# Performance Analysis of Cooperative Non-Orthogonal Multiple Access in Visible Light Communication
## 2 System Model
### VLC Channel Model  
Modeling the VLC channel in this paper, as shown in Figure 1, takes into account only the line of sight (LOS). Although VLC channel has a diffusive component, it can be neglected because the reflected signal has very low energy compared to that of the LOS signal [17]. The direct current (DC) channel gain between user \(U_{k}\) and the LED is given as in [18] by:  
\[h_{k}=\left\{\begin{array}{ll}\frac{A_{k}}{d_{k}^{2}}R_{o}(\phi_{k})T_{s}(\psi _{k})g(\psi_{k})\ \cos\ (\psi_{k}),&0\leq\psi_{k}\leq\psi_{c}\\ 0,&\psi_{k}>\psi_{c}\end{array}\right. \tag{1}\]  
where \(k=1,2,\ldots,M,A_{k}\) denotes the detection area of the PD, \(d_{k}\) donates the Euclidean distance between the LED and the \(k\)-th user. The PD field of view (FOV) is donated by \(\psi_{c}\) while the angle of incidence with respect to the axis normal to the transmitter surface and the angle of irradiance with respect to the axis normal to the receiver surface are denoted by \(\psi_{k}\) and \(\phi_{k}\), respectively. \(T_{s}(\psi_{k})\) represents the gain of the optical filter used at the receiver and \(g(\psi_{k})\) is the gain of the optical concentrator, which is given as in [18] by:  
\[g(\psi_{k})=\left\{\begin{array}{ll}\frac{n^{2}}{\sin^{2}(\
abla_{k})},&0 \leq\psi_{k}\leq\psi_{c}\\ 0,&\psi_{k}>\psi_{c}\end{array}\right. \tag{2}\]  
where \(n\) is the corresponding reflective index of the optical concentrator. Furthermore, \(R_{o}(\phi_{k})\) is the Lambertian radiant intensity of the LEDs, represented as:  
\[R_{o}(\phi_{k})=\frac{m+1}{2\pi}\cos^{m}(\phi_{k}) \tag{3}\]  
where \(m\) is the order of Lambertian emission defined as:  
\[m=-\frac{\ln\ (2)}{\ln\ (\cos\ (\phi_{1/2}))} \tag{4}\]  
where \(\phi_{1/2}\) is the semi-angle at half power of an LED. The noise generated at the PDs has Gaussian distribution of zero mean and variance:  
\[\sigma_{n_{k}}^{2}=\sigma_{sh_{k}}^{2}+\sigma_{th_{k}}^{2} \tag{5}\]  
where \(\sigma_{sh_{k}}^{2}\) and \(\sigma_{th_{k}}^{2}\) are the variances of the shot noise and thermal noise, respectively. The shot noise variance at the \(k\)-th receiver is determined as in [19] by:  
\[\sigma_{sh_{k}}^{2}=2 qR_{p}P_{r_{k}}B+2 qI_{bg}I_{2} \tag{6}\]  
where \(q\) is the electronic charge, \(R_{p}\) is the photodetector responsivity, \(P_{r_{k}}\) is the received optical power for user \(k\), \(B\) is the equivalent bandwidth, \(I_{bg}\) is the photocurrent due to background radiation and \(I_{2}\) is the noise bandwidth factor.  
The thermal noise is generated within the transimpedance receiver circuit. If the noise effect from the gate leakage current is neglected, the thermal variance is expressed as in [19] by:  
\[\sigma_{th_{k}}^{2}=\frac{8\pi\mathrm{x}\mathrm{T}_{k}}{G_{ol}}C_{pd}AI_{2}B^ {2}+\frac{16\pi^{2}kT_{k}\Gamma}{g_{m}}C_{pd}{}^{2}A^{2}I_{3}B^{3} \tag{7}\]  
which consists of feedback-resistor noise, and FET channel noise, where \(k\) is the Boltzmann's constant, \(T_{k}\) is absolute temperature, \(G_{ol}\) is the open-loop voltage gain, \(C_{pd}\) is the fixed capacitance of photodetector per unit area, \(\Gamma\) is the FET channel noise factor, \(g_{m}\) is the FET transconductance and \(I_{3}=0.0868\) is a weighting function that depends on the input optical pulse shape [19].