# Sum Rate Optimization Scheme of UAV-Assisted NOMA under Hardware Impairments
## 2 System Model and Problem Formulation
The received signal in the UAV can be expressed as:  
\[y_{r}=h_{sr}(\sum_{n=1}^{N}\sqrt{p_{n}}\theta_{n}+\eta_{sr})+n_{r}, \tag{3}\]  
where \(p_{n}\) represents the transmitted power of the user \(n\), \(\theta_{n}\) is the information symbol of the user \(n\), \(n_{r}\sim\mathcal{CN}(0,\sigma_{R}^{2})\) is the additive white Gaussian noise (AWGN) at UAV, the HIs at both BS and UAV are characterized as independent distortion noise \(\eta_{sr}\sim\mathcal{CN}\left(0,\kappa_{sr}^{2}P_{s}\right)\), where \(P_{s}\) represents the total power transmitted by BS, \(\kappa_{sr}^{2}\overset{\Delta}{=}\sqrt{\kappa_{s}^{2}+\kappa_{r}^{2}}\) represents the total distortion level from BS to the UAV relay, \(\kappa_{s}\) and \(\kappa_{r}\) characterize the distortion noise at BS and UAV respectively. In the second slot, UAV amplifies and forwards the received signal \(y_{r}\) to ground users with amplification gain \(G\) as:  
\[G=\sqrt{\frac{P_{r}}{P_{s}(1+\kappa_{sr}^{2}){h_{sr}}^{2}+\sigma_{R}^{2}}}, \tag{4}\]  
where \(P_{r}\) denotes the UAV transmitted power. Figure 1: NOMA-based UAV communication model with HIs. The received signal at user \(n\) can be expressed as:  
\[y_{\text{LI}_{n}} = h_{sr}n_{m}G\left(\sum\limits_{i=1}^{n-1}\sqrt{p_{i}}\theta_{i}+ \sqrt{p_{n}}\theta_{n}+\sum\limits_{j=n+1}^{N}\sqrt{p_{j}}\theta_{j}\right)\] \[+h_{sr}h_{rn}G\eta_{sr}+h_{rn}Gnt_{r}+h_{rn}\eta_{rn}+n_{\text{LI}_ {n}},\]  
where \(n_{\text{LI}_{n}}\sim\mathcal{CN}\left(0,\sigma_{\text{LI}_{n}}^{2}\right)\) is the AWGN at the user \(n\), the HIs at both the UAV and user \(n\) is characterized as independent distortion noise \(\eta_{rn}\sim\mathcal{CN}\left(0,\kappa_{rn}^{2}P_{r}\right)\), where \(\kappa_{rn}^{2}\triangleq\sqrt{\kappa_{r}^{2}+\kappa_{n}^{2}}\) represents the total distortion level from UAV to the user \(n\) in which \(\kappa_{r}\) and \(\kappa_{n}\) characterize the distortion noise at the UAV and user \(n\) respectively. According to the NOMA protocol, the signal receiver of weak users adopts SIC technology. Firstly, the weak users perform successive interference cancellations from the received signal, and the strong user's signal is regarded as interference to the weak users. Therefore, we introduced the binary variable \(\alpha_{k,n}\in\{0,1\},\forall k,n\) to denote the SIC decoding order. When \(\alpha_{k,n}=1\), the user \(k\) with a stronger channel gain is treated as interference to decode the user \(n\). \(\alpha_{k,n}=0\) to present other situations. Thereby, \(\alpha_{k,n}\) is defined as:  
\[\alpha_{k,n}=\left\{ \begin{array}{cc}1,d_{k}<d_{n}&\\ 0,d_{k}>d_{n}&,\forall k,n.\\ 0\,or\,1,d_{k}=d_{n}&\end{array} \right. \tag{6}\]  
Nevertheless, \(\alpha_{k,n}\) is an integral binary variable. For ease of explosion, we can rewrite (6) as follows:  
\[\alpha_{k,n}\in\{0,1\},\forall k\
eq n, \tag{7a}\] \[\alpha_{n,n}=0,\forall n,\] (7b) \[\alpha_{k,n}+\alpha_{n,k}=1,\forall k\
eq n,\] (7c) \[\alpha_{k,n}a_{rk}^{2}\leq d_{rn}^{2},\forall k\
eq n. \tag{7d}\]  
Note that (7b) means that the user \(n\) decoding its signal should not regard itself as interference. (7c) means that two different users must be able to be divided into a strong user and a poor user. (7a), (7c), and (7d) indicate that when \(\alpha_{k,n}=1\), the distance from the user \(n\) to the drone is greater than the distance from the user \(k\). Finally, the rate of the user \(n\) is defined as:  
\[R_{n} = \frac{1}{2}\text{log}_{2}\left(1+\frac{p_{n}A_{n}{}^{\prime}}{A_{n} \sum\limits_{k\
eq n}^{N}\alpha_{k,n}p_{k}+A_{n}B_{n}+C_{n}}\right), \tag{8}\]  
where  
\[A_{n} = P_{n}P_{n}l_{sr}^{2}l_{rn}^{2},\] \[A_{n}{}^{\prime} = P_{n}l_{sr}^{2}l_{rn}^{2},\] \[B_{n} = \kappa_{sr}^{2}+\kappa_{rn}^{2}+\kappa_{rn}^{2}\kappa_{sr}^{2},\] \[C_{n} = P_{n}h_{sr}^{2}\left(1+\kappa_{sr}^{2}\right)\sigma_{\text{LI}_{ n}}^{2}+P_{r}l_{rn}^{2}\left(1+\kappa_{rn}^{2}\right)\sigma_{R}^{2}+\sigma_{R}^{2} \sigma_{\text{LI}_{n}}^{2}.\]