## 2 Inverse Solution Algorithms
### SFA and Sfs
\right]\in\mathbb{C}^{\mathrm{K}\times P}\) is the steering matrix from the plane-wave components obtained in the preceding SFA stage to the control points. Therefore, the optimal solution can be written as  
\[\mathbf{s}_{\mathrm{s}}(\omega)=\mathbf{H}^{\#}(\omega)\mathbf{B}(\omega) \mathbf{s}_{p}(\omega), \tag{41}\]  
where \"\(\#\)\" symbolizes some type of inverse operation on the matrix \(\mathbf{H}(\omega)\). In this study, TIKR was utilized to calculate the input signal amplitudes to the secondary sources. GSS can be used to find the optimal regularization parameter. Experiments were conducted to validate the proposed audio analysis and synthesis system. In the SFA stage, a 24-element circular microphone array with a radius of 12 cm was utilized to capture and parameterize the sound field in an anechoic chamber (the recording room), as illustrated in Figure 10. In the SFS stage, a rectangular, 32-loudspeaker array was employed to reproduce in a live room (the reproduction room) the sound field previously encoded in the SFA stage. The walls of the room were lined with acoustically reflective boards (Figure 11). Figure 10: Sound field analysis experimental arrangement in a 5.4 m \(\times\) 3.5 m \(\times\) 2 m anechoic room. To process microphone output and loudspeaker input signals, multichannel analog-to-digital converters (M-32 AD) and digital-to-analog converters (M-32 DA) (RME, Haimhausen, Germany) were used with a sampling frequency of 16 kHz. An audio codec system involves three inverse problems, namely the SFA stage, room response modeling, and the SFS stage. The condition numbers are plotted against the frequencies of three steering matrices in Figure 12a-c. Figure 12c indicates that the ill-posedness encountered in the room response modeling procedures must be addressed, with the aid of appropriate regularization methods. Large regularization parameters can increase the robustness of the inverse problem. In this study, we set the regularization parameter to 10. Figure 11: Sound field synthesis experimental arrangement in a 3.6 m \(\times\) 3.6 m \(\times\) 2 m live room fitted with reflective walls. Figure 12: _Cont._  
In the SFA experiment, loudspeaker sources positioned at the angles \(\theta=60^{\circ},240^{\circ}\) played two 10-s speech clips. After recording the source by CMA, we used three algorithms to extract the source signals. First, we applied the two-stage MPDR and TIKR algorithms. The MPDR spectrum is plotted as a function of angle and frequency in Figure 13a. The resulting frequency-averaged and normalized MPDR spectrum is illustrated in Figure 13b, which peaks at the angles \(\theta=60^{\circ},240^{\circ}\) as desired. The results show that the source was accurately localized using MPDR. Next, the source signals were extracted using the TIKR algorithm. We also applied one-stage CS algorithms and one-stage FOCUSS algorithms, which located sources and separated their amplitudes in a single calculation. Figure 12: Plots of condition numbers versus frequencies of each steering matrix in three inverse problems: (**a**) sound field analysis stage; (**b**) room response modeling; and (**c**) sound field synthesis stage. The signals extracted using different methods were evaluated by using the MOS of the PESQ test. Results confirmed that the TIKR performed well in signal separation with satisfactory audio quality. The results are summarized in Table 7. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Methods**} & \multicolumn{1}{c}{**Two-Stage TIKR**} & \multicolumn{1}{c}{**One-Stages-CVX**} & \multicolumn{1}{c}{**One-Stagefocuss**} \\ \hline PESQ & Source 1 & 2.84 & 3.11 & 1.56 \\  & Source 2 & 2.79 & 2.99 & 1.61 \\ \hline \multirow{2}{*}{segSNR} & Source 1 & 17.40 & 20.74 & 13.54 \\  & Source 2 & 16.59 & 17.34 & 12.68 \\ \hline \multicolumn{2}{c}{CPU time (s)} & 8 & 27,588 & 275 \\ \hline \hline \end{tabular}
\end{table}