# Distributed Satellite Relay Cooperative Communication with Optimized Signal Space Dimension
## 3 Results
### Analysis of MIMO Channel Characteristics  
Assuming that the communication node is in the slowly changing condition of the far-field plane wave in the relay propagation environment, the semi-static channel transmission response \(\mathbf{H}^{[R,l]}\) can be obtained. The ideal propagation conditions make it possible to separate multiple antenna channels at the relay side. At this point, channels \(\mathbf{H}^{[R,l]}\) and \(\mathbf{H}^{[R,j]}\) need to satisfy:  
\[\frac{\left(\mathbf{H}^{[R,l]}\right)^{H}\mathbf{H}^{[R,j]}}{\sqrt{\mathrm{E} \Big{\{}\|\mathbf{H}^{[R,l]}\|^{2}\Big{\}}\Big{\{}\|\mathbf{H}^{[R,j]}\|^{2} \Big{\}}}}\to 0 \tag{15}\]  
The convergence characteristics of different fading channels can be considered by Formula (15). The interpretation of (15) is that the channel directions become asymptotically orthogonal. This is because the propagation paths of the antenna experience independent fading, and the probability of each independent propagation path entering shadow fading at the same time is very low. Spatial diversity gain can be obtained with multiple antennas. The channel matrix \(\mathbf{H}^{[R,l]}\) varies with time and is a random variable. If the variables satisfy Equation (16), the random channel matrix \(\mathbf{H}^{[R,l]}\) exhibits channel hardening properties. That is, the channel fading gradually becomes deterministic and approaches hardening. \[\frac{\left\|\mathbf{H}^{[R,l]}\right\|^{2}}{\mathrm{E}\Big{\{}\|\mathbf{H}^{[ R,l]}\|^{2}\Big{\}}}\to 1 \tag{16}\]  
With the increasing number of antennas, all singular values of the random matrix \(\mathbf{H}^{[R,l]}\) obey a certain distribution function. All eigenvalues have the same distribution and are gradually replaced by the mean. Consider the fading channel hardening phenomenon in \(N\)-dimensional space as shown in Figure 4. The average value of the normalized instantaneous channel gain is \(\|\mathbf{H}\|^{2}/\mathrm{E}\Big{\{}\|\mathbf{H}\|^{2}\Big{\}}\). It can be seen from the comparison of the channel hardening of different antenna number probability and random realization that all schemes gradually converge with the increase of the number of antennas. In particular, when the number of antennas is greater than 50, the convergence tends to be stable. Channel parameters will gradually change from randomness to determinism. This feature can ensure that communication nodes use simple linear precoding to replace complex nonlinear precoding and real-time precoding. Figure 4: Comparison of channel hardening characteristics. In wireless communication, electromagnetic waves may arrive at the receiver via many different paths. In the process of transmission through the wireless channel, it may pass through the direct line of sight (LoS) path or through the plane reflection. Since the multiple copies of the original transmission signal propagate at different distances, they arrive at the receiver at different times and have different average power levels. A traditional fading channel simulation technology, tapped delay line, is used by people to model impulse response. Each tap represents the sum of multipath signals arriving at the same time. Since signals arriving later have greater path loss, the tap amplitude usually decreases over time. The average power and delay of each tap are displayed as channel impulse responses, also known as power delay profile (PDP). Three taps (signal paths) are designed using Jakes' fading simulator. In order to study the propagation characteristics of a Rayleigh fading channel, a 2 \(\times\) 2 MIMO channel is constructed. The power delay profile and response of each channel are measured. The measurement results are shown in Tables 2 and 3. To combat channel fading, data packet retransmission or practical channel error correction codes are generally used. As can be seen from the figure, the fading characteristics of different channels are very different, which has a great impact on data transmission.