \begin{array}{c}S_{\mathcal{I}_{1}}(n,l)\\ S_{\mathcal{I}_{2}}(n,l)\\ S_{\mathcal{I}_{3}}(n,l)\end{array} \right]=\left[ \begin{array}{c}S_{W}(n,l) \\ S_{X}(n,l)\\ S_{Y}(n,l)\end{array}