# A Combined Method for Localizing Two Overlapping Acoustic Sources Based on Deep Learning
## 2 Materials and Methods  
We assume that the problem of localizing two overlapping sound sources is solved in a closed environment (for example, in a room). It is assumed that a planar orthogonal microphone array consists of four omnidirectional microphones located in the room, as shown in Figure 1:  
The microphones \(M_{1}\) and \(M_{3}\) are located along the x-axis orthogonal to the microphones \(M_{2}\) and \(M_{4}\) along the y-axis; \(d=\overline{M_{1}M_{3}}=\overline{M_{2}M_{4}}\) represents the size of the micro  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Study** & **Approach** & **Strengths** & **Limitations** & **Our Advantages** \\ \hline [PERSON] et al. [13] & DNN + 40-mic array & High-resolution separation and localization & Impractical for real-world setups (massive array) & Fewer mics needed (four) + BLSTM separation improves overlap handling & Fewer mics needed (four) + BLSTM separation improves overlap handling & Fewer (four) + BLSTM separation improves overlap handling & Fewer (four) + BLSTM separation improves overlap handling \\ [PERSON] et al. [14] & DNN for multi-speaker localization & Real-time, noise-robust overlapping sources & Fails for 100\% overlapping sources & Separation-first pipeline resolves full overlap \\ [PERSON] and [PERSON] [15] & CNN–DOA (noise-trained) & Generalizable to noisy environments & Limited to partial overlap scenarios & End-to-end overlap handling via BLSTM + localization \\ Our Method & BLSTM separation + single-source DOA & computationally efficient (vs. large arrays and large dataset) & Higher latency due to separation step & Novelty: Unified separation localization \\ \hline \end{tabular}
\end{table}
Table 1: Comparative analysis of multi-source sound localization methods.  
Figure 1: Microphone array configuration.  
phone array, and \(O\) is the center of the array. A far-field propagation model is used [17], where the directions of the two acoustic sources are represented by angles \(\theta_{1}\) and \(\theta_{2}\) defined with respect to the positive x-axis: that is, \(\theta_{1},\theta_{2}\in[-180^{\circ},180^{\circ})\).  
Microphone signals are represented by Formulas (1) and (2) as follows :  
\[x_{i}(t)=a_{i\,1}\left(t\right)*s_{1}(t)+a_{i\,2}\left(t\right)*s_{2}(t)+n_{i} (t), \tag{1}\]  
\[x_{i}(t)=\sum_{\tau=0}^{T-1}a_{i\,1}\left(\tau\right)s_{1}(t-\tau)+\sum_{\tau= 0}^{T-1}a_{i\,2}\left(\tau\right)s_{2}(t-\tau)+n_{i}(t), \tag{2}\]  
where \(x_{i}(t)\) denotes the signal of two overlapping sources received by microphone \(i\), \(s_{1}(t)\) and \(s_{2}(t)\) are the signals of the first and second acoustic source, respectively, \(a_{i\,1}\left(t\right)\) and \(a_{i\,2}\left(t\right)\) are the room impulse responses (RIRs) [18] between microphone \(i\) and the first and second sources, respectively, \(n_{i}(t)\) is background noise and possibly microphone noise, and \(*\) denotes convolution. The signals are digital; therefore, \(t\) and \(T\) are discrete time indices, and \(T\) is the effective length of the RIRs.  
The methodology for localizing two overlapping sound sources is shown schematically in Figure 2:  
Each microphone signal from the array is subjected to an acoustic source separation method implemented using an appropriate model. The output of this model is the separated signals from the sound sources. The signals of the first source obtained by the separation method are the raw input to the single SSL model, which, in turn, estimates the direction of the first source, whereas the signals of the second source obtained by the separation  
Figure 2: Localization scheme of two overlapping sound sources.  
method are the raw input to the localization model, which, in turn, estimates the direction of the second source.