## 2 System Model
### TV-GSC-MVDR Beamformer  
In this sub-section, the TV-MVDR beamformer based on a GSC structure, referred to as TV-GSC-MVDR, is described, and is utilized in the T-F mask generation process as part of the neural network training.  
The MVDR beamformer can be formulated by the following constrained optimization problem:  
\[min_{\mathbf{w}}\mathbf{w}^{H}\mathbf{R_{n}}\mathbf{w}\ s.t.\mathbf{w}^{H}.\mathbf{d_{ster}}=1 \tag{6}\]  
The optimal solution to (6) is given as follows:  
\[\mathbf{w}_{opt}=\frac{(\mathbf{R_{n}})^{-1}.\mathbf{d_{ster}}}{(\mathbf{d_{ster}})^{H}(\mathbf{R_ {n}})^{-1}.\mathbf{d_{ster}}} \tag{7}\]  
In Equations (6) and (7), the term \(\mathbf{d_{ster}}\) denotes the beam-steering vector which is computed from the TDOA (alternatively, DOA). The term \(\mathbf{R_{n}}\) denotes the noise covariance matrix, while the terms \(\mathbf{w}\) and \(\mathbf{w}_{opt}\) refer to MVDR beamformer weights and optimal MVDR beamformer weights, respectively. The selection of the steering vector for the \(k\)th frame (denoted by \(\mathbf{d_{k}}\)) is achieved by exploiting the source state and corresponding label predicted by the MS-GLMB filter. As pointed out in [6], the steering vector can be selected by choosing the column vector from the steering matrix \(D_{k,\mathbf{x}_{k}^{est}}^{(q)}(f)\) for the \(k\)th time frame, where \(D_{k,\mathbf{x}_{k}^{est}}^{(q)}(f)\) can be expressed as follows:  
\[D_{k,\mathbf{x}_{k}^{est}}^{(q)}(f)=\begin{bmatrix}e^{i\omega_{f}(\tau(\alpha_{k,1 }^{pos},\mu(q,1)))}&\dots&e^{i\omega_{f}(\tau(\alpha_{k,\mathbf{x}_{k}^{est}},\mu( q,1)))}\\ \vdots&\ddots&\vdots\\ e^{i\omega_{f}(\tau(\alpha_{k,1}^{pos},\mu(\alpha_{k}q_{k})))}&\dots&e^{i \omega_{f}(\tau(\alpha_{k,\mathbf{x}_{k}^{est}},\mu(\alpha_{k}q_{k})))}\end{bmatrix} \tag{8}\]  
In (8), the terms \(\alpha_{k,1}^{pos}\dots\alpha_{k,N_{k}^{est}}^{pos}\) denote the positions of sources indexed \(1\dots N_{k}^{est}\) at frame 'k', \(u^{(q,1)}\dots u^{(q,M_{q})}\) denote the positions of microphones indexed \((q,1)\dots(q,M_{q})\) for the \(q\)th array and microphone elements \((1\dots M_{q})\), respectively. The term \(\tau\Big{(}\alpha_{k,n}^{pos},u^{(q,\text{m})}\Big{)}\), \(\text{n}=1\dots N_{k}^{est}\), \(m=1\dots M_{q}\) represents the time delay between \(n\)th source located at position \(\alpha_{k,n}^{pos}\) and the \(m\)th microphone element of the \(q\)th array represented by the position \(u^{(q,\text{m})}\). The term \(\omega_{f}\) denotes the angular frequency. Finally, the term \(N_{k}^{est}\) denotes the estimated number of sources in the acoustic environment.  
Figure 3: T-F mask by the proposed residual dense convolutional graph-U-Net framework (testing phase).  
The steering vector selection is achieved by the following operation:  
\[d^{(q)}_{k,l^{est}_{i}}(f){=}\left((D^{(q)}_{k,l^{est}_{i}}(f))^{H}\right)^{\#} \cdot r_{N^{est}_{k}}(l^{est}_{i}) \tag{9}\]  
In (9), as described in [8], the term \(r_{N^{est}_{k}}(l^{est}_{i})\) denotes the selection operator, which selects the steering vector for the target speaker. The dimension of \(r_{N^{est}_{k}}(l^{est}_{i})\) depends on the number of estimated sources. The term \(d^{(q)}_{k,l^{est}_{i}}(f)\) denotes the steering vector selected by the operation in (9) for the T-F point (k, f), where the suffix \(l^{est}_{i}\) denotes the label of the source estimated by the MS-GLMB filtering algorithm, which helps to demarcate the target source from the interfering sources within the time frame. Thus, the dynamic MVDR beamformer weight for the \(k\)th frame can be expressed as follows:  
\[\mathbf{w}_{k,l^{est}_{i}}(q)=\frac{\mathbf{R_{n}}^{-1}.\mathbf{d}^{(q)}_{k,l^{est}_{i}}(f )}{\left(\mathbf{d}^{(q)}_{k,l^{est}_{i}}(f)\right)^{H}\cdot\mathbf{R_{n}}^{-1}.\mathbf{d} ^{(q)}_{k,l^{est}_{i}}(f)} \tag{10}\]