## 4 The Geometries of the Proposed Transmitters
### SNR Performances of Proposed Transmitters  
Figure 19 shows SNR vs. distance curves for each proposed transmitter. As expected, the SNR decreases with distance for all the transmitter configurations. According to the IEEE 802.15.7 standard [43], the minimum acceptable SNR level is 15 dB and the maximum acceptable BER is \(10^{-6}\). Therefore, we considered a communication range with a minimum SNR level of 15 dB. We observe that the distance achievable with this limit is between 70 and 85 m. It is seen that the array transmitters have improved performance compared to the single-LED transmitter system. The communication ranges for each transmitter are given in Table 7. We find that the honeycomb hexagonal array transmitters offer the longest communication range. Specifically, they achieve a 19% longer range than the one provided by the single-LED transmitters. Figure 19 shows that the array geometries including a larger number of LEDs provide longer communication ranges. Including 25 LEDs, the \(5\times 5\) square and honeycomb \begin{table}
\begin{tabular}{l l} \hline \hline
**Transmitter Type** & **Range** \\ \hline Single-LED & 70.5 m \\ \(3\times 3\) square array & 78.7 m \\ Single hexagonal array & 78.6 m \\ Octagonal array & 72.7 m \\ \(5\times 5\) square array & 80.1 m \\ Honeycomb hexagonal array & 83.8 m \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The communication ranges of the proposed transmitter configurations at a 15 dB signal-to-noise ratio (SNR) threshold. Figure 18: The irradiance distributions at \(y=1\) m with the final values of the design parameters: (**a**) with the single-LED transmitters, (**b**) with the \(3\times 3\) square array transmitters, (**c**) with the single hexagonal array transmitters, (**d**) with the octagonal array transmitters, (**e**) with \(5\times 5\) square array transmitters, and (**f**) with the honeycomb hexagonal array transmitters. hexagonal array transmitters have a higher SNR than the \(3\times 3\) square, single hexagonal, and octagonal arrays that include 9 LEDs. To understand why the honeycomb hexagonal arrays outperformed the \(5\times 5\) square arrays, despite both having the same number of LEDs, we need to examine their geometries closely. Each LED in the array transmitters has a half FOV of \(15^{\circ}\), while the transmitter must still illuminate the same area as a \(30^{\circ}\) half FOV LED. We adjusted the LED orientations to satisfy this requirement. The orientation vector \(\mathbf{O}_{0,0}\) of the center LED \(e_{0,0}\) is aligned with the y-axis. In contrast, the orientation vectors of the surrounding LEDs are angled relative to the y-axis. As a result, the light from these LEDs hits the receiver at steeper angles of irradiance, reducing the DC channel gain \(H(0)\) (refer to Equation (7)). When an LED is closer to the center LED, its orientation vector becomes less angled with the y-axis, resulting in a narrower angle of irradiance. In the honeycomb hexagonal array, the distances between the center LED and adjacent LEDs are shorter than those in the \(5\times 5\) square array, leading to a higher SNR in the honeycomb array. Since there are as few as nine LEDs in the \(3\times 3\) square, single hexagonal, and octagonal array transmitters, the orientation vectors of each array make larger angles with the y-axis. In contrast, the \(5\times 5\) square and honeycomb hexagonal arrays include 25 LEDs, allowing their inner LEDs to have more moderate angles with the y-axis. Moreover, the \(3\times 3\) square arrays, with nine LEDs, performed better than the other two nine-LED arrays due to their advantage in illuminating the region between the transmitters, \(-75\,\mathrm{cm}<\mathrm{x}<75\,\mathrm{cm}\). When \(d_{0}\) decreases below a certain value, the LED arrays tend to fail in illuminating this region.