# SIC-Free Based Indoor Two-User NOMA-VLCP System
## 2 System Module and Optimization Algorithm
After MRC, the signal received by each PD from user \(i\) can be expressed as [31]:  
\[S_{\text{MRC},i}=\sum_{j=1}^{N_{r}}\left(\gamma_{j}S_{r,i,j}+N_{r,j,i}\right)x_ {i} \tag{7}\]  
where \(N_{r}\) is the number of merged branches, while in the system proposed in this letter, the number of branches is equal to the number of PDs in the receiver, \(\gamma_{j}\) is the gain of the \(j\)-th merged branch, \(x_{i}\) is the merging coefficient of each branch in the MRC, the branch weighting coefficients are determined by the adaptive feedback threshold (AFT) extracted from the corresponding PDs, \(N_{j,r,i}\) is the noise of the \(j\)-th PD, and the noise model can be regarded as Gaussian white noise, expressed as [32]:  
\[\sigma_{N_{j}}^{2}=\sigma_{\text{shot}}^{2}+\sigma_{\text{thermal}}^{2} \tag{8}\]  
The signal-to-noise ratio (SNR) is the ratio of the received signal power to the noise power, where the noise includes shot noise and thermal noise, with \(\sigma_{\text{shot}}^{2}\) and \(\sigma_{\text{thermal}}^{2}\) respectively. The horizontal PD is set as PD\({}_{1}\) and the received optical power used for positioning is estimated based on the signal threshold of each PD. AFT-based multi-PD positioning is achieved by calculating the Received optical power ratio (ROPR), which can be expressed as:  
\[ROPR_{i,j}=\frac{P_{r,i,j}}{P_{r,i,1}},j=2,\cdots,N_{r} \tag{9}\]  
where \(P_{r,i,j}\) represents the power of the optical signal received by the \(j\)-th PD of the user \(i\). The threshold value of the signal received by each PD is determined after AFT. The threshold value of the received signal and the LOS link gain of the visible light channel show a positive correlation. To facilitate the subsequent calculation formula derivation, the expression is set as follows:  
\[ \begin{cases}\delta_{j}=sin(\alpha)cos(\theta_{j})\\ \theta_{j}=sin(\alpha)sin(\theta_{j})\\ \lambda_{j}=[x_{t}-rcos(\alpha)cos(\
u_{j})]\delta_{j}\\ +[Y_{t}-rcos(\alpha)sin(\
u_{j})]\theta_{j}+hcos(\alpha)\end{cases}  \tag{10}\]  
Figure 1: The proposed integrated baseband VLCP system. where \(\alpha\) is the tilt angle of the tilted PD relative to the horizontal plane, \(\
u_{i}\) represents the angle between the center PD and the line connecting the \(i\)-th tilted PD on the x-y axis plane, constructs the auxiliary parameter azimuth angle \(\theta_{j}\). \(X\) and \(Y_{t}\) are the coordinates of the LED, \(r\) represents the distance from the center of the tilted PD to the center of the horizontal PD in the direction parallel to the tilted plane, and \(h\) is the height between the LED and the receiving PD.The receiver consists of a horizontal PD and four tilted PDs, providing five received optical power parameters. Since the transmitter position is fixed, by adjusting the tilt angles of the four tilted PDs, the incident angles (\(\
abla_{i}\)) are different. Combined with Equation (4), the PD received optical power is different, resulting in four received optical power ratio parameters, and the position information is obtained under the premise that the receiver parameters are known [32]. \[\left\{ \begin{aligned} x_{i,j,r}&=\frac{C_{i}- hROPR_{i}}{A_{i}}\\ i,j&=(1,2),(1,4),(3,2),(3,4)\\ y_{i,j,r}&=\frac{C_{j}-hROPR_{j}}{B_{j}}\\ \end{aligned} \right. \tag{11}\]  
Hence, the current position can be expressed as the coordinates of the four tilted PDs:  
\[\left\{ \begin{aligned} x_{r}&=\frac{h(ROPR_{3}- ROPR_{1})}{2\sin(\alpha)}\\ y_{r}&=\frac{h(ROPR_{4}-ROPR_{2})}{2\sin(\alpha)}\\ \end{aligned} \right. \tag{12}\]  
For VLC, to reduce the interference between users, the user \(i\) signal is represented by the Volterra equalizer as:  
\[ \begin{aligned} S_{\text{rvol},i}&=\sum_{m_{1}=1}^{ \text{LN}}\theta_{1}(n)S_{MRC,i}(n-m_{1})+\\ &\sum_{m_{2}=0}^{\text{NLN}}\sum_{m_{3}=0}^{\text{NLN}}\theta_{2 }(n_{1},n_{2})S_{MRC,i}(n-m_{2})S_{MRC,i}(n-m_{3})\\ \end{aligned}  \tag{13}\]  
The length of the linear part of the Volterra equalizer is represented by LN, while NLN represents the length of the nonlinear part.