# Beam-Hopping-Based Resource Allocation in Integrated Satellite-Terrestrial Networks
## 2 System Model
### Downlink Transmission Model  
Based on the above scenario and spectrum sharing scheme, the received signal-to-interference-plus-noise ratio (SINR) of the \(i\)-th satellite terminal (\(i\in\mathcal{N}_{sat}\)) in time slot \(t\in\Gamma\) can be expressed as  
\[\delta_{i,t}^{sat}=\frac{\alpha_{i}^{sat}P_{b}^{sat}}{N_{0}^{sat}+\sum_{k=1,k \
eq i}^{N_{sat}}x_{k,t}I_{k,i}^{sat}}, \tag{1}\]  
where \(\alpha_{i}^{sat}\) denotes the channel gain between the satellite and the \(i\)-th satellite terminal, \(P_{b}^{sat}\) denotes the transmit power of each satellite beam, \(N_{0}^{sat}\) denotes the noise power, and \(I_{k,i}^{sat}\) denotes the interference power from the spot beam pointing to the \(k\)-th satellite terminal. The noise power \(N_{0}^{sat}=KT_{n}W\), where \(K\) is the Boltzmann constant, \(T_{n}\) is the receiver noise temperature, and \(W\) is the transmission bandwidth. Because of the introduction of the inter-system protection distance, CCI from the terrestrial systems to the satellite terminal can be neglected. It is assumed that the receiving antenna of the satellite terminal can keep pointing to the satellite and the channel gain remains unchanged in a BH time window. Then, \(\alpha_{i}^{sat}\) can be expressed as  
\[\alpha_{i}^{sat}=G_{T,0}^{sat}G_{R,0}^{sat}/L_{i}^{sat}, \tag{2}\]  
where \(G_{T,0}^{sat}\) is the peak transmitting antenna gain of the satellite, \(G_{R,0}^{sat}\) is the peak receiving antenna gain of the satellite terminal, and \(L_{i}^{sat}\) is the free space propagation loss between the satellite and the \(i\)-th satellite terminal. The free space propagation loss is a function of the distance between the transmitter and the receiver. Specifically, \(L_{i}^{sat}\) can be written as  
\[L_{i}^{sat}=\left(\frac{4\pi d_{i}^{sat}}{c/f}\right)^{2}, \tag{3}\]  
where \(d_{i}^{sat}\) is the distance between the satellite and the \(i\)-th satellite terminal, \(c\) is the velocity of light, and \(f\) is the carrier frequency. Because of full frequency multiplexing, the inter-beam interference power \(I_{k,i}^{sat}\) (\(k\
eq i\)) can be expressed as  
\[I_{k,i}^{sat}=G_{T}^{sat}(\theta_{k,i})G_{R,0}^{sat}P_{b}^{sat}/L_{i}^{sat}, \tag{4}\]  
where \(G_{T}^{sat}(\theta_{k,i})\) denotes the transmitting antenna gain of the satellite in the direction of angle \(\theta_{k,i}\), which is given by [31]  
\[G_{T}^{sat}(\theta_{k,i})=G_{T,0}^{sat}\bigg{(}\frac{I_{1}(u(\theta_{k,i}))}{ 2u(\theta_{k,i})}+36\frac{J_{3}(u(\theta_{k,i}))}{u(\theta_{k,i})^{3}}\bigg{)} ^{2}, \tag{5}\]  
where \(\theta_{k,i}\) is the off-axis angle of the \(i\)-th satellite terminal from the bore sight of the satellite beam pointing to the \(k\)-th satellite terminal, \(J_{1}(\cdot)\) and \(J_{3}(\cdot)\) are the first-kind Bessel function of orders 1 and 3, respectively, and \(u(\theta_{k,i})\) is expressed as [31]  
\[u(\theta_{k,i})=2.07123\frac{\sin(\theta_{k,i})}{\sin(\theta_{3 dB})}, \tag{6}\]  
where \(\theta_{3 dB}\) is the 3 dB beam width of the transmitting antenna. It should be noted that as \(\theta_{k,i}\) decreases, the transmitting antenna gain \(G_{T}^{sat}(\theta_{k,i})\) and the inter-beam interference power \(I_{k,i}^{sat}\) increases, which may degrade the SINR of the \(i\)-th satellite terminal. Based on [PERSON]'s theorem, the transmission capacity of the \(i\)-th satellite terminal in a BH time window can be expressed as  
\[C_{i}^{sat}=\sum_{t=1}^{N_{slot}}x_{t,t}WT_{slot}\log_{2}(1+\phi_{t,t}^{sat}). \tag{7}\]In each terrestrial system, the terrestrial BS may provide services to multiple terrestrial terminals. Since the locations of the terrestrial terminals may change from time to time, the worst case is considered where all terrestrial terminals are located at the cell edge of the terrestrial system.