## 2 Methodologies
### Preliminary Signal Estimation  
Because the location of the \(k\)th sound source can be indicated by (10)--in other words, \(r_{m}\) is known--the estimated temporal characteristics of the \(k\)th source can be obtained by taking the average of all the weighted signals:  
\[\hat{x}_{m}^{k}(t)=\frac{1}{M}\sum\limits_{m=1}^{M}4\pi r_{km}p_{m}\bigg{(}t+ \frac{r_{km}}{c_{0}}\bigg{)}. \tag{11}\]  
In (11), the weighted signal \(p_{m}(t)\) is amplified to compensate for the time delay and attenuation of the amplitude due to the travelling distance of the sound wave from the source to each microphone. By substituting (1) into (11), the temporal estimation of the \(k\)th source can be rewritten as  
\[\hat{x}_{m}^{k}(t) =\frac{1}{M}\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{N}r_{km}r_{km }^{-1}s_{n}\Big{(}t-\frac{r_{km}}{c_{0}}+\frac{r_{km}}{c_{0}}\Big{)}+\frac{1}{M }\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{N}4\pi r_{km}\Phi_{n}\Big{(}t+\frac{ r_{km}}{c_{0}}\Big{)}\enspace. \tag{12}\] \[=s_{k}(t)+e_{k}(t)\]  
In (12), the estimated source signals are divided into two parts by simplification: the first part is the real source signal \(s_{k}(t)\), while the second part could be considered the error function \(e_{k}(t)\), which represents the difference between the real source signal and estimated signal:  
\[\begin{array}{ll}e_{k}(t)&=\frac{1}{M}\sum\limits_{m=1}^{M}\sum\limits_{i=1 }^{k-1}r_{km}r_{im}^{-1}s_{i}\Big{(}t+\frac{r_{km}-r_{im}}{c_{0}}\Big{)}+\frac{ 1}{M}\sum\limits_{m=1}^{M}\sum\limits_{j=k+1}^{N}r_{km}r_{jm}^{-1}s_{j}\Big{(} t+\frac{r_{km}-r_{jm}}{c_{0}}\Big{)}\\ &+\frac{1}{M}\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{N}4\pi r_{km}\Phi_{n} \Big{(}t+\frac{r_{km}}{c_{0}}\Big{)}\end{array}\enspace. \tag{13}\]  
In general, the right side of \(e_{k}(t)\) contains three parts: (1) the contribution of the sound sources that have been localized before the \(k\)th source, (2) the contribution of the sound sources that have not been localized after the \(k\)th source, and (3) the contribution of the incoherent noise from each measurement signal. First, compared with the real source signal, \(s_{k}(t)\), a weighted averaging by the difference between \(r_{jm}\) and \(r_{im}\) in the error function \(e_{k}(t)\) would suppress the amplitude of \(s_{i}(t)\) and \(s_{j}(t)\) in (12), which means that the amplitude of \(e_{k}(t)\) is much smaller than \(s_{k}(t)\). Therefore, \(\hat{x}_{m}^{k}(t)\) can be regarded as a rough estimate of \(s_{k}(t)\). This acceptable hypothesis is the precondition of the following separation and reconstruction approach. Subsequently, \(s_{i}(t)\) in the first part of \(e_{k}(t)\) can be replaced by \(\hat{x}_{m}^{i}(t)\). The real source signal, \(s_{k}(t)\), can be estimated using an iteration computation as:  
\[\hat{s}_{k}(t)=\hat{x}_{m}^{k}(t)-\frac{1}{M}\sum\limits_{m=1}^{M}\sum\limits_ {i=1}^{k-1}r_{km}r_{im}^{-1}\hat{x}_{m}^{i}\bigg{(}t+\frac{r_{km}-r_{im}}{c_{0 }}\bigg{)}. \tag{14}\]  
The virtual received signals could be updated by newly estimated source signals. The iteration process stops if no additional sound source appears. Finally, all \(r_{im}\) in the first part are obtained from the updated beamforming maps.  
It is worth mentioning that the estimation of \(\hat{s}_{k}(t)\) by \(\hat{x}_{m}^{k}(t)\) is rather rough, with large errors in (12). However, the errors that cumulate after the iterative calculations could significantly affect the reconstruction signals when the number of sound sources is large. Therefore, DNN technology is introduced to eliminate errors and to acquire an acceptable estimation of \(\hat{s}_{k}(t)\) in the next section.