## 2 Inverse Solution Algorithms
### SFA and Sfs  
Depending on the sparsity of the sound sources, the SFA stage can be implemented in several manners [35; 36; 37; 38]. For the sparse-source scenario, a two-stage algorithm is utilized; the source bearings are estimated using the minimum power distortionless response (MPDR) [7] and the associated amplitudes of plane waves are estimated using the TIKR algorithm. For the nonsparse-source scenario, a one-stage algorithm based on the CS-CVX algorithm or the FOCUSS algorithm is employed. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Methods**} & \multicolumn{1}{c}{**without Mismatch**} & \multicolumn{1}{c}{**with Mismatch**} & \multicolumn{1}{c}{**Extreme Mismatch**} \\ \hline \multirow{2}{*}{PESQ} & Source 1 & 2.52 & 2.48 & 2.14 \\  & Source 2 & 2.81 & 2.83 & 1.70 \\ \hline \multirow{2}{*}{segSNR} & Source 1 & 6.90 & 6.04 & \(-\)0.40 \\  & Source 2 & 6.33 & 7.26 & 0.18 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Separation performance of [PERSON]’s algorithm for two speech sources with different mismatch conditions. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Methods**} & \multicolumn{1}{c}{**TIKR**} & \multicolumn{1}{c}{**CS-CVX**} & \multicolumn{1}{c}{**SD**} & \multicolumn{1}{c}{**NT**} & \multicolumn{1}{c}{**CG**} \\ \hline \multirow{2}{*}{PESQ} & Source 1 & 1.99 & 3.12 & 2.39 & 2.48 & 2.76 \\  & Source 2 & 2.60 & 3.31 & 2.77 & 2.83 & 3.13 \\ \hline \multirow{2}{*}{segSNR} & Source 1 & 2.15 & 7.54 & 5.08 & 6.04 & 0.50 \\  & Source 2 & 2.73 & 8.24 & 7.02 & 7.26 & 1.53 \\ \hline \multicolumn{2}{c}{CPU time (s)} & 201 & 31065 & 377 & 296 & 386 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Separation performance of five algorithms for two speech sources. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Methods**} & \multicolumn{1}{c}{**TIKR**} & \multicolumn{1}{c}{**CS-CVX**} & \multicolumn{1}{c}{**SD**} & \multicolumn{1}{c}{**NT**} & \multicolumn{1}{c}{**CG**} \\ \hline \multirow{2}{*}{PESQ} & Source 1 & 1.82 & 2.83 & 2.22 & 2.38 & 2.59 \\  & Source 2 & 2.12 & 3.20 & 2.53 & 2.53 & 2.88 \\ \hline \multirow{2}{*}{segSNR} & Source 1 & 1.40 & 2.11 & 2.03 & 4.13 & \(-\)0.50 \\  & Source 2 & 2.30 & 6.51 & 2.09 & 1.72 & 1.23 \\ \hline \multicolumn{2}{c}{CPU time (s)} & 211 & 31348 & 358 & 290 & 307 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Separation performance of five algorithms for two speech sources with additive noise. The SFS stage is carried out using a loudspeaker array to reconstruct the sound field with the source bearing and amplitude obtained in the SFA stage. Pressure matching was employed for the SFS purpose in this study by sampling a large number of virtual control points in the interior area surrounded by the loudspeakers. The pressure matching procedure can be described as the following optimization problem:  
\[\underset{\mathbf{s}_{\mathrm{s}}(\omega)}{\text{min}}\|\mathbf{B}(\omega) \mathbf{s}_{p}(\omega)-\mathbf{H}(\omega)\mathbf{s}_{\mathrm{s}}(\omega)\|, \tag{40}\]  
where \(\mathbf{s}_{p}(\omega)=\left[ \begin{array}{cccc}s_{1}(\omega)&\cdots&s_{P}( \omega)\end{array} \right]^{T}\) is the amplitude vector of the \(P\)th primary plane-wave component, \(\mathbf{s}_{\mathrm{s}}(\omega)=\left[ \begin{array}{cccc}s_{1}(\omega)& \cdots&s_{L}(\omega)\end{array} \right]^{T}\) denotes the amplitude vector of the input signals to the \(L\) secondary loudspeaker sources, \(\mathbf{H}(\omega)\in\mathbb{C}^{\mathrm{K}\times L}\) denotes the room response matrix, and \(\mathbf{b}_{d}=\left[ \begin{array}{cccc}e^{-j\mathbf{k}_{d}\cdot\mathbf{y} _{1}}&\cdots&e^{-j\mathbf{k}_{d}\cdot\mathbf{y}_{K}}\end{array} \right]^{T}\) is the steering vector for the \(d\)th primary plane-wave component to the \(n\)th control point, \(\mathbf{y}_{n}\), \(n=1,\ldots,K\). \(\mathbf{B}(\omega)=\left[ \begin{array}{cccc}\mathbf{b}_{1}&\cdots&\mathbf{b }_{d}\end{array}