# Regularized Zero-Forcing Dirty Paper Precoding in a High-Throughput Satellite Communication System
## 2 High-Throughput Satellite Communication System Model
### Dirty Paper Regularized Zero-Forcing Precoding  
Dirty paper precoding is considered to be the sum rate capacity realization technology in the MIMO link, as well as the best precoding technology. However, it is often very complicated to determine the optimal coding order and optimal correlation matrix of precoding. Therefore, this paper proposes a low-complexity dirty paper regularized zero-forcing precoding scheme, which is based on the idea of dirty paper precoding to optimize regularized zero-forcing precoding.  
In the downlink channel of the high-throughput satellite forward link, different users in different beams can be arranged in descending order according to the strength of channel gain, that is, the channel degradation. When the dirty paper precoding algorithm is implemented, the coding order must be determined first. Therefore, we first face the problem of selecting multiple coding sequences. Generally, the first coded user receives a large amount of interference, and the interference received by the coded user decreases gradually with the progress of serial coding. Therefore, from the technical perspective, users with good channel conditions should be coded first, because they have the ability to withstand interference. Moreover, users with high-rate requirements should be coded last, from the demand perspective, because they must avoid interference to achieve the required rate. Therefore, this paper refers to the principle of channel maximum norm selection and combines with the practical application scenarios to sort users according to the descending order of the defined metrics as \(\log_{2}(1+\left\|h_{i}\right\|^{2})\). The following analysis is generalizable. We assume that when \(K=2\), \(\left\|h_{1}\right\|^{2}\leq\left\|h_{2}\right\|^{2}\), the channel conditions of the users in beam 2 are better than those in beam 1, and beam 2 can deal with stronger interference than beam 1. \(s_{1}\) and \(s_{2}\) represent the user signals of beam 1 and beam 2, respectively, and the transmitter signal is the superposition of them: \((s_{1}+s_{2})\). It is assumed that the transmission power of beam 1 and beam 2 is \(p_{1}\) and \(p_{2}\), respectively. Considering the idea of dirty paper coding, beam 2 is coded first, and the signal of beam 1 is regarded as an interference signal. At this time, the rate reached can be expressed as:  
\[R_{2}=\log_{2}\left(1+\frac{p_{2}\left|h_{2}^{H}w_{2}\right|^{2}}{p_{1}\left| h_{2}^{H}w_{1}\right|^{2}+N_{0}B_{W}}\right) \tag{9}\]  
After coding for the user in beam 2, the user in beam 1 is precoded by means of successive interference cancellation, and the achievable user rate can be expressed as:  
\[R_{1}=\log_{2}\left(1+\frac{p_{1}\left|h_{1}^{H}w_{1}\right|^{2}}{N_{0}B_{W}}\right) \tag{10}\]  
Extending the situation of the above two beams to the actual satellite communication system, the application flow chart of dirty paper regularized zero-forcing precoding can be expressed as Figure 3, where the precoding vector \(w_{i}\) is obtained by the regularized zero-forcing precoding.  
After obtaining the coding order, the achievable SINR of the \(i\)-th user can be expressed as:  
\[\Gamma_{i}=\frac{p_{i}\left|\mathbf{h}_{i}^{H}\mathbf{w}_{i}\right|^{2}}{\sum_{j >i}p_{j}\left|\mathbf{h}_{i}^{H}\mathbf{w}_{j}\right|^{2}+N_{0}W} \tag{11}\]  
For the user in the \(i\)-th beam, the achievable user rate can be expressed as:  
\[R_{i}=W\log_{2}(1+\Gamma_{i}) \tag{12}\]