# Resource Allocation in Downlink VLC-NOMA Systems for Factory Automation Scenario
## 2 System Model
### The VLC-NOMA System  
In this subsection, a downlink VLC-NOMA system is described for the BER performance. The system is made up of a single LED as a transmitter, and two users with photodiodes (PDs) as receivers in an NOMA user pair in an indoor environment. The modulation technique adopts OOK, and the OOK signals for NU and FU are superposed for transmission from the transmitter. The line-of-sight VLC channel model was averaged between the LED and users [25]. The channel model is expressed as follows:  
\[h=P\frac{(m+1)cos^{m}(\Phi)}{2\pi d^{2}}AT(\phi)g(\phi)cos(\phi), \tag{1}\]  
\[m=\frac{-ln(2)}{ln(cos(\Phi_{1/2}))}. \tag{2}\]  
The notations of the above equations are described in Table 1. The channel gain between the LED and a user can be calculated by (1). \begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **Definition** \\ \hline \(m\) & Lambertian emission order \\ \(A\) & area of PD \\ \(T\) & optical filter gain \\ \(\Phi\) & angle of irradiance \\ \(\phi\) & angle of incidence \\ \(\Phi_{1/2}\) & Semi-angle of LED \\ \(P_{f}\) & transmission Power ratio for FU \\ \(p_{n}\) & transmission Power ratio for NU \\ \(h_{f}\) & channel gain between FU and LED \\ \(h_{n}\) & channel gain between NU and LED \\ \(y_{f}\) & received signal for FU \\ \(y_{n}\) & received signal for NU \\ \(x_{f}\) & transmitted signal for FU \\ \(x_{n}\) & transmitted signal for NU \\ \(erfc\) & error function \\ \(E_{b}\) & energy per bit \\ \(N_{0}\) & noise power \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Table for notations. NOMA superimposes and transmits signals to users. As described in Figure 1, the system is made up of one LED as a transmitter and two users, including NU and FU. An equation for it is as follows:  
\[y=h(x_{n}+p_{f}x_{f})+n_{o}. \tag{3}\]  
As the system superimposes signals, NU signal \(x_{n}\) and FU signal \(x_{f}\) are superimposed. The signal for NU is regarded as interference for FU. Therefore, the signal \(y_{f}\) is as follows:  
\[y_{f}=h_{f}p_{f}x_{f}+\underbrace{h_{f}p_{n}x_{n}}_{\text{Interference}}+n_{o}. \tag{4}\]  
The signal \(y_{n}\) for NU after SIC is as follows:  
\[y_{n}=h_{n}p_{n}x_{n}+n_{o}. \tag{5}\]  
As the signal for FU is canceled by SIC, only the other signal for NU is residual and then decoded by NU. However, in cases where SIC fails, decoding the signal is hard for NU. Therefore, the success ratio of decoding signals to FU affects the success of SIC. To calculate the error mentioned above, the BER equations for both users are going to be dealt with. There are four cases for bits of NU and FU using OOK in the downlink VLC-NOMA system. \[p(0,0)=\frac{1}{4},\quad p(0,1)=\frac{1}{4},\quad p(1,0)=\frac{1}{4},\quad p(1,1)=\frac{1}{4}, \tag{6}\]  
where \(p(x,y)\) denotes the probability of bits, \(x\) for FU, and \(y\) for NU. The constellation for the received signal is described in Figure 2. We assume \(p_{n}+p_{f}=1\). In the case of received signals, the constellation of the receiver is affected by channel gain \(h\). Therefore, the larger \(h\) is, the lower BER is. The threshold for signals for FU is \(\frac{hE_{h}}{2}\), because FU should decode the signal regardless of the existence of the signal for NU, as shown in Figure 2. This means that irrespective of whether the bit for NU is 0 or 1, the range of signals for FU starts at the same point, \(\frac{hE_{h}}{2}\). Figure 1: Downlink VLC-NOMA system. Therefore, the BER for FU, where \(x\) is 1 and \(y\) is 0, is as below:  
\[p(e|1,0)=\frac{1}{\sqrt{\pi N_{0}}}\int_{-\infty}^{\frac{\frac{k_{b}\sqrt{E_{b}}} {2}}{2}}e^{\frac{-(y-r_{b}\sqrt{E_{b}})^{2}}{N_{0}}}dy. \tag{7}\]  
\(E_{b}\) means energy per bit; \(N_{0}\) is noise power; and as the bit for FU is 1 and the bit for NU is 0, the signal for (7) will be converged at \(P_{f}h\sqrt{EB}\) in the constellation shown in Figure 2. The reason why \(p_{n}\) was not expressed in (7) is that the bit 0 for NU means no power in OOK. When the bit is 0, no power is allocated for transmitting the signal.