# Enhanced Gain Difference Power Allocation for NOMA-Based Visible Light Communications
## 3 Proposed Power Allocation Scheme
### Allocation Principle Formulation
If all users possess equal power allocation factors, the user with superior channel conditions will attain a higher data rate. Moreover, to enhance fairness among users, the channel allocation factor for users with inferior channel conditions is augmented while the power allocation factor for users with superior channel conditions is reduced. Accordingly, the energy efficiency can be calculated by\[\eta_{EE} =\frac{\sum\limits_{k=1}^{K}R_{k}}{P_{\max}} \tag{25}\] \[=\frac{B}{2P_{\max}}\text{log}_{2}\Bigg{(}\Bigg{(}1+\frac{\left(h_{ K}\mu_{K}\right)^{2}}{\sigma^{2}}\Bigg{)}\prod\limits_{k=1}^{K-1}\Bigg{(}1+ \frac{\left(h_{k}\mu_{k}\right)^{2}}{\sum_{i=k+1}^{K}\left(h_{k}\mu_{i}\right)^ {2}+\sigma^{2}}\Bigg{)}\Bigg{)},\]

where \(P_{\max}\) represents the maximum transmit power of the LED. Hence, considering that \(P_{\max}\) is constant, the problem of maximizing energy efficiency can be reduced to the problem of maximizing achievable rates. ### Error Probability

For simplicity, only two users are considered in the NOMA-based MIMO-VLC system. It is assumed that the symbols of both users are mutually independent and equiprobable when evaluating the error probability. Considering the OOK modulation, the BER for \(U_{1}\) (i.e., distant user) can be formulated as

\[P_{e,1}=\frac{1}{4}\{2Q(\gamma_{1}\mu_{2})+Q(\gamma_{1}(\mu_{2}-2\mu_{1}))+Q( \gamma_{1}(2\mu_{1}+\mu_{2}))\}\, \tag{26}\]

where \(\gamma_{1}=\gamma_{oe}h_{2}/\sigma_{n}\), \(Q(x)\triangleq\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}\exp(-u^{2}/2)du\). Let \(\Theta_{i}\) and \(\overline{\Theta}_{i}\) represent the successful and erroneous demodulation of \(U_{i}\) (\(i=1,2\)), respectively. Thus, the BER for \(U_{2}\) (i.e., near user) can be calculated as

\[P_{e,2} =P(\overline{\Theta}_{2},\Theta_{1})+P(\overline{\Theta}_{2}, \overline{\Theta}_{1}) \tag{27}\] \[=(1-P_{2}(\overline{\Theta}_{1}))P(\overline{\Theta}_{2}|\Theta_ {1})+P(\overline{\Theta}_{2}|\overline{\Theta}_{1})P(\overline{\Theta}_{1}),\]

where \(P(\overline{\Theta}_{2},\Theta_{1})\) represents the joint probability when \(U_{1}\) correctly decodes its signal, whereas \(U_{2}\) has incorrect decoding. Similarly, \(P(\overline{\Theta}_{2},\overline{\Theta}_{1})\) denotes the joint probability when both \(U_{1}\) and \(U_{2}\) obtain the incorrect decoding at the same time. \(P(\overline{\Theta}_{2}|\Theta_{1})\) and \(P(\overline{\Theta}_{2}|\overline{\Theta}_{1})\) are the conditional probabilities as \(U_{2}\) decodes its signal incorrectly on the conditions that the correct and incorrect decoding of \(U_{1}\) are achieved, respectively. Replacing \(\gamma_{1}\) with \(\gamma_{2}\) in (26), the BER for decoding the \(U_{1}\)'s signal at \(U_{2}\) can be given as

\[P_{2}(\overline{\Theta}_{1})=\frac{1}{4}\{2Q(\gamma_{2}\mu_{2})+Q(\gamma_{2}( \mu_{2}-2\mu_{1}))+Q(\gamma_{2}(2\mu_{1}+\mu_{2}))\}, \tag{28}\]

where \(\gamma_{2}=\gamma_{oe}h_{1}/\sigma_{n}\). As for the evaluation of \(P_{e,2}\), the decoding process is initially applied to the far user and subsequently followed by the implementation of SIC. After the successful execution of SIC at \(U_{1}\), the error probability for \(U_{2}\) can be derived as

\[P(\overline{\Theta}_{2}|\Theta_{1})=Q(\gamma_{2}\mu_{1}). \tag{29}\]

When the signal of \(U_{1}\) is incorrectly decoded at \(U_{2}\), the joint error probability of \(U_{2}\) can be derived as

\[P(\overline{\Theta}_{2},\overline{\Theta}_{1}) =\frac{1}{4}\{Q(\gamma_{2}\mu_{2})Q(\gamma_{2}(\mu_{1}+2\mu_{2})+ Q(\gamma_{2}\mu_{2})Q(\gamma_{2}(\mu_{1}-2\mu_{2})\] (30) \[+Q(\gamma_{2}(\mu_{1}-2\mu_{2}))Q(\gamma_{2}(\mu_{2}-2\mu_{1}))+ Q The receiving plane has a height of 0.85 m above the floor, and the cell radius \(R\) is 4 m. Let \(\varphi_{\mathrm{C}}\) and \(\phi_{1/2}\) be fixed to 72\({}^{\circ}\) and 50\({}^{\circ}\), respectively, and the modulation index \(\zeta\) be 0.5. The refractive index \(n\) is 1.5, the transmitted optical power \(P_{o}\) is 10 W, whilst the modulation bandwidth \(B\) is configured to 10 MHz.