# Radar-Assisted Multiple Base Station Cooperative mmWave Beam Tracking
## 3 Radar-Assisted Multi-Rsu Cooperative Beam Tracking
### Multi-RSU Cooperative Beam Tracking  
Multi-RSU cooperation can address the problems of low localization accuracy and poor coverage in the single-RSU network. Since the sensing performance of each RSU is related to the relative position of the vehicle to it, various RSUs can provide different position predictions of vehicles which could be jointly exploited to improve the accuracy of kinematic parameter prediction. At the \(n\)-th time slot, the \(L\) RSUs receive \(K\) echo signals reflected from \(K\) vehicles, which are exploited to formulate measurements \(\mathbf{y}_{k,l,n},\)\(k=1,2,...,K,l=1,2,\ldots,L,\) where \(K\) and \(L\) denote the number of vehicles and RSUs, respectively. Since kinematic state prediction \(\hat{\mathbf{x}}_{k,l,n|n-1}\) and its corresponding measurement \(\mathbf{y}_{k,l,n}\) have a high correlation, we calculate the measurement predictions of \(L\) RSUs given \(K\) kinematic state predictions \(\hat{\mathbf{x}}_{p,l,n|n-1},\ p=1,2,\ldots,K\) at \(n\)-th time slot, which can be expressed as  
\[\hat{\mathbf{y}}_{p,l,n}=\mathbf{f}\left(\hat{\mathbf{x}}_{p,l,n|n-1}\right),\ p=1,2,\ldots,K \tag{33}\]  
Before each RSU reports prediction to ES, echoes from different vehicles need to be differentiated. Each RSU calculates the Euclidean distance between \(\mathbf{y}_{k,l,n}\) and each of the \(K\) measurement predictions \(\hat{\mathbf{y}}_{p,l,n}\), thereby associating radar echoes measurements \(\mathbf{y}_{k,l,n}\) with the corresponding vehicles based on the smallest Euclidean distance, which can be given by  
\[p=\ arg\ \min_{p}\mathbf{y}_{k,l,n}-\hat{\mathbf{y}}_{p,l,n}.\mathbf{p}=1,2, \ldots,K \tag{34}\]  
As for multi-RSU cooperation, the ES first obtains different position prediction coordinates of the vehicle based on the kinematic state variables calibrated by EKF reported by each RSU, which can be expressed as  
\[\hat{x}_{k,l,n}^{pre}=\hat{d}_{k,l,n}\cdot\cos\hat{\theta}_{k,l,n} \tag{35}\]  
\[\hat{y}_{k,l,n}^{pre}=\hat{d}_{k,l,n}\cdot\sin\hat{\theta}_{k,l,n} \tag{36}\]  
where \(\hat{d}_{k,l,n}\) and \(\hat{\theta}_{k,l,n}\) denote the distance and angle of \(k\)-th vehicle reported by the \(l\)-th RSU at the \(n\)-th time slot, respectively. \(\hat{x}_{k,l,n}^{pre}\) denotes the calculated x-coordinate of the vehicle based on \(d_{k,l,n}\) and \(\hat{\theta}_{k,l,n}\). Similarly for the y-coordinate of the vehicle \(\hat{y}_{k,l,n}^{pre}\). Since the distance between the RSU and the vehicle will affect the tracking performance of the RSU, the prediction accuracy varies significantly for RSUs distributed in different geographical locations. Therefore, it is necessary to design the weights corresponding to different RSUs during cooperation. In our paper, the weight of each RSU is related to the Euclidean distance between the real and predicted echoes. The weights are calculated as follows  
\[E_{k,l,n}=\min_{p}y_{k,l,n}-\hat{y}_{p,l,n},\mathbf{p}=1,2,\ldots,K \tag{37}\]  
\[w_{k,l,n}=\frac{1}{L-1}\left(1-\frac{E_{k,l,n}}{\sum_{i=1}^{L}E_{k,i,n}}\right) \tag{38}\]  
where \(E_{k,l,n}\) denotes the smallest Euclidean distance which is associated with the accuracy of RSU beam tracking. Smaller Euclidean distances represent more accurate position prediction which leads to effective beam tracking. \(w_{k,l,n}\) denotes the weight of each RSU involved in the cooperation, while smaller Euclidean distances or more accurate predictions of kinematic parameters are associated with larger weights. \(1-\frac{E_{k,l,n}}{\sum_{i=1}^{L}E_{k,i,n}}\) denotes the prediction confidence of the \(l\)-th RSU, which is positively correlated with the weight of cooperation. \(L\) denotes the total number of RSUs involved in the cooperation. As for the limitation of Equation (38), the cooperation weight \(w_{k,l,n}\) is limited between zero and one, and the sum of the weights of all RSUs is limited to one by using the term \(\frac{1}{L-1}\), whichthus guarantees the effectiveness of the cooperation results. In addition, there are no other limitations on the use of Equation (38).