# A Biologically Inspired Sound Localisation System Using a Silicon Cochlea Pair
## 2 Materials and Methods
### Modelling the Medial Superior Olive Using Coincidence Detection  
In this work, the MSO is modelled by instantaneous correlations between the two CAR-FAC delay lines. At each time \(t\), all the channels from the left CAR-FAC are compared with the right CAR-FAC channels to form a 2-D instantaneous correlogram. \[Corlation_{i,j}(i,j,t)=z_{l}(i,t)\times z_{r}(j,t) \tag{2}\]  
\(Corlation_{i,j}(i,j,t)\) is the instantaneous correlation at time \(t\) between channel \(i\) of the left CAR-FAC_L output \(z_{l}(i,t)\) and channel \(j\) of the right CAR-FAC_R output \(z_{r}(j,t)\). Figure 4: **(A)** Delay of CAR-FAC channels against centre frequency (CF) to a speech and delay of each CAR-FAC channel; **(B)** The CAR-FAC IHC (blue) and the lateral inhibition (red) response to a 500 Hz sine wave on its CF channel (CF corresponds to 500 Hz). Figure 3: CAR-FAC basilar membrane (BM) response to **(A)** a 500 Hz sine tone at six channels along the travelling wave structure and **(B)** 70-channel response at an instant in time. **(C)** a click (a short and sharp sound) at seven channels along the travelling wave structure and **(D)** 70-channel response at 3 different times. **(E)** a 200 Hz and 800 Hz tone at seven channels along the travelling wave structure and **(F)** 70-channel response at an instant in time. The instantaneous correlation is approximated by computing the quantised channel outputs:  
\[\hat{Corr}_{i,j}(i,j,t)=\left\{ \begin{array}{l}1,\,if\,z_{l}(i,t)\times z_{r}(j,t)>0\\ -1,\,if\,z_{l}(i,t)\times z_{r}(j,t)<0\end{array} \right. \tag{3}\]  
By quantising the instantaneous correlation into a binary range, the resource costs of the hardware implementation are significantly reduced. The FPGA implementation of the instantaneous correlations is described in Reference [36]. The device utilisation of the binaural CAR-FAC and the correlogram implementation on the cyclone V starter kit FPGA board is shown in Table 1.  
The quantised instantaneous correlation describes the correlations between two channels. Correlations of the same polarity of the two inputs produce a positive correlation signal, and correlations of the opposite polarity of the two inputs produce a negative anti-correlation signal. Figure 5 illustrates the quantised instantaneous correlation calculation. At time \(t\), channel \(m\) of CAR-FAC_L is compared with all the channels of CAR-FAC_R (Only channel \(m\) to channel \(r\) are shown in Figure 5). The rectangular waves represent the quantised channel output. Within the first \(m\) to \(n\) channels, the phase shift of CAR-FAC_R with respect to channel \(m\) of CAR-FAC_L is smaller than \(\pi/2\) (in-phase), then the correlations between channel \(m\) of CAR-FAC_L and channel \(m\) to \(n\) of CAR-FAC_R according to equation (2) are positive, denoting correlation. As the wave propagates further in the cascaded structure, the CAR-FAC_R phase shift from channel \(n\) to channel \(p\) is between \(\pi/2\) and \(3\pi/2\) with respect to channel \(m\) of CAR-FAC_L (counter phase). For these channels, the correlations between the two CAR-FACs are negative, denoting anticorrelation. When the wave travels down from channel \(p\) to channel \(r\), the CAR-FAC_R phase shift is between \(3\pi/2\) and \(5\pi/2\) (in-phase), showing correlation again, and so forth. By computing the quantised instantaneous cross-correlations, the ITD between CAR-FAC_L and CAR-FAC_R are actually decided by the interaural phase difference (IPD) of the two cochlear channels. In the Results and Comparisons section, the performance of the non-quantised instantaneous correlation (2) and the quantised instantaneous correlation (3) are compared. Figure 6 shows the formations of correlograms from different input signals with different ITDs or IPDs. Figure 6A shows an instantaneous correlogram generated from two 200 Hz sine tones with zero delay at time \(t\). Since the two input signals are identical, the two cochlear outputs from each channel have the same phase. This results in the symmetric pattern along the diagonal.