# Service-Driven Dynamic Beam Hopping with Resource Allocation for LEO Satellites
## 2 System Model and Problem Formulation
### Channel Model  
The forward-path channel coefficient between satellite beam \(k\) and beam position \(m\) is defined as \(h_{m}^{k}\), which is composed of path transmission loss \(PL_{km}\), satellite antenna transmit gain \(G_{T}\), ground-end antenna receive gain \(G_{R}\), and the beam hopping pattern \(x_{t}^{m}\) of this beam position in the current time slot. Specifically, the forward-path channel coefficient is expressed as \(h_{m}^{k}=PL_{km}\cdot G_{T}\cdot G_{R}\cdot x_{t}^{m}\). Therefore, \(H=\{h_{m}^{k}\mid m\in\Psi,k\in\mathcal{K}\}\) represents the channel coefficient matrix between on-board beams and ground beam positions. The path loss \(PL_{km}\) can be calculated as follows:  
\[PL_{km}=L_{km}^{prop}+L_{km}^{sh}+L_{km}^{ra}+L_{km}^{mp} \tag{1}\]  
where the components included in Equation (1) can be defined as follows. * \(L_{km}^{prop}\): Free-space loss, a function of carrier frequency \(f_{c}\), and the straight-line distance \(d_{km}\) between the satellite and the beam-positioning center. * \(L_{km}^{sh}\): Shadow fading, modeled as a log-normal random variable. Figure 1: Forward link model of LEO beam hopping satellite system. * \(L_{km}^{ra}\): Rain attenuation, dependent on rain probability \(p_{ra}\). * \(L_{km}^{rp}\): Multipath fading, characterized by a Rayleigh distribution. The mathematical expressions for each component can be represented as follows: \[L_{km}^{prop}=32.45+20\log_{10}d_{km}+20\log_{10}f_{c}\] (2) \[L_{km}^{sh}\sim N(0,\sigma_{sh}^{2})\] (3) \[L_{km}^{ra}=A_{0.01}*f(p_{ra})\] (4) \[L_{km}^{mp}\sim R(\sigma_{mp})\] (5)  
where \(\sigma_{sh}^{2}\) represents the shadow variance, \(A_{0.01}\) represents the attenuation value with an annual average rain-time percentage of 0.01%, and \(\sigma_{mp}\) represents the Rayleigh distribution standard deviation. In this paper, to ensure efficient utilization of the frequency band and enable each beam to share on-board resources, the communication system employs a full frequency reuse scheme for beam allocation. However, this approach means that there is non-negligible co-channel interference among different service beams at the same time. Therefore, it is necessary to model and analyze beam interference. The inter-beam interference value mainly depends on the angle \(\theta\) between the satellite's aiming lines to the centers of beam positions \(m\) and \(i\).The cause of co-channel interference is the radiation from the antenna side-lobes interfering with co-channel beam positions. The co-channel interference model for LEO satellites is shown in Figure 2. Service beams illuminate beam position \(m\) and beam position \(i\) at the same time. The geometric angle formed by the satellite's center to beam position \(m\) and its center to beam position \(i\) is denoted by \(\theta\). In this scenario, assuming that beam position \(m\) is the interfered area, we discuss the co-channel interference that beam position \(i\) causes for beam position \(m\). The interference between beam positions in the downlink can be denoted by  
\[I_{i}=\frac{\sum_{i=1,j\
eq m}^{M}[P_{i}\cdot g_{i}\cdot G_{i}(\theta)]}{(4\pi d /\lambda)^{2}\cdot L_{m}(\theta)} \tag{6}\]  
where \(M\) represents the number of beam positions that share the same frequency as beam position \(m\). \(P_{i}\) is the transmit power of the satellite for beam position \(i\) other than beam position \(m\). \(g_{i}\) is the antenna gain from the satellite to the center point of beam position \(i\). \(d\) is the signal propagation distance. \(\lambda\) is the signal wavelength. \(L_{m}(\theta)\) is the channel  
Figure 2: Co-channel interference in an LEO satellite system. attenuation from the satellite to the center point of the interfered beam position \(m\). \(G_{i}(\theta)\) is the antenna gain of the beam illuminating beam position \(i\) concerning the center point of beam position \(i\), and the expression is as follows.