# Dynamic Divide Grouping Non-Orthogonal Multiple Access in Terrestrial-Satellite Integrated Network
## 5 Joint Resource Allocation Optimization Scheme
### Power Control Factor Optimization  
For many practical scenarios, the total transmission power constraint is a crucial criterion. For example, in a cell with multiple users sharing the same bandwidth, the constraint of the total transmission power within this cell is important to manage inter-cell interference. Another example is hybrid NOMA, where users are paired to perform NOMA and inter-pair interference is canceled by relying on conventional interference management techniques. The use of the total power constraint is therefore useful to measure inter-group interference [45], i.e., \(\rho_{S,I,I}^{(1)}+\rho_{S,I,I}^{(2)}=1\). To reduce the interference of users in the group and ensure the lowest transmission rate of users, the optimal power control factor is designed to achieve higher resource utilization and improve the total throughput of users. On the other hand, the rate of user \(i\) in OMA, such as time division multiple access (TDMA), is given by  
\[R_{\rm T}^{(i)}=\frac{1}{2}{\rm log}_{2}\left(1+\frac{P_{B,i,I}H_{B,i,I}^{(i)}} {I_{\rm S}+r_{B,i,I}N_{0}}\right),i=\{1,2\}. \tag{27}\]  
To assure that the transmission rate with the NOMA scheme always outperforms that with the TDMA scheme, users 1 and 2 in a group need to satisfy that \(R_{B,i,I}^{(i)}\geq R_{\rm T}^{(i)}\), i.e.,  
\[{\rm log}_{2}\left(1+\frac{\rho_{B,i,I}^{(1)}P_{B,i,I}H_{B,i,I}^{(1)}}{\rho_{B,i,I}^{(2)}P_{B,i,I}H_{B,i,I}^{(2)}+I_{\rm S}+r_{B,i,I}N_{0}}\right)\geq\frac{ 1}{2}{\rm log}_{2}\left(1+\frac{P_{B,i,I}H_{B,i,I}^{(1)}}{P_{B,i,I}H_{B,i,I}^{( 2)}+I_{\rm S}+r_{B,i,I}N_{0}}\right). \tag{28}\]  
Figure 5: User pairing for uplink NOMA. Due to \(\rho^{(1)}_{B,i,J}+\rho^{(2)}_{B,i,J}=1\), the range of power allocation factor can be further constrained as  
\[\frac{1+\frac{P_{B,i,J}H^{(2)}_{B,i,J}}{I_{S}+\tau_{B,i,J}N_{0}}}{1+\frac{P_{B,i, J}H^{(2)}_{B,i,J}}{I_{S}+\tau_{B,i,J}N_{0}}+\sqrt{1+\frac{P_{B,i,J}H^{(1)}_{B,i,J}}{I_{S}+ \tau_{B,i,J}N_{0}}+1}}\leq\rho^{(1)}_{B,i,J}\leq\frac{\sqrt{1+\frac{P_{B,i,J}H^ {(2)}_{B,i,J}}{I_{S}+\tau_{B,i,J}N_{0}}}}{\sqrt{1+\frac{P_{B,i,J}H^{(2)}_{B,i,J} }{I_{S}+\tau_{B,i,J}N_{0}}+1}}. \tag{30}\]  
Then, the sum rate of group users can be expressed as  
\[R_{sum}\Big{(}\rho^{(1)}_{B,i,J}\Big{)}=\log_{2}\left(1+\frac{\rho^{(1)}_{B,i, J}P_{B,i,J}H^{(1)}_{B,i,J}}{(1-\rho^{(1)}_{B,i,J})P_{B,i,J}H^{(2)}_{B,i,J}+I_{S}+P_{ B,i,J}N_{0}}\right)+\log_{2}\left(1+\frac{(1-\rho^{(1)}_{B,i,J})P_{B,i,J}H^{(2)}_{B,i,J}}{I_{S}+\tau_{B,i,J}N_{0}}\right). \tag{31}\]  
Let \(k_{1}=P_{B,i,J}H^{(1)}_{B,i,J},k_{2}=P_{B,i,J}H^{(2)}_{B,i,J},b=I_{S}+r_{B,i,J} N_{0}\), the first derivative of sum rate \(R_{sum}(\rho^{(1)}_{B,i,J})\) with respect to \(\rho^{(1)}_{B,i,J}\) is  
\[R_{sum}(\rho^{(1)}_{B,i,J})^{{}^{\prime}}=\frac{(k_{1}k_{2}-{k_{2}}^{2})(1- \rho^{(1)}_{B,i,J})+b(k_{1}-k_{2})}{[(1-\rho^{(1)}_{B,i,J})k_{2}+b](1+\rho^{(1 )}_{B,i,J}k_{1})}. \tag{32}\]  
Due to \(H^{(1)}_{B,i,J}>H^{(2)}_{B,i,J}\), the first derivative of sum rate \(R_{sum}(\rho^{(1)}_{B,i,J})\) with respect to \(\rho^{(1)}_{B,i,J}\) is strictly positive, which implies that the sum rate is increasing with \(\rho^{(1)}_{B,i,J}\). Therefore, taking the constraint of (30) into consideration, the maximized sum rate of group users is obtained when \(\rho^{(1)}_{B,i,J}=\frac{\sqrt{1+\frac{P_{B,i,J}H^{(2)}_{B,i,J}}{I_{S}+\tau_{B,i,J}N_{0}}}}{\sqrt{1+\frac{P_{B,i,J}H^{(2)}_{S,i,J}}{I_{S}+\tau_{B,i,J}N_{0}} +1}}\).