# Radar-Assisted Multiple Base Station Cooperative mmWave Beam Tracking
## 3 Radar-Assisted Multi-Rsu Cooperative Beam Tracking
### Extend Kalman Filtering for Linear and Curvilinear Mobility Scenarios  
In this subsection, we propose an EKF scheme for beam prediction and tracking. Firstly, kinematic state variables at the \((n-1)\)-th slot are used to predict those at the \(n\)-th slot based on the kinematic evolution model. Then, the predicted AOD is exploited to formulate a transmitted beam pointing towards the vehicle and thereby transmitting the DFRC signal. The partial signal is reflected by the vehicle and arrives at the RSU side, which is used to calibrate the kinematic state variables based on the radar measurement model. Normally, calibrated state variables are used as inputs directly for the EKF prediction at the next time slot, however, we reduce the gap between calibration results and true kinematic parameters of the vehicle by adding multi-RSU cooperation. Through a process of iterative prediction of kinematic parameters and beam tracking, the RSU can perform simultaneous sensing and communication with the vehicle. Figure 4: The proposed multi-RSU cooperation-based beam tracking scheme in ISAC scenario. Figure 3: The Frame Structure for radar-assisted communication in V2I scenario. The kinematic evolution model and radar measurement model are needed to enable EKF prediction and calibration, which are given by  
\[\left\{ \begin{array}{l}KinematicEvolutionModel:\mathbf{x}_{n}=\mathbf{p}(x_{n-1})+ \mathbf{q}_{n}\\ RadarMeasurementModel:\mathbf{y}_{n}=\mathbf{f}(x_{n})+\mathbf{w}_{n}\end{array} \right. \tag{22}\]  
where \(x_{n-1}\) is a kinematic state variable at time slot \(n-1\), \(\mathbf{p}()\) and \(\mathbf{f}()\) denote compact forms of predictive and measurement models, respectively. \(\mathbf{q}_{n}\) and \(\mathbf{w}_{n}\) represent predicting and measuring Gaussian noise with zero mean and corresponding variance. For linear mobility scenario, according to [16], \(x=\left[\theta,d,v,\beta\right]^{T}\), \(y=\left[\tilde{r}^{T},\tau,\mu\right]^{T}\), \(\mathbf{p}()\) is defined in (1), with \(\mathbf{q}=\left[q_{\theta},q_{d},q_{v},q_{\beta}\right]^{T}\), \(\mathbf{f}()\) is defined as (12)-(14), with \(\mathbf{w}=\left[w_{\theta}^{T},w_{\tau},w_{\mu}\right]^{T}\). Thus, the variance matrix in the EKF is expressed by  
\[\mathbf{Q}_{s}=diag\Big{(}\sigma_{\theta}^{2},\sigma_{d}^{2},\sigma_{v}^{2}, \sigma_{\beta}^{2}\Big{)} \tag{23}\]  
\[\mathbf{Q}_{m}=diag\Big{(}\sigma_{1}^{2}\mathbf{1}_{N_{\tau}}^{T},\sigma_{2}^ {2},\sigma_{3}^{2}\Big{)} \tag{24}\]  
where \(\mathbf{1}_{N_{\tau}}\) denotes a column vector consisting of all ones with a length of \(N_{\tau}\). For curvilinear mobility scenario, on the top of [16], \(x=\left[\theta,d,v_{y},v_{x},\beta\right]^{T}\), \(y=\left[\tilde{r}^{T},\tau,\mu\right]^{T}\), \(\mathbf{p}()\) is defined in (3), with \(\mathbf{q}=\left[q_{\theta},\,q_{d},\,q_{v_{y}},\,q_{v_{x}},\,q_{\beta}\right] ^{T}\), \(\mathbf{f}()\) is defined as (12)-(15), with \(\mathbf{w}=\left[w_{\theta}^{T},w_{\tau},w_{\mu}\right]^{T}\). Thus, the variance matrix in the EKF is expressed by  
\[\mathbf{Q}_{s}=diag\Big{(}\sigma_{\theta}^{2},\sigma_{d}^{2},\sigma_{v_{y}}^ {2},\sigma_{v_{x}}^{2},\sigma_{\beta}^{2}\Big{)} \tag{25}\]  
\[\mathbf{Q}_{m}=diag\Big{(}\sigma_{1}^{2}\mathbf{1}_{N_{\tau}}^{T},\sigma_{2}^ {2},\sigma_{3}^{2}\Big{)} \tag{26}\]  
We have acquired the kinematic evolution model and radar measurement model which can enable the EKF technique. In accordance with the typical process of Kalman filtering [21], the kinematic parameters prediction and calibration are given by  
1. Predict Kinematic State Variable \[\mathbf{\hat{x}}_{n|n-1}=\mathbf{p}(\mathbf{\hat{x}}_{n-1}),\mathbf{\hat{x}} _{n+1|n-1}=\mathbf{p}\Big{(}\mathbf{\hat{x}}_{n|n-1}\Big{)}\] (27)
2. Linearize Kinematic Evolution Model and Radar Measurement Model \[\mathbf{P}_{n-1}=\frac{\partial\mathbf{p}}{\partial\mathbf{x}}|_{\mathbf{x} =\mathbf{\hat{x}}_{n-1}},\mathbf{F}_{n}=\frac{\partial\mathbf{f}}{\partial \mathbf{x}}|_{\mathbf{x}=\mathbf{\hat{x}}_{n|n-1}}\] (28)
3.