## 2 System Model
### MS-GLMB Filtering  
Multi-sensor (MS) RFS filtering-based tracking provides superior performance than single sensor tracking when multiple acoustic sources are in motion [6; 31; 32]. MS-GLMB filtering is applied in this work, whereby the propagation of the filtering density is achieved through a recursive procedure comprising two steps, namely the time update and the data update. In this work, the time-updated and data-updated filtering densities can be expressed in closed-form via the GLMB density [6; 31; 32]:  
\[\pi(\mathbf{X}_{k})=\Delta(\mathbf{X}_{k})\sum\
olimits_{\theta_{1:k}\in\theta_{1:k}} \omega^{(\theta_{1:k})}(\mathcal{L}(\mathbf{X}_{k}))\prod\
olimits_{\mathbf{x}_{k} \in\mathbf{X}_{k}}p^{(\theta_{1:k})}(\mathbf{x}_{k}) \tag{2}\]  
In (2), \(\theta_{1:k}\in\theta_{1:k}\) represents the history of multi-array association mappings up to frame '\(k\)', i.e., \(\theta_{1:k}\hat{=}(\theta_{1},\ \theta_{2}\ldots\theta_{k})\). The term \(\mathcal{L}(\mathbf{X}_{k})\) can be further expressed as follows:  
\[\mathcal{L}(\mathbf{X}_{k})\hat{=}\{l:(x_{k},l)\in\mathbf{X}_{k}\} \tag{3}\]  
Furthermore, in (2), the term \(\Delta(. )\) denotes a distinct label indicator where \(\Delta(\mathbf{X}_{k})=1\) if and only if the cardinality of the label set denoted by \(|\mathcal{L}(\mathbf{X}_{k})|\) equals the cardinality of \(\mathbf{X}_{k}\) denoted by \(|\mathbf{X}_{k}|\). Each weight term \(\omega^{(\theta_{1:k})}(\mathcal{L}(\mathbf{X}_{k}))\) in (2) is a non-negative weight, such that their sum equals 1:  
\[\sum\
olimits_{L\subseteq\Pi_{0:k}}\sum\
olimits_{\theta_{1:k}\in\theta_{1:k }}\omega^{(\theta_{1:k})}(L)=1 \tag{4}\]  
Each of the non-negative weights in (3) and (4), represented by \(\omega^{(\theta_{1:k})}(\mathcal{L}(\mathbf{X}_{k}))\), can be interpreted as a probability of the sources with label set \(\mathcal{L}(\mathbf{X}_{k})\) being active, while at the same time being associated with the detections indicated by the history of multi-array association mappings up to the frame '\(k\)', denoted by \(\theta_{1:k}\). Finally, the term \(p^{(\theta_{1:k})}(\mathbf{x}_{k})\) in (2) represents the probability density of the source (with label '\(l\)') that is located at the state \(x_{k}=\left(\alpha_{k}^{pos}\,,\alpha_{k}^{vel}\right)\). A detailed explanation of the terms in (2) can be found in [6; 31; 32]. The estimated position and label corresponding to the sources at each frame '\(k\)' can be expressed as follows:  
\[X_{k}^{est}=\left\{\left(\alpha_{k,1:est}^{pos},l_{1}^{est}\right),\left( \alpha_{k,2:est}^{pos},l_{2}^{est}\right)\ldots\left(\alpha_{k,N_{k}:est}^{ pos},l_{N_{k}}^{est}\right)\right\} \tag{5}\]  
In (5), the term \(\,N_{k}=\left|X_{k}^{est}\right|\) denotes the estimated number of sources at the \(k\)th frame. The estimated position and its corresponding label \(\left(\alpha_{k,i:est}^{pos},l_{i}^{est}\right)\), \(i=1,2\ldots N_{k}\) can be used to construct the steering vector corresponding to the direction of the target source represented by the position-label pair \(\left(\alpha_{k,dest}^{pos},l_{d}^{est}\right)\) while directing spectral nulls in the directions of the interfering sources denoted by the position-label pairs, \(\left(\alpha_{k,i:est}^{pos},l_{i}^{est}\right)\), \(\,\forall i\
eq d\). As reported in [6], such a beamformer can dynamically enhance the speech corresponding to the target acoustic source by utilizing the estimates from the GLMB filter. In this work, we focus on utilizing MS-GLMB filtering to assist the construction of the target label set (in this case, the T-F mask specific to the target speech source) for the neural architecture. Due to the association of a unique label with every speech source within the environment, the MS-GLMB filtering helps to correctly discriminate between the target source and the interfering sources, which helps to train the subsequent deep neural architecture with an appropriately constructed label set specifically to the target source in motion under noisy reverberant conditions.