# Dynamic Divide Grouping Non-Orthogonal Multiple Access in Terrestrial-Satellite Integrated Network
## 3 System Modeling
### Problem Formulation  
According to the principle of NOMA, within each group, the NOMA scheme is implemented by applying SIC at the user. The decoding order of SIC is in the order of the increasing channel gain, and thus any user can decode the signals of other users that experience poorer channels than itself. As stated before, the near user experiences a better channel than the far user. The near user will first decode the signal of the far user for interference cancellation, and then decode its own signal after subtracting the far user's signal from the received signal. Thus, the signal to interference noise ratios (SINR) of two users in the satellite user group \(J\) (near user and far user) can be calculated as  
\[\gamma_{S,I,J}^{(1)}=\frac{\rho_{S,J,I}^{(1)}P_{S,I,J}H_{S,I,J}^{(1)}}{\rho_{ S,J,I}^{(2)}P_{S,I,J}H_{S,J,I}^{(2)}+I_{B}+r_{S,I,J}N_{0}}, \tag{3}\]  
Figure 1: The system model of the non-orthogonal multiple access based integrated terrestrial-satellite network. \[\gamma^{(2)}_{S,l,l}=\frac{\rho^{(2)}_{S,l,l}P_{S,l,l}H^{(2)}_{S,l,l}}{I_{B}+r_{S,l,l}N_{0}}, \tag{4}\]  
where \(I_{B}=\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{N}\left(\rho^{(1)}_{B,i,j}P_{B,i, j}H^{(1)}_{B,i,j}+\rho^{(2)}_{B,i,j}P_{B,i,j}H^{(2)}_{B,i,j}\right)\), and the rate expressions of near user and far user in satellite NOMA group can be expressed as  
\[C^{(1)}_{S,l,l}=r_{S,l,l}\text{log}_{2}\left(1+\frac{\rho^{(1)}_{S,l,l}P_{S,l, l}H^{(1)}_{S,l,l}}{\rho^{(2)}_{S,l,l}P_{S,l,l}H^{(2)}_{S,l,l}+I_{B}+r_{S,l,l}N_{0 }}\right), \tag{5}\]  
\[C^{(2)}_{S,l,l}=r_{S,l,l}\text{log}_{2}\left(1+\frac{\rho^{(2)}_{S,l,l}P_{S,l, l}H^{(2)}_{S,l,l}}{I_{B}+r_{S,l,l}N_{0}}\right). \tag{6}\]  
Furthermore, we can observe that the BS user is interfered by the signals of other satellite users and the BS users. The SINR of the near user and far user in the BS group \(J\) can be calculated as  
\[\gamma^{(1)}_{B,i,l}=\frac{\rho^{(1)}_{B,i,l}P_{B,i,l}H^{(1)}_{B,i,l}}{\rho^{( 2)}_{B,i,l}P_{B,i,l}H^{(2)}_{B,i,l}+I_{S}+r_{B,i,l}N_{0}}, \tag{7}\]  
\[\gamma^{(2)}_{B,i,l}=\frac{\rho^{(2)}_{B,i,l}P_{B,i,l}H^{(2)}_{B,i,l}}{I_{S}+r _{B,i,l}N_{0}}. \tag{8}\]  
Then, the capacity of the BS near user and BS far user are  
\[C^{(1)}_{B,i,l}=r_{B,i,l}\text{log}_{2}\left(1+\frac{\rho^{(1)}_{B,i,l}P_{B,i, l}H^{(1)}_{B,i,l}}{\rho^{(2)}_{B,i,l}P_{B,i,l}H^{(2)}_{B,i,l}+I_{S}+r_{B,i,l}N_{0 }}\right), \tag{9}\]  
\[C^{(2)}_{B,i,l}=r_{B,i,l}\text{log}_{2}\left(1+\frac{\rho^{(2)}_{B,i,l}P_{B, i,l}H^{(2)}_{B,i,l}}{I_{S}+r_{B,i,l}N_{0}}\right), \tag{10}\]  
where, \(I_{s}=\sum\limits_{j=1}^{K_{s}/2}\left(\rho^{(1)}_{S,l,l}P_{S,l,j}H^{(1)}_{S, l,j}+\rho^{(2)}_{S,l,j}P_{S,l,j}H^{(2)}_{S,l,j}\right)\). From the above derivation, we can see that the BSs and the satellite serve multiple ground users cooperatively while NOMA techniques are applied to reuse the entire bandwidth. As the service capability of the BS is limited, the satellite can be utilized to provide extra service for those users that cannot be served by the BSs. However, due to frequency reuse, the terrestrial BSs and the satellite will interfere each other. Furthermore, within each network, there also exists interference among users. Thus, it is of great importance to design the total system reasonably to achieve optimal capacity performance. Considering the communication rate constraints and the transmission power constraints of users, the optimization problem can be formulated as \[ \begin{array}{ll}\max\limits_{r,p}&\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{N} \left(C_{B,i,j}^{(1)}+C_{B,i,j}^{(2)}\right)\ +\sum\limits_{l=1}^{L}\sum\limits_{j=1}^{K_{l}/2}\left(C_{S,l,j}^{(1)}+C_{S,l, j}^{(2)}\right),\\ s.t&C1:\ \sum\limits_{j=1}^{K_{l}/2}r_{S,l,j}\ \leq r_{S,p},\\ &C2:\ \sum\limits_{j=1}^{N}r_{B,j,j}\ \leq r_{B,j,\max},\\ &C3:\rho_{B,j,l}^{(1)}+\rho_{B,j,l}^{(2)}=1,\rho_{S,l,j}^{(1)}+\rho_{S,l,j}^{( 2)}=1\,\\ &C4:\ r_{S,l,j},r_{B,j,l}>0,\\ &C5:\ \gamma_{S,l,j}^{(K)},\gamma_{B,j,l}^{(K)}>\gamma_{\min},k=\{1,2\},\\ &C6:\ P_{S,l,j},P_{B,j,l}\leq P_{\max}.\end{array}