# Performance Analysis of Cooperative Non-Orthogonal Multiple Access in Visible Light Communication
## 2 System Model
### Cooperative PD-NOMA
#### 2.3.2 Cooperative Phase
\begin{table}
\begin{tabular}{c l l l} \hline \hline
**Time Slot** & & **Cooperative Phase Process** & \\ \hline \multirow{4}{*}{1} & \(U_{M}\) transmits messages & \(U_{M-1}\) combines the received signals from light-emitting diode and \(U_{M}\) using the maximum maximum ratio combining technique. & \(U_{M-1}\) decodes the maximum ratio combining signal to extract \(U_{1}\), \(U_{2}\), \(\dots\), \(U_{M-1}\), messages using successive interference cancellation. \\ \hline \multirow{4}{*}{2} & \(U_{M-1}\) transmits messages & \(U_{M-2}\) combines the received signals from light-emitting diode, \(U_{M}\) and \(U_{M-1}\), using the maximum ratio combining technique. & \(U_{M-2}\) decodes the maximum ratio combining signal to extract \(U_{1}\), \(U_{2}\), \(\dots\), \(U_{M-2}\), messages using successive interference cancellation. \\ \hline \multirow{4}{*}{2} & \(\bullet\) & \(\bullet\) & \(\bullet\) \\ \cline{2-4}  & \(\bullet\) & \(\bullet\) & \(\bullet\) \\ \cline{1-1} \cline{2-4}  & \(\bullet\) & \(U_{1}\) combines the received signals form light-emitting diode, \(U_{M}\), \(\dots\), \(U_{2}\), using maximum ratio combining technique. & \(U_{1}\) decodes its own message. \\ \hline \multirow{4}{*}{\(M-1\)} & \(U_{2}\) transmits message intended for \(U_{1}\). & \(U_{1}\) combines the received signals form light-emitting diode, \(U_{M}\), \(\dots\), \(U_{2}\), using maximum ratio combining technique. & \(U_{1}\) decodes its own message. \\ \cline{1-1} \cline{2-4}  & \(U_{1}\) decodes its own message. & & \\ \cline{1-1} \cline{2-4}  & \(U_{1}\) decodes its own message. & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Time slot analysis of the cooperative phase. The \((M-i+1)\)-th user transmits in the \(i\)-th time slot a combination of \((M-i)\) messages where \(1\leq i\leq(M-1)\), with power allocation coefficients \(q_{M-i+1,l}\), where \(\sum_{l=1}^{M-i}q_{M-i+1,l}^{2}=1\). The received signal at the \(k\)-th user for \(k<(M-i+1)\) is given by:  
\[y_{2,k}=\sum_{l=1}^{M-i}\sqrt{P_{\textit{elec}}}\;g_{M-i+1,l}q_{M-i+1,l}s_{l}+ n_{i+1,k}. \tag{21}\]  
The SINR for the \((M-i)\)-th user to decode the \(k\)-th user's message signal for \(1\leq k<(M-i)\) is expressed as in [16] by:  
\[\textit{SINR}_{M-i,k}=\frac{\left(h_{M-i}a_{k}\right)^{2}}{\left(h_{M-i} \right)^{2}\sum_{l=k+1}^{M}a_{l}^{2}+\frac{1}{\rho_{k}}}+\sum_{j=1}^{i}\frac{ \left(g_{M-i+1,M-i}q_{M-j+1,k}\right)^{2}}{\left(g_{M-j+1,M-i}\right)^{2} \sum_{l=k+1}^{M-j}q_{M-j+1,l}^{2}+\frac{1}{\rho_{k}}}. \tag{22}\]  
The SINR for the \((M-i)\)-th to decode its own message is expressed as:  
\[\textit{SINR}_{M-i,M-i}=\frac{\left(h_{M-i}a_{M-i}\right)^{2}}{ \left(h_{M-i}\right)^{2}\sum_{l=k+1}^{M}a_{l}^{2}+\frac{1}{\rho_{k}}}+\sum_{j =1}^{i-1}\frac{\left(g_{M-j+1,M-i}q_{M-j+1,M-i}\right)^{2}}{\left(g_{M-j+1,M- i}\right)^{2}\sum_{l=M-i+1}^{M-j}q_{M-j+1,l}^{2}+\frac{1}{\rho_{k}}} \tag{23}\]  
Finally, we can see from Equation (23) that the second and third terms represent the effect of cooperative PD-NOMA for enhancing the link reliability. These two terms are responsible for increasing the \(k\)-th user's SINR.