## 4 The Geometries of the Proposed Transmitters
### Honeycomb Hexagonal Array
In the proposed array transmitter configurations, this relationship is exploited to enhance the SNR by using LEDs with smaller half FOVs. To validate our expectation, we chose two of the proposed configurations and plotted the SNR against distance \(y\) in MATLAB with different values of \(\phi^{\prime}_{\frac{1}{2}}\). The simulation parameters are listed in Figure 14:1  
Figure 16 shows the SNR vs. distance for the \(5\times 5\) square array and the octagonal array with \(d_{0}=130\) and \(\phi^{\prime}_{\frac{1}{2}}=15^{\circ},30^{\circ}\), and \(45^{\circ}\). As expected, we observe that the SNR increases as the half FOV decreases. Therefore, the proposed LED array transmitter configurations are expected to yield a higher SNR than the reference configuration with single-LED transmitters, since the single LEDs have a half FOV of \(\phi_{\frac{1}{2}}=30^{\circ}\), whereas the LEDs in the array transmitter have a narrower half FOV of \(\phi^{\prime}_{\frac{1}{2}}=15^{\circ}\). We expect the SNR to increase with \(d_{0}\) because the orientation vectors of the array make smaller angles with the y-axis as \(d_{0}\) increases (refer to Equations (20), (24), (30), (31), (34), (40) and (41)). Therefore, the light from each LED falls on the receiver with smaller angles of irradiance \(\phi\), which results in higher DC channel gain \(H(0)\), as seen from Equation (7). Since the value of \(d_{0}\) should affect all the array geometries similarly, two of the proposed array transmitter configurations are selected and the SNR is plotted versus distance \(y\) in MATLAB with different values of \(d_{0}\) to validate our expectation. Figure 17 shows the SNR versus distance for the \(5\times 5\) square array and the octagonal array with \(d_{0}=130\) cm, \(160\) cm, and \(190\) cm. We observe that the SNR increases with \(d_{0}\), which validates our expectation. The same result is expected for the other transmitter configurations, as well. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Modulation scheme & OOK \\ Modulation frequency & 1 MHz \\ Receiver area \(A\) & 1 cm\({}^{2}\) \\ Optical filter gain \(T_{s}\) & 1 \\ Optical concentrator gain \(g\) & 10 \\ Transmitted power \(P_{t}\) & 10 W \\ Transmitter half FOV \(\phi_{\frac{1}{2}}\) & 30\({}^{\circ}\) \\ Array transmitter half FOV \(\phi^{\prime}_{\frac{1}{2}}\) & 15\({}^{\circ}\) \\ Receiver responsivity \(\gamma\) & 0.54 A/W \\ Receiver FOV \(\psi_{c}\) & 60\({}^{\circ}\) \\ [PERSON]’s constant \(k\) & 1.3806488 \(\times 10^{-23}\) J/K \\ Absolute temperature \(T_{k}\) & 300 K \\ Fixed capacitance of photodetector & \(1.12\times 10^{-10}\) F/cm\({}^{2}\) \\ per unit area \(\delta\) & 0.562 \\ Noise bandwidth factor \(I_{2}\) & 0.562 \\ \(I_{3}\) & 0.0868 \\ Noise bandwidth \(B\) & 10\({}^{8}\) Hz \\ Open-loop voltage gain \(G\) & 10 \\ FET channel noise factor \(E\) & 1.5 \\ FET transconductance \(g_{m}\) & \(3\times 10^{-2}\) S \\ Electron charge \(q\) & \(1.6\times 10^{-19}\) C \\ Background current \(I_{bg}\) & \(5.1\times 10^{-3}\) A \\ Photodiode dark current \(I_{0}\) & \(10^{-8}\) A \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulation parameters and values. Since a larger value of \(d_{0}\) is required for enhanced system performance, the final values of the design parameter \(d_{0}\) are maximized while ensuring the \(H\) and \(W\) values are never smaller in any array geometry than those of the single-LED transmitters. The final values are determined and listed in Table 5. The irradiance distributions of each configuration at \(y=1\) m with these final values are given in Figure 18, and the \(H\) and \(W\) values for each configuration are given in Table 6. It is seen that we can ensure that a desired area is illuminated by a particular LED array geometry by properly adjusting the value of \(d_{0}\).