## 4 Computer Simulation Results
### Evaluation Metrics
#### 4.4.1 Beamformer Baselines for Performance Comparison  
In this section, an overview of the state-of-the-art beamformer approaches is presented. The proposed approach is compared with these approaches for target speech enhancement under reverberant conditions involving moving sources. The evaluated baseline beamformers can be denoted by time invariant MVDR (TL_MVDR), online MVDR (online_MVDR) involving online computation of SCM, block MVDR (BLK_MVDR) involving block-wise processing of time-varying SCM, direction-of-arrival (DOA)-based MVDR (DOA_MVDR) baseline, mask-based MVDR using attention mechanism (ATT_MASK_MVDR), residual dense GAN-based beamformer (RD-GAN) [25] and GCN-based MVDR (GCN_MVDR). Outlines of such beamformer baselines can be obtained from [8; 25; 29]. TI-MVDR Beamformer Baseline  
The TL_MVDR system computes the SCM in a time-invariant fashion, where the assumption is that the transfer function is static within an utterance. The time-invariant SCMs can be computed as follows:  
\[\mathbf{\Phi}_{f}^{v}=\sum_{\tau=1}^{T}\frac{\mathbf{1}}{\sum_{\tau^{\prime}=1}^{T}m_{ \tau^{\prime},f}^{v}}\mathbf{m}_{\tau,f}^{v}\mathbf{Y}_{\tau,f}\mathbf{Y}_{\tau,f}^{T} \tag{43}\]  
In (43), the term \(\mathbf{Y}_{\tau,f}\mathbf{Y}_{\tau,f}^{T}\) can be represented by \(\mathbf{\Psi}_{\tau,f}^{v}\) which can be referred to as an instantaneous SCM (SCM) at the T-F bin. The term \(\mathbf{m}_{\tau,f}^{v}\in[\mathbf{0},\ \mathbf{1}]\) represents the T-F mask. In (43), the superscript \(\mathbf{v}\in\{\mathbf{S},\mathbf{N}\}\) denotes the indices of speech (S) and noise (N). Online MVDR Beamformer Baseline  
The online_MVDR system involves the application of a recursive approach to compute a time-varying (T-V) SCM, \(\mathbf{\Phi}_{tf}^{v}\). Such an approach can lead to the computation of SCMs for the MVDR beamformer in an online manner. The recursive computation of the online SCMs can be expressed as follows:  
\[\mathbf{\Phi}_{tf}^{v}=\mathbf{\alpha}.\mathbf{\Phi}_{t-1,f}^{v}+\mathbf{\Psi}_{tf}^{v}=\sum _{\tau=1}^{t}\mathbf{\alpha}^{t-\tau}.\mathbf{\Psi}_{\tau,f}^{v} \tag{44}\]  
The online_MVDR approach enables the computation of SCMs as well as MVDR beamformer weights at each time frame, which in turn allows for the tracking of a speech source. Block MVDR Beamformer Baseline  
The BLK_MVDR system involves the computation of SCM corresponding to each time block of the signal into which it is divided. The block-wise computation of the time-varying SCMs can be expressed as follows:  
\[\mathbf{\Phi}_{tf}^{v}=\sum_{\tau=t-L}^{t+L}\frac{1}{\sum_{\tau^{\prime}=t-L}^{t+ L}m_{\tau^{\prime},f}^{v}}.\mathbf{\Psi}_{\tau,f}^{v} \tag{45}\] \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Method** & \multicolumn{2}{c}{**RT60 = 550 ms**} & \multicolumn{2}{c}{**RT60 = 250 ms**} & \multicolumn{2}{c}{**RT60 = 50 ms**} \\ \hline  & **PESQ** & **SI-SNR (dB)** & **PESQ** & **SI-SNR (dB)** & **PESQ** & **SI-SNR (dB)** \\ \hline MS-GLMB filter-assisted MVDR & 3.28 & 5.65 & 3.44 & 6.55 & 3.51 & 7.66 \\ DOA-MVDR & 3.43 & 8.85 & 3.49 & 9.75 & 3.55 & 10.36 \\ Mask-MVDR with self-attention & 3.57 & 10.37 & 3.60 & 10.55 & 3.65 & 12.66 \\ U-Net with embedded GCN & 3.56 & 10.54 & 3.65 & 11.66 & 3.67 & 12.78 \\ RD-GAN & 3.58 & 10.58 & 3.67 & 11.72 & 3.69 & 12.82 \\ RD-Graph-U-Net-GAN (proposed approach) & 3.60 & 10.77 & 3.70 & 11.88 & 3.85 & 12.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: _Cont._In (45), the term \(L\) denotes the block-size parameter and \((\mathbf{2}L+\mathbf{1})\) frames were utilized for the computation of SCM for each block.