# Distributed Satellite Relay Cooperative Communication with Optimized Signal Space Dimension
## 3 Results
### Spectral Efficiency
Therefore, correlated Rayleigh fading channels such that \(\mathbf{h}^{[i,j]}\sim\mathbb{N}_{\mathbb{C}}\Big{(}0_{M},\quad\mathbf{R}^{[i,j]} \Big{)}\), where \(\mathbf{R}^{[i,j]}\in\mathbb{C}^{M\times M}\) is the positive semi-definite spatial correlation matrix and it is also the covariance matrix due to the zero mean. The normalized trace \(\beta^{[i,j]}=tr\Big{(}\mathbf{R}^{[i,j]}\Big{)}/M\) determines the average channel gain from one of the antennas at transmitter \(I\) to receiver \(j\). Uncorrelated Rayleigh fading with \(\mathbf{R}^{[i,j]}=\beta^{[i,j]}\mathbf{I}_{M}\) is a special case of this model. Figure 8 shows the cumulative distribution function (CDF) curves of SE changes at any node in the transmission network. The direction of arrival of the array elements in the MIMO channel is non-omnidirectional and non-uniform, so the correlation characteristics of the channel transfer function and the performance of the wireless channel are closely related to the corresponding spatial distance vector. Through the simulation comparison of uncorrelated Rayleigh fading channel and Gaussian local scattering channel model, the CDF curve with spatial correlation is on the right. The CDF curve of uncorrelated channels is steeper, and the channel statistical capacity is larger. Therefore, it can be concluded that the stronger the correlation between the antennas, the smaller the channel capacity, and the slower the CDF curve of the transmission capacity. The transmission capacity is less than that of an independent uncorrelated Rayleigh channel. The best detection algorithm for nonlinear detection is Maximum Likelihood Received Detection (ML), which is especially beneficial for obtaining the smallest error rate. However, the complexity of ML grows exponentially with transmit antenna and modulation order. When the antenna dimension and modulation order are large, it cannot be applied in practical systems. The detection algorithms of linear detection mainly include zero forcing (ZF) algorithm and minimum mean square error (MMSE) algorithm. They are simple to implement but have poor bit error rate performance. When the transmission channel \(\mathbf{H}\) is an ill-conditioned matrix, that is, when there is a correlation, the interference between the data streams is relatively large. The channel matrix \(\mathbf{H}\) is subjected to sorted QR decomposition \(\mathbf{H}=\mathbf{Q}\mathbf{R}\) to generate a set of permutation sequences, and at the same time, the unitary matrix \(\mathbf{Q}\) and the upper triangular matrix \(\mathbf{R}\), whose elements above the diagonal are ordered, are obtained. In particular, the upper diagonal elements contain all the information of the channel correlation. Wideband signals will cause more serious nonlinear effects, which is not conducive to the transmission of high-order modulated signals. At the same time, in order to meet the needs of satellite communication for high-speed data transmission, 16 QAM modulation is adopted in this paper. The transmitter uses an 8-order raised cosine roll off filter with a roll off factor of 0.3 to realize pulse shaping filtering. At the receiving end, a corresponding raised cosine roll off filter is used for matched filtering. It can be seen from Figure 9 that MMSE usually has better decoding performance than ZF due to the lack of amplification noise. When linear correlation is introduced between channel matrix columns, an ill conditioned matrix is formed. The small error in the coefficients has a great harmful effect on the decoding, and the decoding performance of MMSE and ZF decreases sharply. In order to reduce the noise enhancement, we use QR decomposition to reduce correlated condition. In particular, the ZF decoder based on QR decomposition provides performance improvement in case of ill-conditioned channels due to the way it considers only the well-conditioned elements of the channel. This method is robust to correlated channels.