# Enhanced Integrated Satellite-Terrestrial NOMA with Cooperative Device-to-Device Communication
## 2 System Model
### Transmission Parameters and Achievable Rates  
In case that the satellite applies the NOMA scheme based on user pairing, for each pair of users, the satellite transmits a superimposed signal as:  
\[x_{s}=\sqrt{G_{i}^{s}}\left(\sqrt{\alpha_{i}P_{s}}s_{i}+\sqrt{\alpha_{j}P_{s}} s_{j}\right). \tag{3}\]  
where \(s_{i}\) and \(s_{j}\) are the signals for the users UE\({}_{i}\) and UE\({}_{j}\), respectively, and \(a_{i}\) and \(a_{j}\) denote the fraction of the total satellite transmit power \(P_{s}\) that is assigned to each user, with \(a_{i}+a_{j}=1\). Then, each UE\({}_{i}\) with \(l=\{i,j\}\) receives the satellite signal:  
\[y_{l}^{s}=\sqrt{\frac{G_{r}^{s}}{L_{FS_{l}}}}h_{l}^{s}x_{s}+z_{n}^{s}, \tag{4}\]where \(z_{n}^{s}\) is the AWGN noise in the satellite link. The channel gain \(\Gamma_{l}\) for each UE\({}_{l}\) including additional gains, losses and the noise power of the satellite receiver \(N_{s}\) is given as:  
\[\Gamma_{l}=\frac{G_{t}^{s}G_{r}^{s}}{L_{FS_{l}}N_{s}}|h_{i}^{s}|^{2}. \tag{5}\]  
Assume that UE\({}_{i}\) and UE\({}_{j}\) form a pair with \(\Gamma_{i}\leq\Gamma_{j}\). Thus, the UE\({}_{j}\) is the strong channel satellite user and UE\({}_{i}\) is the weak channel satellite user. According to the NOMA principle, the fraction of the total satellite transmit power assigned to UE\({}_{j}\) should be less than the fraction of the power assigned to UE\({}_{i}\) i.e., \(a_{j}\leq a_{i}\). As a result, UE\({}_{j}\) performs SIC and decodes their signal \(s_{j}\). At the same time, the weak channel satellite user UE\({}_{i}\) directly decodes the \(s_{i}\), and \(s_{j}\) is treated as noise. Thus, the achievable rates for each UE\({}_{i}\) and UE\({}_{j}\) that form a pair in a satellite sub-channel are given from the following equations:  
\[R_{j}^{NOMA_{(i,j)}}=B_{c}^{NOMA}\log_{2}\left(1+\alpha_{j}P_{s}\Gamma_{j} \right), \tag{6}\]  
\[R_{i}^{NOMA_{(i,j)}}=B_{c}^{NOMA}\log_{2}\left(1+\frac{(1-\alpha_{j})P_{s} \Gamma_{i}}{\alpha_{j}P_{s}\Gamma_{i}+1}\right), \tag{7}\]  
where \(B_{c}^{NOMA}\) is the satellite channel bandwidth in the case of NOMA. In particular, when the satellite applies NOMA, user pairing allocates to each pair a sub-channel on the satellite and within this sub-channel different power allocation factors are assigned to each user of the pair. Therefore, the total number of sub-channels in this case is equal to \(N/2\) and the total available satellite bandwidth is equally distributed to each sub-channel. Thus, the satellite channel bandwidth for NOMA is equal to \(B_{c}^{NOMA}=2B_{s}/N\). On the contrary, for OMA, each user is allocated a separate channel. Thus, the available satellite bandwidth is equally assigned to each UE\({}_{k}\). As a result, the channel bandwidth in this case is equal to the channel bandwidth in case of NOMA, multiplied by \(1/2\), i.e., \(B_{c}^{OMA}=B_{c}^{NOMA}/2\), and the achievable rate for each UE\({}_{k}\) is given as:  
\[R_{k}^{OMA}=\frac{1}{2}B_{c}^{NOMA}\log_{2}\left(1+P_{s}\Gamma_{k}\right). \tag{8}\]  
Regarding the terrestrial communication, the channel gain for the D2D link \(\Lambda_{i,j}\) between UE\({}_{i}\) and UE\({}_{j}\), including the channel coefficient \(|h_{i,j}^{d}|^{2}\), additional gains, losses and the noise power of the terrestrial receiver \(N_{d}\), is modeled as:  
\[\Lambda_{i,j}=\frac{G_{t}^{d}G_{r}^{d}}{PL(d)_{i,j}N_{d}}|h_{i,j}^{d}|^{2}, \tag{9}\]  
where \(PL(d)_{i,j}\) is equal to the antilogarithm of the result that is calculated via (2). The achievable rate for UE\({}_{i}\) through the D2D link with UE\({}_{j}\) it is equal to:  
\[R_{i}^{D2D_{(i,j)}}=B_{c}^{D2D}\log_{2}\left(1+P_{d}\Lambda_{i,j}\right), \tag{10}\]  
where \(B_{c}^{D2D}\) is the channel bandwidth that is allocated to a pair of users forming a terrestrial D2D link. Since D2D communication between a pair of users takes place in a different sub-channel, we do not take into account interference between the satellite and the D2D links.