# Performance Improvement by FRFT-OFDM for Visible Light Communication and Positioning Systems
## 2 Principle
### System Model
This paper considers a VLC and VLP system. As illustrated in Figure 3, four LEDs are mounted on the ceiling of the room. They are symmetrically positioned relative to the center of the ceiling and serve both communication and positioning functions. To enhance spectrum utilization, we multiplex the signals from the four LEDs in the fractional Fourier domain, with the subcarriers corresponding to LED\({}_{1}\)-LED\({}_{4}\) illustrated in Figure 4. The receiver, equipped with a PD, is mounted on the observation plane to receive light signals from the LEDs. In this scenario, the indoor wireless optical communication VLC channel comprises both line-of-sight (LOS) and diffuse or non-line-of-sight (NLOS) paths, which can be expressed as  
\[h_{\rm VLC}(t)\ =\ h_{\rm LOS}(t)+h_{\rm NLOS}(t) \tag{19}\]  
Here, we consider only the LOS component. Figure 4: Subcarrier allocation in the fractional Fourier domain for 4-LED VLC and VLP systems. Figure 3: VLC and VLP link with LOS and NLOS paths. For the LOS path, the impulse response is given by  
\[h_{\text{LOS}}(t)\ =\ \frac{A}{d^{2}}R(\phi)\cos(\theta)\delta(t-\frac{d}{c}) \tag{20}\]  
where \(\phi\) and \(\theta\) denote the radiation angle of the transmitter and the incidence angle of the receiver, respectively. \(A\) represents the surface area of the PD, \(d\) is the LOS path length, \(c\) is the speed of light, and \(R(\phi)\) is the radiation intensity of the light source. Considering the optical characteristics of the LED, the angular distribution of the light source's radiation intensity follows the Lambertian radiation intensity model, characterized by the following distribution:  
\[R(\phi)\ =\ \frac{m+1}{2\pi}\cos^{m}(\phi) \tag{21}\]  
where \(m\) denotes the Lambertian emission order, which is related to the semiangle at half power \(\phi_{1/2}\) of the LED light emission and is defined as  
\[m\ =\ \frac{-\ln(2)}{\ln(\cos(\phi_{1/2}))} \tag{22}\]  
For the VLP module, the optical power \(P_{rx}\) received by the PD at the receiver can be used to calculate the channel gain \(G\). The channel gain can be expressed as  
\[G\ =\ \frac{P_{rx}}{P_{tx}} \tag{23}\]  
where \(P_{tx}\) represents the transmitter's signal power. It is assumed that only the LOS link component in the VLC channel is considered, and the normal vector of the PD at the receiver is perpendicular to the horizontal plane. When the incidence angle of the PD is smaller than its field of view (FOV), the channel gain can be expressed as  
\[ \begin{split} G&=\frac{A}{d^{2}}R(\phi)\cos(\theta )\\ &=\frac{(m+1)A}{2\pi d^{2}}\cos^{m}(\phi)\cos(\theta)\end{split}  \tag{24}\]  
From this, we can derive the estimated distance \(d_{est}\) from the LED to the PD as  
\[d_{est}\ =\ \sqrt{\frac{(m+1)AP_{tx}}{2\pi P_{rx}}\cos^{m}(\phi)\cos(\theta)} \tag{25}\]  
We calculate the estimated distance from each LED to the PD individually. With the specific positions of each LED and the corresponding estimated distances to the PD, we can determine the PD's position at the receiver using the least squares method. For the VLC system, the received optical signal is converted back into the electrical format using an optical Rx, which is a PD and a trans-impedance amplifier (TIA). The signal at the receiver can be expressed as  
\[y(t)\ =\ G_{\text{TIA}}R_{\text{PD}}(x_{out}(t)\otimes h_{\text{VLC}}(t)+n(t)) \tag{26}\]  
where \(n(t)\) represents the additive white Gaussian noise (AWGN) generated by the PD, \(R_{\text{PD}}\) denotes the response of the PD, and \(G_{\text{TIA}}\) is the gain of the TIA. The output \(y(t)\) is then processed by applying the RPFRFT following analog-to-digital (A/D) conversion and ZP removal. This output is merged into a complex bit stream, and the final data are obtained through QAM demodulation. The resulting data are compared bit by bit with the original data from the transmitter to calculate the bit error rate, thereby validating the effectiveness of the FRFT within this system configuration.