# A Combined Method for Localizing Two Overlapping Acoustic Sources Based on Deep Learning
## 4 Results and Discussion  
Following training of the proposed source separation model, we integrated it with the single-source localization model developed in [16] (which achieves \(10^{\circ}\) spatial resolution). In order to evaluate the effectiveness of the two-source localization model, the prediction accuracy (localization accuracy) metric was used as a performance measure, which is defined as follows [16]:  
\[PA=\frac{N_{p}}{N_{s}}\times 100(\%), \tag{9}\]  
where \(N_{s}\) represents the total number of source directions being evaluated and \(N_{p}\) is the number of source directions correctly recognized. The direction of the source is considered to be correctly recognized if the predicted direction is within the spatial resolution of the model: that is, the deviation of the predicted direction from the actual direction is within \(\mp\theta_{0}\) for the spatial resolution \(\theta_{0}\)[26]. The total number of candidate source directions was \(K=36\), which were uniformly distributed in the range \([-180^{\circ},180^{\circ})\) with a step of \(10^{\circ}\). To synthesize the test data, 36 audio files with a duration of 500 ms and a sampling frequency of 16 kHz were randomly selected from the TIMIT test set without repetition, each audio file corresponding to one of the 36 directions. The audio files associated with the directions represented the acoustic sources, which were convolved with the corresponding RIRs to form a set of 36 samples (each sample contained a signal corresponding to each of the four microphones). This set was randomly divided into pairs; then, the signals corresponding to each microphone were mixed to form a single sample containing the mixed signals of the four microphones. Thus, the final size of the raw test data set (mixed signals) was 18 samples, which was equal to the number of pairs in the original samples. Each of these mixed signals in each sample represented the raw input to the source separation model. The separated signals representing the first source were raw input to a single SSL model, as were the separated signals representing the second source, which were also raw input to a similar model. The generalization ability of the model is validated by forming the test samples in such a way that all possible directions of the sound source within the range \([-180^{\circ},180^{\circ})\) are covered, where each direction corresponded to a different sound source signal, and the mixed signals had variable SSR. The data preprocessing pipeline, including raw data preparation and feature extraction, was implemented in MATLAB R2020a. The machine learning model was subsequently developed using Python 3. * within the Google Colab cloud computing environment. This  
Figure 5: Model testing scheme. hybrid implementation approach leveraged MATLAB's robust signal processing capabilities for feature engineering while utilizing Python's extensive deep learning ecosystem for model development. Table 2 shows the true values (\(\theta_{1}\) and \(\theta_{2}\)) and predicted values (\(\hat{\theta_{1}}\) and \(\hat{\theta_{2}}\)) of the directions of each of the two sources, the SSR, and the values of the prediction accuracy metric (PA) for each of the test samples. The SSR was calculated using the following formula:  
\[SSR=10 log_{10}\left(\frac{E\big{[}s_{1}^{2}(t)\big{]}}{E\big{[}s_{2}^{2}(t) \big{]}}\right), \tag{10}\]  
where \(E\) is the average value. The prediction accuracy for each test sample was considered 100% when the model correctly estimated the directions of both sound sources. If only one direction was correctly predicted while the other was incorrect, the accuracy for that sample was set at 50%. The proposed localization methodology demonstrated high effectiveness in simultaneously estimating the directions of two overlapping sources, achieving an average localization accuracy of 86.1% on the test dataset, which included source signals with varying SSRs.