# A Multi-Source Separation Approach Based on DOA Cue and DNN
## 3 Multiple Sound Source Separation Framework
### Sparse Component Point Recovery  
In this paper, the DOA of sound source is used to determine the sparse component points and obtain the separated component from the mixed signals. To obtain the DOA estimates of real sound sources, we explore the distribution of DOA estimates of total TF points. For the mixed signals recorded by the sound-field microphone, DOA estimates of total TF points can be directly calculated from the B-format recordings [31; 32]. The DOA estimation method proposed in [22] is employed to obtain DOA estimates of real sound sources. To be specific, the single source time-frequency point in the mixed signal is detected by diffuseness measurement; the active sound intensity obtained by the sound field microphone recordings is used to obtain DOA estimates of the single source time-frequency points. So, the sound sources number and the multi-source DOA estimates can be estimated by peak searching on the statistical histogram formed by the DOA estimates of the single source time-frequency point. Theoretically, if the TF point is dominated by the direct component of the sound source \(s_{i}\), the DOA estimate of this TF point may be same as that of the \(i_{th}\) sound source. Therefore, the TF point whose DOA estimate is closest to the DOA of the real sound source is considered the sparse component point. In fact, to achieve a satisfactory tolerance, the restriction on DOA estimates of TF points is relaxed. The TF point whose DOA estimates are within the range of the DOA estimate of source \(\pm\Delta\) is the sparse component point. The remaining TF points are detected as non-sparse  
Figure 5: Schematic diagram for sparse and non-sparse component points. The rectangular boxes with a single color represent sparse component points; Rectangular boxes with multiple colors represent non-sparse component points. component points. So, the set of sparse component points and non-sparse component points are defined as follows:  
\[\mathbf{\mathcal{R}_{s}}\,:=\,\left\{(n,k)\big{|}\theta_{n,k}\in\left[\omega_{i}- \Delta,\omega_{i}+\Delta\right]\right.\,\ \ \ \ \ i=1,\dots,N\} \tag{18}\]  
\[\mathbf{\mathcal{R}_{ns}}\,:=\,\left\{(n,k)\big{|}\theta_{n,k}\
otin\left[\omega_{i} -\Delta,\omega_{i}+\Delta\right]\right.\,\ \ \ \ \ i=1,\dots,N\} \tag{19}\]  
where \(\omega_{i}\) and \(\theta_{n,k}\) denote the DOA estimate of the \(i_{th}\) real sound source and the TF point \((n,k)\), respectively. \(\mathbf{\mathcal{R}_{s}}\) and \(\mathbf{\mathcal{R}_{ns}}\) represent the set of sparse component points and non-sparse component points, respectively. For convenience, \(n\) and \(k\) are omitted in \(\theta_{n,k}\) in the following. For example, Roomsim was used to obtain the recorded signal. A sound field microphone was utilized to record a mixed signal with three sound sources and \(T_{60}=300\) ms. The real DOAs of the three sound sources are \(100^{\circ}\), \(160^{\circ}\) and \(250^{\circ}\). Figure 6a is the normalized statistical histogram of the DOA estimates of the total TF points obtained by the DOA estimation algorithm proposed in [22]. The horizontal axis is the DOA estimate, and the vertical axis is the normalized statistical amplitude. There are three peaks which represent the DOAs of the three real sound sources and are delineated by the black stems. From Figure 6a, it is clear that the distribution of DOA estimates is relatively concentrated around the real DOAs of the sound sources. If the DOA estimate of a TF point is closest to the real DOA of a sound source, the TF point can be considered to be dominated by this sound source, which is determined to be a sparse component point. Figure 6b shows the distribution of DOA estimates of \(\mathbf{\mathcal{R}_{s}}\) where \(\Delta=10^{\circ}\). Figure 6: The normalized statistical histogram of DOA estimates: (**a**) is for total TF points; (**b**) is for sparse component points. According to the above analysis, the separated component from the sparse component point is the component of the dominant sound source.