## 4 The Geometries of the Proposed Transmitters
### SNR Performances of Proposed Transmitters
The \(3\times 3\) square array can sustain a larger \(d_{0}\) before failing to illuminate the middle region because, as seen in Figure 5, the \(3\times 3\) square array has six LEDs, three from each transmitter, aimed towards the middle region. Thus, it can illuminate the desired area with a larger value of \(d_{0}\), which results in less steep orientation vectors, causing this configuration to perform better than the single hexagonal and octagonal arrays. Finally, the single hexagonal array outperformed the octagonal array because its elements are closer together near the center, again allowing for less angled orientation vectors. Initially, the receiver was misaligned horizontally by 75 cm from each transmitter, Tx1 and Tx2, with zero vertical misalignment. To assess the impact of misalignment on system performance across the six proposed configurations, we plotted the SNR against distance  
Figure 19: An SNR vs. distance plot for the proposed transmitters. with the receiver positioned at \(\mathrm{P}(350\,\mathrm{cm},y,30\,\mathrm{cm})\). After this adjustment, the misalignment of the receiver was as follows:  
* **From Tx1:*
* Horizontal: 425 cm
* Vertical: 30 cm
* **From Tx2:*
* Horizontal: 275 cm
* Vertical: 30 cm  
As expected, Figure 20 shows that all of the six configurations perform worse when the misalignment is more severe. Nevertheless, the array transmitter configurations still outperform the single-LED transmitters. Although we expected poorer system performance with more severe misalignment, we do not anticipate any performance change in the long range. This is because the vectors from Tx1 and Tx2 to the receiver will deviate by very small angles when the distance is large, making the receiver appear to the transmitters as if its location did not change. The dominant factor limiting system performance is the long distance between the transmitters and the receiver. Figure 21 validates our expectation, showing that the results with the new misalignment conditions are almost identical to the results with the initial misalignment conditions. The BER of the VLC system is computed using Equations (16) and (17), and it is plotted against the SNR, as depicted in Figure 22. It is observed that the BER becomes  
Figure 21: SNR vs. distance for the proposed transmitters in the long range: (**a**) with the initial receiver position and (**b**) with the receiver positioned at \(\mathrm{P}(350\,\mathrm{cm},y,30\,\mathrm{cm})\). Figure 20: SNR vs. distance for the proposed transmitters in the short range: (**a**) with the initial receiver position and (**b**) with the receiver positioned at \(\mathrm{P}(350\,\mathrm{cm},y,30\,\mathrm{cm})\). \(10^{-6}\) at 13.2 dB SNR. This result suggests that the system achieves a BER lower than the maximum acceptable level specified in the IEEE 802.15.7 standard [43] at 15 dB SNR, which was chosen as the minimum SNR threshold for determining the communication ranges. In Figure 23, the BER for the six proposed transmitter configurations is plotted against distance. As expected, the BER increases with distance for all transmitter configurations. The distances at which the BER reaches \(10^{-6}\) are given in Table 8. Notably, the honeycomb hexagonal transmitters achieve a 19% improvement compared to the single-LED transmitters. To assess the design's ease of implementation in the automotive industry, we compare the size of our transmitter designs to that of Samsung's PixCell LED module, shown in Figure 2, as this module has a light-emitting surface that is \(1/16\) the size of a traditional one. When we decrease the value of \(s\) while keeping \(d_{0}\) constant, we expect more angled orientation vectors, though we do not anticipate significant deterioration in system performance. Figure 23: A bit error rate (BER) vs. distance plot for the proposed transmitters. Figure 22: A BER vs. SNR plot for the communication link. To validate this prediction, simulations were conducted with \(s\) decreased from 1 cm to 1 mm.