# Performance Analysis of Dual-Hop Hybrid RF-UOWC NOMA Systems
## 2 System Model  
In this paper, we propose a downlink NOMA-based dual-hop hybrid RF-UOWC system depicted in Figure 1, where the source (\(S\)) is equipped with an RF interface that aims to communicate with two destinations (\(D_{1}\) and \(D_{2}\)) equipped with UOWC interface via an intermediate decode and forward relay (\(R\)). The relay has an RF interface to receive from \(S\) and then transmit to \(D_{1}\) and \(D_{2}\) through the UOWC interface, where \(D_{1}\) is the far or weak user and \(D_{2}\) is the near or strong user. Such a scenario can find applications in many areas in the UIoT [19] (e.g., offshore oil field exploration, oceanic monitoring, and data collection). The \(S\)-\(R\) channel (\(h_{w}\)) is assumed to be a RF channel characterized by Rayleigh fading with AWGN and the \(R\)-\(D_{i}\) channels (\(h_{i}\)) are assumed to be UOWC channels characterized by EGG fading with AWGN, where \(i\in\{1,2\}\). For the sake of improving the spectral efficiency, we assume that \(S\) and \(R\) adopt PD-NOMA for multiplexing their messages. The communication is initiated at \(S\) by multiplexing the two messages \(x_{1}\) and \(x_{2}\) intended for \(D_{1}\) and \(D_{2}\), respectively. The \(S\)-to-\(R\) message is \(x_{S}=\sqrt{a_{1}P_{S}}x_{1}+\sqrt{a_{2}P_{S}}x_{2}\), where \(P_{S}\) is the total transmitted power at \(S\) and \(a_{i}\) is the NOMA power allocation factor for \(D_{i}\) at \(S\). Without loss of generality, we assume that \(a_{1}>a_{2}\) and \(a_{1}+a_{2}=1\). The received message at \(R\) through the RF link is \(y_{R}=h_{w}d^{\frac{-1}{2}}x_{S}+n_{\omega}\), where the expectation of RF channels gain is \(E[|h_{w}|^{2}]=1\), \(d\) is the  
Figure 1: Downlink NOMA-based hybrid RF-UOWC system model. \(S\)-to-\(R\) link distance, \(v\) is the RF channel path-loss exponent, and \(n_{\omega}\) represents AWGN with \(n_{\omega}\sim\mathcal{CN}(0,\ \sigma_{\omega}^{2})\). Utilizing NOMA concept, \(R\) decodes \(x_{1}\) first, then applies the SIC operation, which is assumed to be imperfect, to decode \(x_{2}\). So, the signal-to-interference-plus noise ratios (SINRs) for decoding \(x_{1}\) and \(x_{2}\) are expressed as \(\gamma_{R}^{1}=\dfrac{a_{1}\rho_{s}d^{-v}|h_{w}|^{2}}{a_{2}\rho_{s}d^{-v}|h_{w} |^{2}+1}\) and \(\gamma_{R}^{2}=\dfrac{a_{2}\rho_{s}d^{-v}|h_{w}|^{2}}{a_{1}\rho_{s}\eta d^{-v}| h_{w}|^{2}+1}\), respectively, where \(\rho_{s}=P_{S}/\sigma_{\omega}^{2}\), and \(0\leq\eta\leq 1\) is the residual power factor of the imperfect SIC. In the second phase, \(R\) retransmits the received messages over the UOWC channels that are characterized by independent but not necessarily identical mixture EGG distribution [5]. The relay multiplexes the detected messages using PD-NOMA again, such that \(x_{R}=\sqrt{b_{1}P_{R}x_{1}}+\sqrt{b_{2}P_{R}}x_{2}\), where \(P_{R}\) is the total transmitted power at \(R\) and \(b_{i}\) is the NOMA power allocation factor for \(D_{i}\) at \(R\). Without loss of generality, \(b_{1}>b_{2}\) and \(b_{1}+b_{2}=1\). The received message at \(D_{1}\) through the UOWC link \(h_{1}\) is \(y_{D1}=\varepsilon h_{1}x_{R}+n_{u}\), where \(h_{1}\) is the EEG fading of UOWC channel from \(R\)-to-\(D_{1}\) with expectation \(E[|h_{1}|^{2}]=1\), \(\varepsilon\) is responsivity that is considered to be unity, and \(n_{u}\) represents AWGN with \(n_{u}\sim\mathcal{CN}(0,\ \sigma_{u}^{2})\). Utilizing NOMA concept, \(D_{1}\) decodes \(x_{1}\) first. So, the SINR for decoding \(x_{1}\) at \(D_{1}\) is expressed as \(\gamma_{D1}^{1}=\dfrac{b_{1}\rho_{R}|h_{1}|^{2}}{b_{2}\rho_{R}|h_{1}|^{2}+1}\), where \(\rho_{R}=P_{R}/\sigma_{u}^{2}\). The received message at \(D_{2}\) through the UOWC link \(h_{2}\) is \(y_{D2}=\varepsilon h_{2}x_{R}+n_{u}\), where \(h_{2}\) is the EEG fading of UOWC channel from \(R\)-to-\(D_{2}\) with expectation \(E[|h_{2}|^{2}]=1\). Following the NOMA principle, \(D_{2}\) decodes \(x_{1}\) first and then applies the SIC operation, which is assumed to be imperfect, to decode \(x_{2}\).