# Enhanced Gain Difference Power Allocation for NOMA-Based Visible Light Communications
## 2 System Model  
As illustrated in Figure 1, we consider a MIMO-VLC system comprising \(I\) LEDs and \(K\) users. Therein, each user is equipped with \(J\) photodetectors (PDs). The coverage area radius for the cell is \(R\) and \(U_{1}\) is positioned at the center of the cell. We assume that all users are arranged in straight lines, and the distance between the central user and the edge user is denoted as \(r\), while the distance between the central user and \(U_{k}\) is denoted as \(r_{k}\). We define \(L_{i}\) as the \(i\)-th LED transmitter, and \(D_{j}\) as the \(j\)-th PD. In addition, a DC bias \(I_{DC}\) is always added to the signal to obtain the non-negative waveform \(x_{i}\) to drive \(L_{i}\), which can be expressed by  
\[x_{i}=\sum\limits_{k=1}^{K}\sqrt{\mu_{i,k}P_{i}s_{i,k}}+I_{DC}, \tag{1}\]  
where \(P_{t}\) is the electrical power of the emitter, \(\mu_{i,k}\) represents the normalized power allocation factor at the \(i\)-th LED transmitter for \(U_{k}\), and \(s_{i,k}\) denotes the zero-mean OOK modulated signal prepared for \(U_{k}\) at \(L_{i}\). To maintain constant total electrical power, the power allocation factors should satisfy \(\sum\limits_{k=1}^{K}\mu_{i,k}=1\). After VLC channel transmission, the optical signal is captured by the PD at \(U_{k}\) and then converted to an electrical current based on optical-electrical conversion. Since the DC signal does not convey any useful information, it is always eliminated from the received signal before demodulation. Therefore, the received signal for \(U_{k}\) can be given by  
\[y_{k}=\gamma_{\alpha\rho}P_{\phi}\xi H_{k}\mathbf{x}+n_{k}, \tag{2}\]  
Figure 1: Indoor NOMA-based MIMO-VLC system with \(I\) transmitters and \(K\) users, where each receiver is equipped with \(J\) PDs. where \(\gamma_{\sigma e}\) is the optical-electrical responsivity of the PD, \(P_{o}\) represents the output optical power of the emitter, \(\zeta\) denotes the modulation index, the channel matrix for \(U_{k}\) is denoted as \(H_{k}\in\mathbb{C}^{J\times M}\), and the vector \(\mathbf{x}\) represents the transmitted signal vector from all LEDs. Additionally, \(n_{k}\) denotes the additive noise vector with zero mean and variance \(\sigma_{noise}^{2}\), which comprises shot noise and thermal noise [13; 28] and can be expressed by  
\[ \begin{split}\sigma_{noise}^{2}&=\sigma_{shot}^{2} +\sigma_{thermal}^{2}\\ &=\left(2 qI_{bg}I_{2}B+2q\gamma_{\sigma e}P_{o}BH\right)+\left( \frac{8\pi kT}{G}\eta A_{PD}I_{2}B^{2}+\frac{16\pi KTT}{g_{m}}\eta^{2}A_{PD}^{ 2}I_{3}B^{3}\right),\end{split}  \tag{3}\]  
where \(q\) is the electronic charge, \(I_{bg}=5100\)\(\mu\)A represents background noise, and the equivalent bandwidth of noise is denoted as \(B\). \(k\) is the Boltzmann constant, \(G=10\) denotes the open-loop voltage, \(\eta=112\)\(\mathrm{pF}/\mathrm{cm}^{2}\) denotes the input capacitance of the PD, \(\Gamma=1.5\) represents the field-effect transistor (FET) channel noise factor, and \(g_{m}\) captures the FET transconductance [14]. In this paper, we only focus on the LOS component in VLC systems because the power of the NLOS links is relatively lower than that of the LOS components. Considering the Lambertian radiation of LEDs, the LOS channel gain between \(L_{i}\) and \(D_{j}\) for \(U_{k}\) can be formulated by  
\[h_{ji,k}=\frac{(\bar{m}_{0}+1)A_{PD}T_{s}}{2\pi(\bar{\mu}_{j,k})^{2}}\mathrm{ cos}^{\bar{m}_{0}}\left(\varphi_{ji,k}\right)g_{s}\left(\psi_{ji,k}\right)cos( \psi_{ji,k}),\quad 0\leq\psi_{ji,k}\leq\psi_{C}, \tag{4}\]  
where the Lambertian order of the LED is \(\bar{m}_{0}=-1/\log_{2}(\cos(\phi_{1/2}))\) and \(\phi_{1/2}\) denotes the semi-angle of the LED. \(A_{PD}\) captures the area of detection of the PD, and \(d_{ji,k}\) represents the distance between \(L_{i}\) and \(D_{j}\) for \(U_{k}\). \(\varphi_{ji,k}\) and \(\psi_{ji,k}\) are the irradiance angle and the incident angle of the optical link for \(U_{k}\), respectively, while \(\psi_{C}\) is the field of view (FOV).