# Performance Analysis of Cooperative Non-Orthogonal Multiple Access in Visible Light Communication
## 2 System Model
### Cooperative PD-NOMA
#### 2.3.2 Cooperative Phase  
In this phase, the users cooperate with each other to improve the link reliability as shown in Figure 3 by dedicating a certain time slot for each user that acts as a relay. All users located far away from the main LED have another PD facing them that acts a relay. While all users located near the main LED source that act as relays are equipped with an LED to transmit in the cooperative phase. We need \((M-1)\) PDs and \((M-1)\) LEDs to achieve user cooperation, exactly similar in the specification as the main LED and PDs used in the direct transmission phase. The cooperative VLC channel model in the case of two users is shown in Figure 4. The DC channel gain between \(U_{2}\) and \(U_{1}\) is expressed as:  
\[g_{2,1}=\left\{ \begin{array}{ll}\frac{A_{k}}{d_{2,1}^{2}}R_{o}(\phi_{2,1})T_ {s}(\psi_{2,1})g(\psi_{2,1})\ \cos\ (\psi_{2,1}),&0\leq\psi_{2,1}\leq\psi_{c}\\ 0,&\psi_{2,1}>\psi_{c}\end{array} \right. \tag{17}\]  
where \(d_{2,1}\) donates the Euclidean distance between \(U_{2}\) and \(U_{1}\). The angle of incidence with respect to the axis normal to the \(U_{2}\) LED surface and the angle of irradiance with respect to the axis normal to the \(U_{1}\) PD surface are denoted by \(\psi_{2,1}\) and \(\phi_{2,1}\), respectively. Figure 4: Cooperative VLC system model with two users. Figure 3: Cooperative PD-NOMA system model. During the cooperative phase, \((M-1)\) time slots are needed where each user has a dedicated time slot to transmit, except for the user farthest from the main LED. Using time slots allows all users except the nearest to the main LED to take advantage of the cooperative phase. Otherwise, the farthest user can only improve its signal reception in the case of using only one time slot in the cooperative phase. The \(M\)-th user transmits in the first time slot a combination of \((M-1)\) messages with power allocation coefficients \(q_{M}\), where \(\sum_{l=1}^{M-1}q_{M,l}^{2}=1\). The received signal at the \(k\)-th user for \(k<M\) is given as in [16] by:  
\[y_{2,k}=\sum_{l=1}^{M-1}\sqrt{P_{\mathit{ele}}}g_{M,k}q_{M,l}s_{l}+n_{2,k} \tag{18}\]  
where \(g_{M,k}\) denotes the channel gain between user \(M\) and the \(k\)-th user. Maximum ratio combining (MRC) is used to combine the received signal from both phases as shown in Table 1.  
We should take into consideration that increasing the number of users leads to increasing the number of time slots used in the cooperative phase which decreasing the transmission rate. Therefore, if the number of users exceeds the limitation of the system, communication between users in a realistic scenario may be lost. The SINR for the \((M-1)\)-th user to decode the \(k\)-th user's message signal for \(k<(M-1)\) is expressed as in [16] by:  
\[SINR_{M-1,k}=\frac{\left(h_{M-1}a_{k}\right)^{2}}{\left(h_{M-1}\right)^{2}\sum _{l=k+1}^{M}a_{l}^{2}+\frac{1}{\rho_{k}}}+\frac{\left(g_{M,M-1}q_{M,k}\right)^ {2}}{\left(g_{M,M-1}\right)^{2}\sum_{l=k+1}^{M-1}q_{M,l}^{2}+\frac{1}{\rho_{k}}}. \tag{19}\]  
The SINR for the \((M-1)\)-th user to decode its own message is expressed as:  
\[SINR_{M-1,M-1}=\frac{\left(h_{M-1}a_{M-1}\right)^{2}}{\left(h_{M-1}a_{M} \right)^{2}+\frac{1}{\rho_{k}}}+\rho_{k}(g_{M,M-1}q_{M,M-1})^{2}. \tag{20}\]