# Regularized Zero-Forcing Dirty Paper Precoding in a High-Throughput Satellite Communication System
## 2 High-Throughput Satellite Communication System Model
### High-Throughput Satellite System Channel Model  
For the LOS (line of sight) satellite channel with the center frequency above 10 GHz, the performance and availability of the high-throughput satellite system will be greatly reduced due to various atmospheric effects in the troposphere. Among various atmospheric effects, rainfall attenuation is the most critical and dominant factor, especially in the Ka-band. Therefore, the influence of rainfall attenuation should be emphatically considered in modeling. Due to the high frequency of the Ka-band and the excessive free space loss of electromagnetic waves, the transmitting and receiving gain of omnidirectional antennas configured for ground mobile users is not enough to meet the link budget. Therefore, the Ka-band high-throughput satellite system is only suitable for providing fixed user services, and satellite mobile services are generally provided by L-band satellites [13]. In this paper, when establishing the channel model of a high-throughput satellite system, the effects of free space loss, rainfall attenuation, and beam gain are considered at the same time. The channel matrix can be expressed as:  
\[H=\widetilde{H}B \tag{4}\]  
where the \(K\times K\) dimension rainfall attenuation matrix \(\widetilde{H}\) simulates the influence of rainfall attenuation, and the \(K\times K\) dimension multi-beam antenna gain matrix \(B\) simulates the influence of satellite antenna radiation mode, receiving antenna gain, path loss, and noise power. The size of the multi-beam antenna gain matrix depends on the radiation pattern of the antenna beams and the position of each user terminal antenna in the beam. For a high-throughput satellite communication system providing fixed user services, the multi-beam antenna gain matrix is fixed, whose \((i,j)\) item \(b_{i,j}\) can be expressed as:  
\[b_{i,j}=\frac{\sqrt{G_{R}^{i}G_{i,j}}}{4\pi\frac{\sqrt{g_{0}^{i}+d^{2}}}{ \lambda}\sqrt{\kappa_{B}TB_{W}}} \tag{5}\]  
where \(G_{i,j}\) is the multi-beam antenna gain from the \(j\)-th satellite feed to the \(i\)-th user terminal, \(G_{R}^{i}\) is the antenna gain of the user terminal, \(\kappa_{B}\) is the Boltzmann constant, \(T\) is the receiver noise temperature, \(B_{W}\) is the user link bandwidth, \(\lambda\) is the carrier wavelength, \(d_{0}\) is the distance from the satellite to the user-located beam center, and \(d\) is the distance from the ground user terminal to the user-located beam center. Due to the curvature of the earth and the wide-area coverage characteristics of satellites, the introduction of \(d\) can more accurately describe the difference of free space loss between different beams of the high-throughput satellite system. The multi-beam antenna gain \(G_{i,j}\) from the \(j\)-th satellite feed to the \(i\)-th user terminal mainly depends on the radiation mode of satellite transmitting antennas and user position, which can be expressed as:  
\[G_{i,j}=G_{T}^{j}\left(\frac{I_{1}(u_{i,j})}{2u_{i,j}}+36\frac{J_{3}(u_{i,j})}{u _{i,j}^{3}}\right)^{2} \tag{6}\]  
where \(G_{T}\) is the transmit antenna gain, intermediate quantity \(u_{i,j}=2.07123\frac{\sin(\theta_{i,j})}{\sin(\theta_{i,j})}\). \(\theta_{i,j}\) is the angle of the path from the \(i\)-th user to the \(j\)-th beam center relative to the satellite, and \(\theta_{i,3 dB}\) is the 3 dB beam angle of the \(i\)-th beam; that is, the beam where the user is. \(J_{1}(\cdot)\) and \(J_{3}(\cdot)\) are the first-order and third-order Bessel functions, respectively. The method of modeling rainfall attenuation adopts the latest empirical model in the ITU-R P.618 recommendation [14], and the power gain \(\xi_{dB}\) in dB follows lognormal distribution, that is, \(\ln(\xi_{dB})\sim N\big{(}\mu,\sigma^{2})\). \(\mu\) and \(\sigma^{2}\) are the mean and variance of the corresponding normal distribution, respectively, which are related to the receiver position, operating frequency, polarization, and satellite elevation angle.