## 4 Conclusions  
This study developed algorithms for solving inverse problems generally encountered in spatial audio signal processing. The TIKR algorithm was shown to solve overdetermined problems. However, the regularization parameter in the TIKR method was not effectively chosen. This study thus presents a guideline for choosing the optimal regularization parameter \(\beta\) in the TIKR method. Specifically, choosing the optimal \(\beta\) involves dividing the maximal singular value at low frequency by the threshold and then running the simulation forward and backward for powers of 10 of \(\beta\). Optimization methods such as GSS can be used by observing the trend of an objective function (such as PESQ). Some trade-offs must be made between localization performance and voice quality. In general, a high \(\beta\) results in a small solution norm with high voice quality, whereas a low \(\beta\) yields a small residual norm with high localization performance.  
Inverse problems in noise sound source localization and separation problems can be solved by 1-stage and 2-stage (overdetermined and underdetermined), each of the 1-stage and 2-stage methods has its advantages and disadvantages. In general, the 1-stage methods provide both localization and separation results with good performance. The 2-stage methods give slightly better separation performance than the 1-stage methods. From our experience, PESQ correlates better with separability and segSNR correlates better with distortion.  
For 1-stage (underdetermined) problems, iterative CS algorithms have been developed for solving acoustic inverse problems, with applications to localization and separation. The results demonstrate that the CS-CVX method was effective in solving CS problems, despite being computationally expensive. Iterative CS methods achieved comparable performance to the CS-CVX method for CS problems, in far less computation time. The FOCUSS-TIKR and CG methods attained high PESQ, whereas the SD and NT methods attained high segSNR. In general, iterative CS methods were determined to perform better than the TIKR method. For 1-stage methods, FOCUSS-TIKR attains the highest MOS value of PESQ for clean signals, while the Newton method performs the best. Both methods require less CPU time than the CS-CVX.  
Inverse solution approaches are also useful in solving SFA and SFS problems. In this study, three inverse problems were solved for implementing an audio codec system. Because of the ill-posed yield at low frequencies, particularly in the room response modeling stage, choosing an appropriate regularization parameter \(\beta\) was crucial. Therefore, this stage required a larger regularization parameter  
Figure 16: Localization results in the sound field synthesis experiment with three different approaches. (**a,b**) first method, (**c,d**) second method, and (**e,f**) third method.  
than the SFA and SFS stages required. In the analysis stage, the one-stage CS algorithm was determined to be more computationally expensive than the two-stage TIKR algorithm. In the synthesis stage, the first method performed well in localization, but did not perform well in reproduced voice quality. As compared with the third method, the second method reproduced signals with boosted high-frequency content above 952 Hz with poor localization. The third method had the highest performance in terms of voice quality and localization performance.  
The work was supported by the Ministry of Science and Technology (MOST) of Taiwan, Republic of China, under project number 102-2221-E-007-029-MY3.  
[PERSON] and [PERSON] conceived and designed the experiments; [PERSON] and [PERSON] performed the experiments; [PERSON] and [PERSON] analyzed the data; [PERSON] and [PERSON] wrote the paper.  
The authors declare no conflict of interest.