## 3 Proposed Method
### Separation of Sparse Components  
Under the sparse assumption discussed in [20], one source will have the same mixing parameter pairs \(A\) and \(\mu\). For a given TF instant of \(i_{th}\) source, this pair is approximated by:  
\[[A_{i}(n,l),\mu(n,l)]=\left[\left|\frac{I_{Y}(n,l)}{I_{X}(n,l)}\right|,\cdot \frac{I_{Y}(n,l)}{I_{X}(n,l)}\right] \tag{7}\]  
The separated sources can be derived by grouping the TF instants using these parameter pairs. If the estimated DOA of \(i_{th}\) source is \(\mu_{i}\), the task is to determine a range around \(\mu_{i}\) (i.e., \([\mu_{i}-\Delta\mu,\mu_{i}+\Delta\mu]\)) such that the TF instants having the DOA estimates within this range are considered as the source \(i\). It should be noted that if the threshold is set small enough, less interference from other sources may be contained in the separation. However, this may fail to derive many TF components whose DOA estimates are slightly different to the true source DOA due to the low fault tolerance of the estimation approach. If the threshold is larger, this may lead to the inclusion in the  
Figure 3: System block diagram of the proposed method. DOA: direction of arrival; TF: time-frequency. separated source of TF components from other sources. Hence, an efficient clustering method is needed to dynamically achieve the separation of sparse components. Based on the directional characteristic of B-format microphone, it can be seen that a strong correlation between the recording signals of adjacent channels in a certain TF zone implies that there is only one source active, while a weaker correlation means it is a region where multiple sources exist [21]. Hence, in this paper, we proposed a dynamic threshold clustering method of sparse components based on the inter-correlation of the raw recording (A-format) signals [21], and the threshold \(\Delta\mu\) is dynamically set by:  
\[\Delta\mu=\frac{1}{1+e^{-\alpha(\overline{\gamma}-\beta)}}\cdot\mu_{0} \tag{8}\]  
where \(\mu_{0}\), \(\alpha\), and \(\beta\) are initial thresholds for the user to define; \(\mu_{0}\) is the threshold to control the dynamic range of the \(\Delta\mu\), \(\alpha\) is a threshold for controlling the \(\Delta\mu\) change curve, and \(\beta\) is a symmetric point corresponding to the change curve. \(\gamma\) is the average of normalized cross-correlation coefficients [21] among four recording signals of the B-format microphone. More specifically, for any pair of soundfield microphone-recorded signals (\(S_{chi}(n,l)\) and \(S_{c}hj(n,l)\)), the function is defined as:  
\[R_{i,j}(K)=\sum_{(n,l)\in\mathcal{K}}|S_{chi}(n,l)\cdot S_{chi}(n,l)| \tag{9}\]  
where \(i\
eq j\), \(S_{chi}(n,l),S_{chi}(n,l)\in\{S_{chi}(n,l),S_{chi}(n,l),S_{chi}(n,l),S_{chi}(n,l),\}\). The normalized cross-correlation coefficient can be obtained by:  
\[\gamma_{i,j}(\mathcal{K})=\frac{R_{i,j}(\mathcal{K})}{\sqrt{R_{i,j}(\mathcal{ K})\cdot R_{j,j}(\mathcal{K})}} \tag{10}\]  
It can be seen from Equation (8) that the value of \(\Delta\mu\) is proportional to the correlation coefficient. Specifically, if the correlation coefficient is close to one, the threshold will be large to obtain more extractions of the corresponding source, while in other TF zones, it will get smaller with the average correlation coefficient being smaller in order to get rid of the interference of other sources. Considering this issue and the value range of the cross-correlation coefficient, Equation (8) should be a function whose independent and dependent variables are both with a value range in [0, 1]. By adjusting the value of \(\alpha\) and \(\beta\), we find \(\alpha=10\) and \(\beta=0.5\) can perfectly meet the requirement mentioned above. Future work will investigate alternative methods for optimizing the choice of these values and find whether there might be a more efficient function that can describe the relation between \(\overline{\gamma}\) and \(\Delta\mu\).