# Distributed Satellite Relay Cooperative Communication with Optimized Signal Space Dimension
## 3 Results
### Spectral Efficiency  
The key idea of the method proposed in this paper is to reasonably choose the beamforming vector within the signal space such that the transmitted signals from two nodes are aligned to the 1-dimension of the intersecting subspace. The relay performs decoding and retransmission of separable superimposed signals. The relay designs a precoding vector with zero interference to eliminate inter-node interference during broadcast transmission. In a 3-user channel, TDMA scheme requires six time slots to implement data  
Figure 5: Spatial distribution of alignment signal angle. exchange between users through the relay, and its multiplexing gain is \(\eta=1\). The multiuser MIMO (MU-MIMO) scheme requires four orthogonal time slots to implement data exchange between users through beamforming vector optimization. The multiplexing gain is \(\eta=1.5\). The proposed scheme achieves data exchange between users in two time slots through signal space compression, and its multiplexing gain is \(\eta=3\). Using these proposed signaling methods, it is shown that the capacity of the MIMO channel is characterized as \(\eta M\log(SNR)+o(\log(SNR))\) in the interference limited environment [29]. Then we compare the achievable capacity of different transmission techniques with that of the signal space under the non-saturated SNR regime. The slope of the curve in Figure 6 represents the multiplexing gain that the scheme can achieve, further reflecting the ability of data exchange and transmission. Throughput refers to the number of information bits correctly transmitted in a unit time, and its transmission capacity can be characterized by channel capacity. It can be seen from Figure 6 that capacity curve of proposed scheme has the steepest slope. Meanwhile, The TDMA curve is the flattest. The corresponding content has been added to the upper paragraph of Figure 6. In Figure 6, the proposed method can achieve greater potential than conventional TDMA and multi-user MIMO (MU-MIMO) schemes. In the following simulations, a cooperative transmission network with K = 10 nodes and different numbers of antennas is considered. In the Gaussian local scattering channel model, the angular standard deviation (ASD) \(\varphi=10^{\circ}\). Figure 7 shows the effect of the number of relay antennas on the average sum spectral efficiency (SE). When the relay receives the superimposed signal aligned in the signal space, it performs local decoding. We chose multi-node minimum mean-squared error (M-MMSE), single-node minimum mean-squared error (S-MMSE), regularized zero-forcing (RZF), ZF, and maximum ratio (MR) combining as the decoding method. In Figure 8, the performance of M-MMSE is the best. In order to reduce the computational complexity, approximate calculation is usually adopted. The performance of RZF and ZF is very close when the number of antennas is large. But when the number of antennas is insufficient, the ZF performance drops sharply. This is mainly because there are not enough degrees of freedom against interference. Also note that the performance of the S-MMSE scheme is only slightly lower than that of the M-MMSE scheme. Although the MR scheme has the worst performance, it does not require matrix inversion and thus has the lowest computational complexity. Figure 6: Capacity of different schemes. A fading channel \(\mathbf{h}\in\mathbb{C}^{M}\) is _spatially uncorrelated_ if the channel gain \(\|\mathbf{h}\|^{2}\) and the channel direction \(\mathbf{h}/\|\mathbf{h}\|\) are independent random variables, and the channel direction is uniformly distributed over the unit-sphere in \(\mathbb{C}^{M}\). The channel is otherwise _spatially correlated_. Figure 8: CDF of different schemes. Figure 7: Average sum SE as a function of the number of relay antennas for different combining schemes. (**a**) DoF = 1, (**b**) DoF = 2, (**c**) DoF = 4. Practical channels are generally spatially correlated, also known as having space-selective fading.