## 4 Numerical Results
### Transmission Reliability  
Aiming at the bottleneck problem of system performance, this paper proposes that the channels between user nodes and relay nodes in the system should be sorted according to transmission quality before solving the appropriate precoding vector. The new channels with poor quality rank first, and the channels with good quality rank last. Then, the spatial alignment direction formed by the interactive signals corresponding to the channel and its transmission precoding vector are solved step-by-step according to the ordered channels. Therefore, how to judge the quality of channel transmission is also a problem to be considered in this paper. The quality of channel transmission is essentially determined by the channel transmission matrix. Therefore, the judgment standard of channel transmission quality is determined from the factors that determine the channel matrix. The size of the singular value obtained after the singular value decomposition of the channel matrix can represent the transmission gain of the equivalent channel. At the same time, the number of nonzero singular values is related to the channel capacity of the MIMO-Y channel model. Whether the distribution of singular values is uniform depends on the degrees of freedom of the channel matrix. Therefore, it can be inferred from the above analysis that the quality of channel transmission requires comprehensive consideration of two factors: the size of the singular value obtained from the singular value decomposition of the channel matrix and the distribution degree of the singular value. Taking these two determinants into account, this paper is based on the normal distribution probability density of the singular value of the channel matrix. The integration of the density function is taken as the standard to measure the quality of channel transmission. Specific judgment methods are as follows:  
(i) The upstream channel matrix from each user node to the relay node in the channel model is decomposed into singular values, such as SVD decomposition of \(\mathbf{H}_{r,1}\). In the three-user MIMO-Y channel model, three singular values can be obtained after decomposition. (ii) A group of singular values obtained from the singular value decomposition of each channel matrix is represented by \(X\), and the mean and variance of \(X\) are calculated. Figure 5: Relationship between antenna average correlation factor and jamming node azimuth. \(X\) follows the normal distribution, i.e., \(X\sim N\left(\mu,\sigma^{2}\right)\). The probability density function of normal distribution is:  
\[f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}} \tag{31}\]  
where, \(\mu\) is the mean value of data \(X\), and \(\sigma^{2}\) is the variance of data \(X\). The integral of the positive half axis of the \(x\)-axis for function \(f(x)\) is:  
\[t=\int_{0}^{\infty}f(x) \tag{32}\]  
The value of \(t\) is used to measure the quality of the channel transmission. Calculate the \(t\) value corresponding to the upstream channel matrix from each user node to the relay node, and then compare the \(t\) value. The larger the \(t\) value is, the better the channel transmission quality is. In addition to the normal distribution method based on the singular value of the matrix proposed above, there are two common methods: measure according to the mean value of the singular value of the channel matrix and measure according to the variance of the singular value of the channel matrix. Figure 6a shows the simulation results of the bit error rate of the traditional SSA algorithm and the signal space orthogonal alignment algorithm varying with the signal to noise ratio. The simulation results show that the signal space orthogonal alignment algorithm further reduces the bit error rate of the system compared with the traditional signal space alignment algorithm. Therefore, the performance advantages of signal subspace orthogonality are verified.