# Enhanced Gain Difference Power Allocation for NOMA-Based Visible Light Communications
## 3 Proposed Power Allocation Scheme
### Allocation Principle Formulation  
A block diagram of a MIMO-VLC system with the proposed power allocation scheme is depicted in Figure 2. At the transmitter, the data of all users are superimposed in the power domain according to an allocation strategy and then combined with a DC signal to drive the LEDs. After passing through the VLC channel, the user captures the received signals by using \(J\) PDs. According to MIMO demultiplexing and the SIC procedure, the signals are finally demodulated into useful data for each user. With the dynamic adjustment of the power allocation factors, the transmission rate for the MIMO-VLC system employing NOMA can be enhanced. It is worth noting that a constant channel gain for fixed transmitter and receiver positions can be obtained according to (6). The GRPA scheme ranks the user channel gains and calculates the power allocation coefficients based on the numerical channel gain relationship between adjacent users in the channel gain ranking. As for GRPA, the relationship between the power factors assigned to \(U_{k}\) and \(U_{k+1}\) at \(L_{i}\) can be described by  
\[\mu_{i,k}=\left(\frac{h_{1i,k+1}+h_{2i,k+1}}{h_{1i,1}+h_{2i,1}}\right)^{k+1} \mu_{i,k+1}. \tag{7}\]  
In the single-cell NOMA-VLC scenario, the primary problem is multi-user interference, where the power assigned to the demodulated user is supposed to surpass the total power allocated to the previously demodulated users, in particular at a high SNR. The problem can be alleviated by a modified fixed-power allocation (MFPA) scheme, which relies on the remaining power after the allocation to the previous user rather than the power initially assigned to them. The corresponding allocated power can be described as  
\[P_{k}=\left\{ \begin{aligned} &(1-\alpha)^{k-1}\alpha P_{t},& k=1,2,\ldots,K-1\\ &(1-\alpha)^{k-1}P_{t},& k=K\end{aligned} \right.. \tag{8}\]  
where \(\alpha\) is the fixed power factor of the scheme. Equation (8) is referred to as the residual power principle. In this paper, by combining (7) and (8), an efficient power allocation scheme with enhanced gain difference power allocation (EGDPA) is proposed to further enhance the achievable data rate. The main idea is that the allocation factors are determined by the users' channel gain differences, which can offer higher flexibility than fixed allocation factors and is better suited to the practical needs. Then, the residual allocation principle is employed to further reduce residual interference from previously demodulated users to the intended users. In particular, the power allocation factor for \(U_{k}\) at \(L_{i}\) can be formulated as  
\[\mu_{i,k}=\left\{ \begin{aligned} &\alpha_{i,1},& k=1\\ &\prod\limits_{q=1}^{k-1}(1-\alpha_{i,q})\alpha_{i,k},& k=2,\ldots,K-1\\ &\prod\limits_{q=1}^{K}(1-\alpha_{i,q}),& k=K\end{aligned} \right.. \tag{9}\]  
where \(b_{i,k}\) can be expressed by  
\[b_{i,k}=\left(\frac{h_{1i,1}+h_{2i,1}-h_{1i,k+1}-h_{2i,k+1}}{h_{1i,1}+h_{2i,1 }}\right)^{k}b_{i,k+1}, \tag{10}\]  
and \(\alpha_{i,k}\) is represented as  
\[\alpha_{i,k}=\frac{b_{i,k}}{1+\sum\limits_{k=2}^{K}b_{i,k}}. \tag{11}\]  
Comparing with (7) and (8), we find that the adjacent power allocation factors formulated by (9) exhibit smaller differences, which results in less residual interference from the previous user to the subsequent user in the SIC process. The proposed power allocation algorithm is shown in Algorithm 1. With I LEDs and K users, the computational complexity required for \(b_{i,k}\) can be approximated as \(O(IK)\) based on (10) and lines 2-6 in Algorithm 1. According to (11), \(\alpha_{i,k}\) can be calculated using the results of (10) without additional complexity. Based on (9), the computational complexity for \(\mu_{i,k}\) can be approximated as \(O(IK^{2})\). Therefore, the overall computational complexity of the proposed algorithm can be approximately denoted as \(O(IK^{2})\).