## 2 Materials and Methods
### Binaural Model  
A binaural model inspired by the work presented in [18; 19] is used to compute the temporal and frequency-dependent level differences between the signals arriving at the left and right ears of a listener. Both the temporal and spectral feature spaces provide directionally-dependent cues, produced by path differences between ears and acoustic shadowing produced by the presence of the head, which allow the human auditory system to localise a sound source in an environment [6; 20]. These directionally-dependent feature spaces are used in this study to produce a feature vector that can be analysed by an NN to estimate the direction of arrival. Prior to running the analysis of the binaural signals, the signal vectors being analysed are zero-padded by 2000 samples accounting for signal delay introduced by the application of a gammatone filter bank. This ensures that no part of the signal is lost when dealing with small windows of sound, where the filter delay would push the signal outside of the represented sample range. The zero-padded signals are then passed through a bank of 64 gammatone filters spaced equally from 80 Hz to 22 kHz using the equivalent rectangular bandwidth scale. The gammatone filter implementation in [PERSON]'s 'Auditory Toolbox' [21] was used in this study. The output of the cochlea is then approximated using the cochleagram function in [22] with a window size of six samples and an overlap of one sample; this produces an \(F\times N\) map of auditory nerve firing rates across time-frequency units, where \(N\) is the number of time samples and \(F\) is the number of gammatone filters. The cochleagram is calculated as:  
\[x_{l}(f,n)=y_{l}(f,\tau)*y_{l}(f,\tau)^{\top} \tag{1}\]  
where \(x_{l}(f,n)\) is the cochleagram output for the left channel for gammatone filter \(f\) at frame number \(n\), \(y_{l}(f,\tau)\) is the filtered left channel of audio at gammatone filter \(f\) and time frame \(\tau\), which is six samples in length and \((. )^{\top}\) signifies vector transposition [22]. The cochleagram was used to extract the features as opposed to extracting directly from the gammatone filters, as it was found to produce more accurate results when passed to the NN. The interaural cues are then computed across the whole cochleagram producing a single set of interaural cues for each binaural signal being analysed. The first of these interaural cues is the interaural cross-correlation (IACC) function, which is computed for each frequency band as the cross-correlation between the whole approximated cochlea output \(x_{l}\) and \(x_{r}\) for the left and right channel, respectively,with a maximum lag of \(\pm 1.1\) ms. The maximum lag of \(\pm 1.1\) ms was chosen based on the maximum time delays suggested by [PERSON] et al. for their binaural model proposed in [18]. The cross-correlation function is then normalised by,  
\[IACC=\frac{xc_{f}}{x_{l,f}x_{l,f}^{\top}x_{r,f}x_{r,f}^{\top}} \tag{2}\]  
where \(xc_{f}\) is the cross-correlation between the left and right approximated cochlea outputs for gammatone filter \(f\). The \(IACC\) is then averaged across the 64 gammatone filters, producing the temporal feature space for the analysed signal. The maximum peak in the IACC function represents the signal delay between the left and right ear. The decision to use the entire IACC function as opposed to the ITD was based on the findings presented in [15], which suggested that features within the IACC function, such as the relationship between the main peak and any side bands, varied with azimuthal direction of arrival. The ILD is then calculated from the cochleagram output in decibels as the loudness ratio between the two ears for each gammatone filter \(f\) such that,  
\[ILD_{f}=10*log_{10}\left(\frac{\sum_{t=1}^{T}xl_{f,t}}{\sum_{t=1}^{T}x_{fr,t}} \right)dB \tag{3}\]  
where \(xl_{f,t}\) and \(xr_{f,t}\) are the approximated cochlea output for gammatone filter \(f\) for signal \(x\), for the left (\(l\)) and right (\(r\)) ear at time window \(t\), and \(T\) is the total number of time windows.