# Enhanced Gain Difference Power Allocation for NOMA-Based Visible Light Communications
## 3 Proposed Power Allocation Scheme
### Allocation Principle Formulation
It should be noted that (16) is conditioned on the fact that \(U_{k}\) can detect all messages from \(U_{j}\), for \(\forall\ j\ \leq\ k\). The rate at which \(U_{k}\) detects the messages sent to \(U_{j}\) is denoted as \(R_{k\to j}\), and the target rate for \(U_{j}\) is \(\tilde{R}_{j}\). This condition can be formulated as

\[R_{k\to j}= \begin{cases}\frac{\beta}{2}\text{log}_{2}\Bigg{(}1+\frac{\left(h_ {k}\mu_{j}\right)^{2}}{\kappa\sum_{p=1}^{j-1}\left(h_{k}\mu_{p}\right)^{2}+ \sum_{v=j+1}^{k}\left(h_{k}\mu_{v}\right)^{2}+\sigma^{2}}\Bigg{)}\geq\tilde{R} _{j},&j\leq k,\ j\
eq K\\ \frac{\beta}{2}\text{log}_{2}\Bigg{(}1+\frac{\left(h_{k}\mu_{k}\right)^{2}}{ \kappa\sum_{p=1}^{j-1}\left(h_{k}\mu_{p}\right)^{2}+\sigma^{2}}\Bigg{)}\geq \tilde{R}_{j},&j=k=K\end{cases} . \tag{17}\]

If (17) is satisfied, it can be inferred that the perfect SIC in the decoding chain can be achieved. Otherwise, communication interruption will occur at \(U_{k}\). It is assumed that each user has not specified any requirements for the target data rate but instead strives to maximize their communication performance using the allocated power (i.e., \(\tilde{R}_{j}=R_{j}\)). **Lemma 1**. : _The rate at which \(U_{k}\) detects the messages sent to \(U_{j}\) is always higher than the achievable rate of \(U_{j}\)._

Proof. : As for \(j=k\), the rate at which \(U_{k}\) detects the messages sent to \(U_{j}\) equals the achievable rate for \(U_{j}\) based on (16) and (17). As for \(j<k\), according to (16) and (17), \(R_{k\to j}\) can be expressed as

\[R_{k\to j}=\frac{B}{2}\text{log}_{2}\left(1+\frac{\left(h_{k}\mu_{j} \right)^{2}}{\kappa\sum_{p=1}^{j-1}\left(h_{k}\mu_{p}\right)^{2}+\sum_{o=j+1}^ {K}\left(h_{k}\mu_{o}\right)^{2}+\sigma^{2}}\right), \tag{18}\]

and \(R_{j}\) is calculated by

\[R_{j}=\frac{B}{2}\text{log}_{2}\left(1+\frac{\left(h_{j}\mu_{j}\right)^{2}}{ \kappa\sum_{p=1}^{j-1}\left(h_{j}\mu_{p}\right)^{2}+\sum_{o=j+1}^{K}\left(h_{ j}\mu_{o}\right)^{2}+\sigma^{2}}\right). \tag{19}\]

To simplify the subsequent analysis, let \(S_{k\to j}\) and \(S_{j}\) replace \(R_{k\to j}\) and \(R_{j}\) according to (19) and (18), respectively. This can be represented as

\[S_{k\to j}=\frac{\left(h_{k}\mu_{j}\right)^{2}}{\kappa\sum_{p=1}^{j-1}\left( h_{k}\mu_{p}\right)^{2}+\sum_{o=j+1}^{K}\left(h_{k}\mu_{e}\right)^{2}+ \sigma^{2}}, \tag{20}\]

\[S_{j}=\frac{\left(h_{j}\mu_{j}\right)^{2}}{\kappa\sum_{p=1}^{j-1}\left(h_{j} \mu_{p}\right)^{2}+\sum_{o=j+1}^{K}\left(h_{j}\mu_{o}\right)^{2}+\sigma^{2}}. \tag{21}\]

We arrange all users according to (12); thus, we have \(h_{j}\leq h_{k}\). Based on the aforementioned analysis, the comparison of \(R_{k\to j}\) and \(R_{j}\) is comparable to that of \(S_{k\to j}\) and \(S_{j}\). Consequently, the difference between \(S_{k\to j}\) and \(S_{j}\) can be expressed as

\[ \begin{split} S_{k\to j}-S_{j}&=\frac{\left(h_{k} \mu_{j}\right)^{2}}{\kappa\sum_{p=1}^{j-1}\left(h_{k}\mu_{p}\right)^{2}+\sum_{ o=j+1}^{K}\left(h_{k}\mu_{o}\right)^{2}+\sigma^{2}}\\ &-\frac{\left(h_{j}\mu_{j}\right)^{2}}{\kappa\sum_{p=1}^{j-1} \left(h_{j}\mu_{p}\right)^{2}+\sum_{o=j+1}^{K}\left(h_{j}\mu_{o}\right)^{2}+ \sigma^{2}}.\end{split}  \tag{22}\]

Let \(S^{\prime}\) denote (22), which can be further represented by

\[ \begin{split} S^{\prime}=&\left(h_{k}\mu_{j}\right)^ {2}\kappa\sum_{p=1}^{k-1}\left(h_{k}\mu_{p}\right)^{2}+\sum_{o=k+1}^{K}\left( h_{k}\mu_{o}\right)^{2}+\sigma^{2}\\ -&\left(h_{j}\mu_{j}\right)^{2}\kappa\sum_{p=1}^{j-1} \left(h_{j}\mu_{o}\right)^{2}+\sum_{o=j+1}^{K}\left(h_{j}\mu_{o}\right)^{2}+ \sigma^{2}.\end{split}  \tag{23}\]

Therefore, the rate difference can be expressed by

\[S^{\prime}=[\left(h_{k}\mu_{j}\right)^{2}-\left(h_{j}\mu_{j}\right)^{2}]\sigma ^{2}\geq 0. \tag{24}\]

The proof is completed. Consequently, each user can achieve a data rate determined by (16), and the total system rate is a summation of all users' data rates. Furthermore, it can be observed from (16) that \(R_{k}\) can be increased with the value of \(h_{k}\) for a given power allocation factor.