## 2 Formulation of The Problem
### Limitation of the W-DO Assumption  
W-disjoint orthogonality (W-DO) [23] reveals the sparsity speech sources, which means that only one source is active at a certain TF bin. Besides, the spatial information of source \(S\) is preserved in the B-format signal by [24; 25]:  
\[\left\{ \begin{array}{c}S_{W}(n,l)=\frac{\sqrt{2}}{2}\cdot S(n,l)\\ S_{X}(n,l)=\cos\mu\cdot\cos\eta\cdot S(n,l)\\ S_{Y}(n,l)=\sin\mu\cdot\cos\eta\cdot S(n,l)\\ S_{Z}(n,l)=\sin\eta\cdot S(n,l)\end{array} \right. \tag{2}\]  
where \(S\) is the sound source signal, and \(\mu\) and \(\eta\) are the azimuth and elevation of the sound source, respectively, with respect to the center point \(O\). If the W-DO property is valid, similar to [21], the estimated DOA \(\hat{\mu}(n,l)\) of each TF bin can be obtained by:  
\[\hat{\mu}(n,l)=arctan\frac{I_{Y}(n,l)}{I_{X}(n,l)} \tag{3}\]  
where \(I_{Y}\) and \(I_{X}\) can be calculated by:  
\[\left\{ \begin{array}{c}I_{X}(n,l)=Re\{S_{W}^{*}(n,l)\cdot S_{X}(n,l)\}\\ I_{Y}(n,l)=Re\{S_{W}^{*}(n,l)\cdot S_{Y}(n,l)\}\end{array} \right. \tag{4}\]  
where \(Re\{\cdot\}\) denotes taking a real part of the argument and \({}^{*}\) denotes conjugation operation. However, simultaneously occurring speech signals are more likely to overlap in the TF domain, which means that more than one source is active at a TF bin with certain probability [21]. Hence, if the overlapped TF bins of multiple sources occupy a excessive proportion, the localization of multiple sources will be badly influenced. Further, the source separation method based on the localization procedure is not valid again. To solve this problem, some efficient localization methods have been proposed based on single-source bins or zone detection [26]. However, these methods can not eliminate the aliasing of TF components of the multiple source signals. As a result, the aliasing of the TF components cannot be recovered completely, which leads to poor separation quality in the case of the multiple sources. These overlapped TF components are defined as non-sparse components (which cause the famous cocktail problem), while the other TF components derived from one source are defined as sparse components. In order to verify the W-DO assumption has reduced accuracy with an increasing number of simultaneously occurring sources, we examine how many TF bins are overlapped in the TF domain. Figure 1: Illustration of the multi-source model and configuration of recording scene. The surrounding sources are numbered \(S_{1}\) to \(S_{K}\). The ratio of overlapped TF bins (ROTF) can be defined as the measure to detect the proportion of non-sparse components, i.e.,  
\[ROTF=1-\frac{1}{K}\sum\limits_{l=1}^{K}\frac{1}{N}\sum\limits_{n=1}^{N}\left( \frac{||S_{i}^{{}^{\prime}}(n,l)||_{0}}{||S_{i}(n,l)||_{0}}\right) \tag{5}\]  
where \(N\) is the number of total frames and quasi-norm \(||\cdot||_{0}\) counts the number of non-zero components in its argument. \(S_{i}^{{}^{\prime}}(n,l)\) can be obtained by:  
\[S_{i}^{{}^{\prime}}(n,l)=\left\{ \begin{array}{c}S_{i}(n,l),if\frac{|S_{i}(n,l)|}{\left|\sum\limits_{l=1}^{K}S_{i}(n,l)\right|}>\xi,l\in[1,L]\\ 0,else\end{array} \right. \tag{6}\]  
where \(L\) is the number of STFT points in a frame. We detect the TF instant where only one source of energy is dominant among all sources, and then calculate the proportion of the these instants. Eventually, the ROTF is defined by subtracting the proportion. The ROTF implies the ratio of overlapped components among \(K\) simultaneously occurring sources. Obviously, a higher ROTF means weaker sparsity among these sources. In order to examine the average ROTF among simultaneously occurring speech signals, statistical analysis is performed. In total, 36 sentences (the sampling frequency is 16 kHz) from the NTT [27] speech database are used for testing. In the following evaluation, all the test data is from the NTT [27] database unless otherwise stated.