# A Multi-Source Separation Approach Based on DOA Cue and DNN
## 3 Multiple Sound Source Separation Framework
### Separated Signal Match and Post-Process  
In Sections 3.1 and 3.2, the separated components from sparse and non-sparse component points can be obtained, respectively. However, we need to determine whether the separated components from the sparse component points and the non-sparse component points come from the same sound source. For sparse component point recovery, the DOAs of the sound sources are used to determine which sound source the separated sparse component comes from. For non-sparse component point recovery, the \(i_{th}\) sound source is selected from the N sound sources as the target sound source in advance. Then, according to Formula (22), the DNN is used to predict the IRM for the target sound source to obtain the separated components of the target sound source from the non-sparse component points. In the case of multiple (at least three) sound sources, we cannot predict the IRMs of  
\begin{table}
\begin{tabular}{c c} \hline
**Items** & **Parameter Value** \\ \hline Batch Size & 256 \\ Epochs & 40 \\ Learning Rate & \(2^{-0.001}\) \\ Drop Factor for learning rate & 0.9 \\ \hline \end{tabular}
\end{table}
Table 1: The parameter configuration for DNN training.  
Figure 7: Illustration of network structure.  
all the sound sources simultaneously. Hence, we repeat the above process for N times, and N DNN models are obtained to estimate the IRMs for the N sound sources, respectively. The separated components of the total sound sources from the non-sparse component points are then obtained. However, the target sound source is set randomly, not according to the DOA of the sound source. Therefore, the separated components of each sound source from the sparse component points \(\mathbf{\mathcal{R}_{s}}\) and the non-sparse component points \(\mathbf{\mathcal{R}_{ns}}\) need to be matched to obtain the separated components coming from the same sound source.  
Considering that a temporal correlation exists in the speech signals from the same sound source, the correlation coefficient can be used to determine whether the separated components from the sparse component points and the non-sparse component points are matched. The sparse component separated signal of one sound source consists of the total separated components of the sound source from the sparse component points. In the same way, the non-sparse component separated signal of one sound source consists of the total separated components of the sound source from the non-sparse component points. The correlation coefficient between the different separated signals can be defined by the following formula:  
\[\rho_{i,j}=\frac{\mathrm{COV}\left(o_{i},o_{j}^{\prime}\right)}{\sqrt{\mathrm{ D}(o_{i})}\sqrt{\mathrm{D}\left(o_{j}^{\prime}\right)}}\,\ \ i=1,\ldots,N\ and\ j=1,\ldots,N \tag{24}\]  
where \(o_{i}\) represents the sparse component separated signal of the \(i_{th}\) source, and \(o_{j}^{\prime}\) represents the non-sparse component separated signal of the \(j_{th}\) source. Here, \(\rho_{i,j}\) is the correlation coefficient between the sparse component separated signals of the \(i_{th}\) source and the non-sparse component separated signals of the \(j_{th}\) source. \(\mathrm{COV}(\cdot)\) and \(\mathrm{D}(\cdot)\) are covariance and variance, respectively.  
Therefore, if the sparse component separated signal comes from the \(i_{th}\) sound source, the following formula can be used to determine that the non-sparse component separated signal from the \(j_{th}\) sound source comes from the same sound source.  
\[m(i)=arg\underset{j}{\max}\{\rho_{i,j},\ j=1,\ldots,N\} \tag{25}\]  
Eventually, we obtain \(N\) separated signals \(\widetilde{s}_{i}=o_{i}+o_{m(i)}^{\prime},i=1,\ldots,N\).