## 2 Inverse Solution Algorithms
### Random Array  
A simulation was conducted for localization and separation of two point sources located at \((0,0,-1\,\mathrm{m})\) and \((0.8,\,0.3,-1\,\mathrm{m})\), both of which were emitting clean speech signals, as illustrated \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Methods**} & **TIKR** & **FOCUSS-PINV** & **FOCUSS-TIKR** & **CS-CVX** \\ \hline \multirow{3}{*}{PESQ} & Source 1 & 2.034 & 3.966 & 3.350 & 3.251 \\  & Source 2 & 1.696 & 3.146 & 2.879 & 2.783 \\  & Source 3 & 1.912 & 3.818 & 3.394 & 3.347 \\ \hline segSNR & Source 1 & 0.554 & 11.92 & 11.27 & 8.817 \\ \hline CPU time (s) & 53 & 810 & 487 & 16302 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Separation performance of the Tikhonov regularization (TIKR) and focal underdetermined system solver (FOCUSS) methods for three sources in the underdetermined system. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Methods**} & **TIKR** & **FOCUSS-PINV** & **FOCUSS-TIKR** & **CS-CVX** \\ \hline \multirow{3}{*}{PESQ} & Source 1 & 2.135 & 1.170 & 2.941 & 2.412 \\  & Source 2 & 1.810 & 1.121 & 2.234 & 1.321 \\  & Source 3 & 1.723 & 1.168 & 2.841 & 2.241 \\ \hline segSNR & Source 1 & 0.52 & \(-\)10.00 & 1.558 & 0.946 \\ \hline CPU time (s) & 58 & 817 & 497 & 16431 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Separation performance of the Tikhonov regularization (TIKR), focal underdetermined system solver (FOCUSS), and compressive sampling-convex optimization (CS-CVX) methods for three sources with additive white noises (40 dB SNR). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Methods**} & **TIKR** & **FOCUSS-PINV** & **FOCUSS-TIKR** & **CS-CVX** \\ \hline \multirow{3}{*}{PESQ} & Source 1 & 1.512 & 1.168 & 2.512 & 1.436 \\  & Source 2 & 1.401 & 1.118 & 2.012 & 1.712 \\  & Source 3 & 1.489 & 1.044 & 2.487 & 1.324 \\ \hline segSNR & Source 1 & \(-\)0.12 & \(-\)13.11 & 0.12 & \(-\)1.112 \\ \hline CPU time (s) & 57 & 815 & 494 & 16421 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Separation performance of the TIKR, FOCUSS and CS-CVX methods for three sources with additive white noises (20 dB SNR). in Figure 7. A 30-element random array with aperture dimension 0.48 m \(\times\) 0.4 m situated at \(z=0\) was utilized to capture the signals emitted by these two sources. To set up the propagation matrix, 100 (10 \(\times\) 10) equivalent sources were distributed on the image plane located 1 m away from the array. Figure 8 shows the condition numbers of different frequencies. The problem is ill-conditioned at low frequencies. Figure 9a-f show the source localization results obtained using six approaches. Two sources were correctly located on the noise map with varying degrees of resolution by all methods. A conventional method delay and sum (DAS) method (a) gave the poorest resolution, whereas the CS-CVX method provided the highest resolution. The SD method (d) and CG method (f) were acceptable but did not perform quite as well as the CS-CVX method. The NT method (e) yielded accurate source locations with a slightly increased sidelobe level, but it was the most computationally efficient (Table 2). Figure 8: Condition number versus frequency. Figure 7: Arrangement for simulation of localization and separation of two sources. Figure 9: _Cont._  
Figure 9: Localization results of two point sources. (**a**) Delay and sum algorithm, (**b**) [PERSON] regularization algorithm, (**c**) compressive sampling-convex optimization algorithm, (**d**) steepest descent algorithm, (**e**) [PERSON]’s algorithm, and (**f**) conjugate gradient algorithm. Table 4 presents a comparison of the separation results obtained using five methods. CS-CVX displayed the highest scores for PESQ and segSNR, despite being extremely time-consuming. Iterative CS approaches were determined to be far more computationally efficient than the CS-CVX method. The CG method attained the highest PESQ, but the lowest segSNR.