# Deep Learning-Based Low-Frequency Passive Acoustic Source Localization
## 3 Test Case Setups  
### Case 1: Stationary Source Localization on a Scanning Plane  
This case takes inspiration from the work by [PERSON] et al. [18], [PERSON] and [PERSON] [27], [PERSON] et al. [18], in particular, carried out source localization on a scanning plane parallel to the microphone array plane for a wide frequency range of 200 to 20,000 Hz, using a deep neural network called DenseNet-201. They were able to show the superiority of DenseNet-201 over other classic acoustic imaging methods. However, DenseNet needed a significant amount of simulated data, memory, and processing power to train. In this paper, we focused on ASL at two frequencies in the low-frequency range (100 Hz and 300 Hz) using limited training data and a shallow CNN to prove that ASL is possible using a computationally less expensive approach. The setup is shown in Figure 7.  
We used a scanning plane of dimensions 1.2 m \(\times\) 1.2 m. The scanning plane was located 1.2 m above the array plane containing the 64-microphone logarithmic spiral array, as described in the previous section. To generate the training dataset, \(S\) monopole sources (S1, Equation (1)) with unit source strength were randomly distributed throughout the \(N\times N\) scanning grid. The acoustic pressure of all sources was calculated at all the microphones (\(M\)) to form the combined pressure vector, \(P\), given as:  
\[P=\left[\sum_{s=1}^{S}P_{s}(1),\sum_{s=1}^{S}P_{s}(2),\sum_{s=1}^{S}P_{s}(3), \ldots,\sum_{s=1}^{S}P_{s}(M)\right] \tag{3}\]  
Vector \(P\), in turn, was used to obtain the cross-spectral matrix (CSM) as:  
\[\text{CSM}=PP^{H} \tag{4}\]  
where \(P^{H}\) is the conjugate transpose of \(P\). The dimensions of the CSM are \(M\times M\), which in this case is 64 \(\times\) 64. The CSM represents the power distribution over frequency, and we used the real part of it as an input feature to the CNN. The CSM here serves a different purpose than what it traditionally serves in signal processing. In the context of deep learning, the CSM is an image-like input feature that encapsulates the acoustic pressure information needed to train our CNN. The CNN was trained against the ground truth,  
Figure 7: Case 1: Acoustic source localization on a scanning plane. The scanning plane is discretized into an \(N\times N\) grid of uniform resolution. The finer the grid size is, the higher is the computational cost.  
i.e., the source distribution from which the CSM was derived, to predict the strength and location of the sources. For the ground truth, the \(N\times N\) scanning grid was converted to a \(N^{2}\times 1\) binary vector in which the elements corresponding to the sources were assigned the value 1 (equal to the source strength), and the elements without a source were assigned the value 0. We defined the output layer as a dense layer with \(N^{2}\) nodes without an activation function, thus treating this as a regression problem. We use the Rectified Linear Unit (ReLU) activation function everywhere throughout the network. The network prediction was a vector of output layer dimensions (\(N^{2}\times 1\)) that was compared against the ground truth vector to calculate the loss. The loss function used was the mean squared error (MSE), defined as:  
\[MSE=\frac{\sum_{i=1}^{n}(y_{pred}-y_{gt})^{2}}{n} \tag{5}\]  
where, \(y_{pred}\) is the predicted vector given by the network, \(y_{gt}\) is the ground truth vector, and \(n\) is the length of the two vectors, which in this case is \(N^{2}\). The optimizer used was _ADAM_[33] with the default values of the hyperparameters. Details of the CNN and the training process are given in Section 4.