# Performance Improvement by FRFT-OFDM for Visible Light Communication and Positioning Systems
## 3 Simulation Results and Discussion
For instance, with 4-QAM at \(E_{b}/N_{0}=30\) dB, the BERs for the FRFT-OFDM scheme and FT-OFDM scheme are \(4.1\times 10^{-5}\) and \(2.6\times 10^{-4}\), respectively. In the case of 16-QAM, although both systems experience degraded BER performance due to the increased number of modulation bits, the FRFT-OFDM scheme still demonstrates a notable advantage in BER performance compared to the FT-OFDM scheme in higher-\(E_{b}/N_{0}\) scenarios. In the positioning system, we evaluated the localization error for both the FRFT-OFDM scheme and the FT-OFDM scheme at the optimal order \(p~{}=~{}1.04\) with the receiver positioned at (1, 1, 0.8). As shown in Figure 7, the FRFT-OFDM scheme demonstrates a significant reduction in localization error compared to the FT-OFDM scheme in higher-\(E_{b}/N_{0}\) environments. Notably, when \(E_{b}/N_{0}\geq 20\) dB, the decline in localization errors for both the FRFT-OFDM scheme and the FT-OFDM scheme becomes more gradual as \(E_{b}/N_{0}\) increases, leading to a negligible effect of \(E_{b}/N_{0}\) on localization error beyond a certain point. Specifically, at \(E_{b}/N_{0}~{}=~{}20\) dB, the localization errors are 6.6 cm for the FRFT-OFDM scheme and 8 cm for the FT-OFDM scheme. This results in a 1.4 cm improvement in positioning accuracy for the FRFT-OFDM scheme over its FT counterpart. Figure 5: BER performance comparison at various FRFT orders \(p\). Figure 6: BER performance against \(E_{b}/N_{0}\) for FRFT-OFDM and FT-OFDM schemes using 4-QAM and 16-QAM. In addition, we investigated the impact of varying the number of lights on localization error at a fixed receiving location. The four-LED positioning scheme was modified to use three LEDs positioned symmetrically around the ceiling's center: (0, 1, 3), (\(-\sqrt{3}/2\), \(-1/2\), 3), and (\(\sqrt{3}/2\), \(-1/2\), 3). As shown in Figure 7, at the optimal order, the three-LED FRFT-OFDM scheme shows minimal differences compared to the four-LED scheme. This indicates that, when the number of LEDs is sufficient, further increasing the number of LEDs has a limited impact on the positioning accuracy of the VLP system, as each LED's position estimation for the receiver is primarily influenced by its position relative to the receiver. Additionally, as shown in Figure 7, both the three-LED and four-LED FRFT-OFDM schemes perform significantly better than their corresponding FT-OFDM schemes. Both the three-LED and four-LED FRFT-OFDM schemes achieve optimization over 1 cm compared to the FT-OFDM scheme when \(E_{b}/N_{0}>20\) dB. To investigate the positioning accuracy of the FRFT-OFDM scheme at various locations in the room and to compare it with the FT-OFDM scheme, we selected a PD with a FOV of \(70^{\circ}\). This ensures that the angle of incidence on the PD remains within its FOV, regardless of the receiver's position within the observation plane, allowing it to capture all direct light emitted by the LEDs. The relationship between localization error and the PD's location is depicted in Figure 8. As the receiver approaches a wall, the localization error increases significantly. This is because the LED is a Lambertian transmitter, and its radiation characteristics are typically angle-dependent. Due to the effect of the semiangle at half power, when the receiver is near the wall, the angle between the receiver and the more distant LED increases, leading to a significant reduction in the power received from that LED and, consequently, a higher localization error. As shown in Figure 8, the localization error can reach up to 17 cm in the four corners of the room. Conversely, when the receiver is positioned at the center of the room, the localization error approaches 0, which is attributed to the symmetrical arrangement of the positioning LEDs relative to the center of the ceiling. As shown in Figure 8, a circular area with a radius of 2.8 m from the room's center ensures that the localization error remains below 12 cm, while a smaller circle with a radius of 2 m guarantees that the error is less than 8 cm.