# Deep Learning-Based Low-Frequency Passive Acoustic Source Localization
## 2 Method
### Microphone Array and Source Simulation  
Two types of microphone arrays and sources are simulated. The first type of array, used for case 1, is a 64-microphone array in the shape of a logarithmic spiral. A spiral is naturally irregular and has unique inter-microphone spacing, which is needed to avoid spatial aliasing, thus making it a popular architecture for microphone arrays [31]. The specific type  
Figure 3: Summary of the test cases. All the cases use the framework shown in Figure 2.  
Figure 2: Flow of data in the method. This structure is followed by all the test cases developed in this paper.  
of logarithmic spiral used in this paper is called the _Arcondoulis_ spiral, which performs well over a range of frequencies. The equation of the spiral is \(r=ae^{b\theta}\), wherein the parameters \(a\) and \(b\) are set as \(a=0.018\) and \(b=0.002\) to obtain the microphone array as shown in Figure 4, similar to the array used by [PERSON] et al. (2018).  
Using a large number of microphones may not always be economically viable nor is it necessary to capture low-frequency content, thus we also simulated a more minimalistic array. The microphone array used for cases 2 and 3 is a four-microphone array, as shown in Figure 5. The design is inspired by the triangular array used by [PERSON] et al. (2011). The addition of a fourth microphone introduces variation in inter-microphone spacing and provides more data and a better-sized input feature to work with. The chosen array depends on whether the case is modeled as a regression or classification task. Case 1 is modeled as a regression problem to accurately predict the location of the sources. Due to its complexity, the array with more microphones is chosen. Cases 2 and 3 are classification tasks that involve estimating the DoA of the signal. They are relatively less complex and thus use the four-microphone array.  
We simulated two types of sources with different levels of complexity. Source 1 (S1) (Equation (1)) is an analytically-defined monopole that oscillates at a fixed frequency, \(f\). \(P_{\mathrm{s}}(m)\) (Equation (1)) shows the complex acoustic pressure due to a monopole, \(s\), as detected by a microphone, \(m\), located at a distance \(r_{\mathrm{s}}\) from the source. Here, \(Q\) is the complex source strength, and \(c_{0}\) is the speed of sound in the air, both assumed to be constant. S1 exhibits  
Figure 4: A 64-channel, simulated, logarithmic spiral-shaped microphone array.  
Figure 5: Four microphone array. All coordinates are in meters. The three outer microphones form an equilateral triangle while the fourth microphone is at the midpoint of the altitude.  
geometric decay through distance-based attenuation of the complex pressure signal. It is computationally inexpensive and allows us to test models rapidly.  
\[P_{s}(m)=Q\ \frac{e^{-j2\pi f_{rs}/c_{o}}}{4\pi|r_{s}|} \tag{1}\]  
The second type of source, source 2 (S2) (Equation (2)), is also an analytical source: a time-domain sinusoidal plane wave of amplitude \(A\), frequency \(f\), and phase difference \(\phi\). The phase difference for one microphone with respect to another is given in terms of the time delay (\(\Delta t\)) in the signal received by the pair of microphones. The plane wave signal is defined at the microphone location, which is fixed, and thus varies sinusoidally with time. The signal is polluted with noise that can be correlated, uncorrelated noise, or a combination of both:  
\[P(t)=A\sin(2\pi ft+\phi)+\textit{Noise} \tag{2}\]  
where \(\phi=2\pi f\Delta t\). Being a plane wave, S2 does not undergo any geometric or viscous decay. Unlike S1, which is strictly defined in the frequency domain, S2 can be used for both time- and frequency-domain processing.