# Multi-User Scheduling for 6G V2X Ultra-Massive MIMO System
## 3 Ultra-Massive MIMO Multi-User Scheduling Based on BD Precoding
### Multi-User Scheduling Based on BD Precoding  
Precoding is used to solve the problem of interference between users. However, after analyzing the BD precoding algorithm, we know that if there is a strong correlation between channels, a large part of their useful signals will be eliminated when eliminating the interference of other users. The received signal at the receiving end is weak and cannot be demodulated normally, which will greatly affect the capacity of the system. If the correlation between the user channel matrices is relatively low, this situation will not occur. The pre-coded users do not need to worry about interference from other users and do not have to worry about their own cancellation effects, and the system capacity can also be significantly improved. Therefore, the correlation between users can be used as part of the scheduling criteria to schedule each new user. The theory proves that when the column vectors of the splicing matrix \(\mathbf{H}_{i,j}=\left[\mathbf{H}_{i},\mathbf{H}_{j}\right]\) composed of two matrices matrix \(\mathbf{H}_{i}\) and matrix \(\mathbf{H}_{j}\) are more orthogonal, the correlation between matrix \(\mathbf{H}_{i}\) and matrix \(\mathbf{H}_{j}\) is lower. Therefore, the Pearson coefficient after matrix vectorization is used in this paper to express the correlation of the user channel matrix. First, vectorize matrix \(\mathbf{H}_{i}\) and matrix \(\mathbf{H}_{j}\) respectively  
\[\mathbf{H}_{i}\in\mathbb{C}^{m\times n}\rightarrow\mathbf{H}\mathbf{V}_{i}\in C ^{1\times(m\times n)} \tag{16}\]  
\[\mathbf{H}_{j}\in\mathbb{C}^{m\times n}\rightarrow\mathbf{H}\mathbf{V}_{j}\in C ^{1\times(m\times n)} \tag{17}\]  
The Pearson coefficient between vectors \(\mathbf{H}\mathbf{V}_{i}\) and vectors \(\mathbf{H}\mathbf{V}_{j}\) is given as  
\[ \begin{array}{l}coor_{ij}(\mathbf{H}\mathbf{V}_{i},\mathbf{H}\mathbf{V}_{j} )\\ \end{array}  =\frac{N_{\Sigma}\mathbf{H}\mathbf{V}_{i}\mathbf{H}\mathbf{V}_{j}- \mathbf{\Sigma}\mathbf{H}\mathbf{V}_{j}}{\sqrt{\Sigma}\mathbf{H}\mathbf{V}_{j} ^{2}-(\mathbf{\Sigma}\mathbf{H}\mathbf{V}_{j})^{2}}\mathbf{\Sigma}\mathbf{H} \mathbf{V}_{j}^{2}-(\mathbf{\Sigma}\mathbf{H}\mathbf{V}_{j})^{2}} \tag{18}\]where the absolute value of \(coor(\mathbf{HV}_{i},\mathbf{HV}_{j})\in[-1,1]\) denotes the Pearson correlation coefficient of channel vector\(\mathbf{HV}_{i}\) and vector \(\mathbf{HV}_{j}\). When the absolute value of \(coor_{ij}(\mathbf{HV}_{i},\mathbf{HV}_{j})\) is larger, it means that the channel correlation between user \(i\) and user \(j\) is higher, and thus the interference between the two users is more serious, and vice versa. Therefore, when the absolute value of \(coor_{ij}(\mathbf{HV}_{i},\mathbf{HV}_{j})\) is smaller, the spatial correlation between the two matrices \(\mathbf{H}_{i}\) and \(\mathbf{H}_{j}\) is lower. The relationship between \(coor\) and correlation is shown in Table 1[15]:  
The greedy algorithm calculates the noise enhancement factor \(l_{k}\) and selects the largest user that can guarantee the maximum \(\sum_{k=1}^{K}{{{\left|{{l_{k}}}\right|}^{2}}}\) of the current system. It can be found that \(\sum_{k=1}^{K}{{{\left|{{l_{k}}}\right|}^{2}}}\) is determined by the channel gain of each user in the set \(\mathbf{S}\). Therefore, the user's noise enhancement factor \(l_{k}\) is also a major factor affecting the system capacity. In this paper we define the scheduling factor \(\rho_{k}=l_{kk}-\sum\limits_{m\leq s}{{{l_{k}}}{{m_{l}}{m_{mk}}}}\), which measures the conditions of the channel based on the noise enhancement factor with \(l_{kk}=\left\|{{\mathbf{P}}_{k}^{\perp}\mathbf{H}_{k}}\right\|_{2}\) and \(\mathbf{P}_{v}^{\perp}=\mathbf{I}-\mathbf{H}_{k}^{H}(\mathbf{H}_{k}\mathbf{H} _{k}^{H})^{-1}\mathbf{H}_{k}\). In summary, \(\rho_{k}coor_{ks}\) is used as the criterion of the multi-user scheduling algorithm in this paper, and the scheduled user set is selected as the effective transmission channel of the system to obtain better system user gain.