## 2 Inverse Solution Algorithms
### SFA and Sfs

Table 7: Mean opinion score of perceptual evaluation of speech quality for source signal separation at the angles 60\({}^{\circ}\), 240\({}^{\circ}\) with speech signals using Tikhonov regularization (TIKR), compressive sampling-convex optimization (CS-CVX), and focal underdetermined system solver (FOCUSS). Figure 13: Localization of one music source signal with sampling rate 16 kHz, located at the angles 60\({}^{\circ}\), 240\({}^{\circ}\) in an anechoic chamber. By using a uniform circular array with a radius of 12 cm, the source direction can be identified by the peak in the angular spectrum. (**a**) Minimum power distortionless response (MPDR) spectrum plotted versus angle and frequency. (**b**) Frequency-averaged and normalized MPDR spectrum. One sample coherence function between one loudspeaker and one microphone is shown in Figure 14, indicating the signal quality to be poor below 200 Hz. Therefore, band-limited processing was applied for all frequencies up to 200 Hz in the SFS stage. In this frequency range, pressure matching was used on the basis of the room response model. The SFS stage was conducted for three different methods. The coherence between the loudspeaker and the microphone was poor below 200 Hz; therefore, the signals below 200 Hz were not processed. Method 1, band-limited processing, was applied from 200 Hz to the spatial aliasing frequency, 952 Hz, in the SFS stage. In this frequency range, pressure matching was performed on the basis of the room response model. Below 200 Hz, unprocessed audio signals were fed directly to the loudspeakers. Above 952 Hz, a simple vector panning [39] approach was adopted. The optimal regularization parameter \(\beta\) achieving the highest MOS in room response modeling was calculated using GSS [21] as \(\beta=0.0008634\). In the second method, instead of a vector panning method, we used DAS to process signals above 952 Hz. In the third method, we used pressure matching to obtain signals above 200 Hz. The use of different regularization parameters in pressure matching results in different levels of localization performance and audio quality. Figure 15a,c shows the MPDR spectrum and the normalized MPDR spectrum obtained using the third method for \(\beta=0.01\) and \(\beta=10\), respectively. Low values of the regularization parameter \(\beta\) yielded higher localization performance than high values did. These two signals were compared with the clean signal through the PESQ test. The results showed that the high \(\beta\) ensured satisfactory voice quality, whereas the low \(\beta\) impaired voice quality. The results of three localization methods are presented in Figure 16a-f. The MPDR spectra are plotted as functions of angle and frequency in Figure 16a,c.f. The resulting frequency-averaged and normalized MPDR spectra are shown in Figure 16b,d,f. Figure 14: Coherence curve measured from one loudspeaker to one microphone in a live room. A PULSE analysis platform made from Brüel & Kjaer was arranged to measure the coherence curve. A white noise signal with sampling rate 16 kHz was used as the driving signal of the loudspeaker. Figure 15: _Cont_. Figure 16: _Cont_. Figure 15: Localization results in the sound field synthesis experiment by the third method with different regularization parameters. \(\beta=10\) (**a**,**b**) and \(\beta=0.01\) (**c**,**d**). Figure 16: _Cont._