# Performance Improvement by FRFT-OFDM for Visible Light Communication and Positioning Systems
## 3 Simulation Results and Discussion  
In the previous section, we outlined the component architectures of the VLC and VLP systems and presented the relevant mathematical formulas. In this section, we will simulate the performance of the FRFT-OFDM scheme, compare it to that of the FT-OFDM scheme, and draw conclusions from the results. We assume that all LEDs are mounted on the ceiling, with LED\({}_{1}\)-LED\({}_{4}\) positioned at (\(-1\), \(-1\), \(3\)), (\(-1\), \(1\), \(3\)), (\(1\), \(1\), \(3\)), and (\(1\), \(-1\), \(3\)), respectively. These LEDs are driven by OFDM signals using QAM. The specific parameters of the entire system are detailed in Table 1. To simplify the calculations, we assume that there are no obstacles between the transmitter and receiver and only consider the LOS path. Notably, we set the PD's FOV to 60\({}^{\circ}\) to minimize the impact of NLOS on the BER and ensure robust overall system communication performance. Consequently, the PD will not be placed too close to the surrounding walls; positioning it near the walls may result in some direct light from the LEDs being undetectable, thus adversely affecting system performance. For accurate positioning throughout the room, a larger FOV for the PD would be necessary. We first assume that the receiver's PD is located at (1, 1, 0.8). From Section 2.1 and Equation (2), we know that the FRFT is a periodic function. Specifically, when the transform order \(p\) spans a range of 2, the transform can cover the entire time-frequency domain, transitioning from the time domain to the frequency domain. Based on this, we selected a range of 0.01 \(\leq p\leq\) 1.99 for traversal and used 4-QAM to determine the optimal transform order \(p\) in this environment. We transmitted 20,000 FRFT-OFDM signals, each containing 64 bits, resulting in 12,80,000 bits sent for each \(p\) calculation. Figure 5 shows the relationship between the BER performance and the transform order \(p\) at fixed \(E_{b}/N_{0}\) levels of 20 dB and 30 dB. Notably, the BER trends are similar for different \(p\) values with both \(E_{b}/N_{0}\) settings. From the trend of the BER curves in Figure 5, it is evident that the transform order \(p\) corresponding to relatively low BER values is concentrated around \(p=1\). This suggests that the nonlinear effects of the LED and the limited bandwidth have a minimal impact on the overall BER of the VLC system. In other words, the BER corresponding to the optimal order \(p\) does not differ significantly from the BER at \(p=1\). Specifically, the lowest BER observed at \(E_{b}/N_{0}=30\) dB is \(4.1\times 10^{-5}\) for \(p=1.04\), while the BER for \(p=1\) is approximately \(2.6\times 10^{-4}\). Thus, the optimal order \(p\) in this scenario is determined to be 1.04. Figure 6 presents a comparison of BER performance between the FRFT-OFDM scheme at the optimal order \(p\) and the FT-OFDM scheme across various modulation modes. As illustrated in the figure, both systems exhibit similar BER performance when \(E_{b}/N_{0}\leq 8\) dB, \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Modulation type & QAM \\ Total number of SCs & 128 \\ Number of data SCs & 64 \\ Number of ZP samples & 16 \\ Maximum value of current, \(I_{max}\) & 1 A \\ DC-bias current, \(I_{\text{DC}}\) & 0.55 A \\ Number of LEDs & 4 \\ Room size & \(6\times 6\times 3\) m\({}^{3}\) \\ Height of receiver & 0.8 m \\ Modulating bandwidth of LED, \(B_{m}\) & 4.5 MHz \\ Total bandwidth of OFDM, \(B\) & 5 MHz \\ Semiange at half power, \(\phi_{1/2}\) & 70\({}^{\circ}\) \\ PD’s FOV, \(\theta_{c}\) & 60\({}^{\circ}\) \\ Active area of PD, \(A\) & 1 cm\({}^{2}\) \\ Responsivity of PD, \(R_{\text{PD}}\) & 0.6 A/W \\ TIA gain, \(G_{\text{TIA}}\) & 50 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The system parameters of the simulation setup. regardless of the modulation scheme used. However, in higher-\(E_{b}/N_{0}\) environments, the FRFT-OFDM scheme shows significant improvements over the FT-OFDM scheme.