# Radar-Assisted Multiple Base Station Cooperative mmWave Beam Tracking
## 2 System Model
### Communication Signal Model  
At time slot \(n\), the \(k\)-th vehicle formulates receiving beam to receive the downlink signal from the RSU, which can be expressed as  
\[c_{k,n}(t)=\sqrt{p_{k,n}}\alpha_{k,n}\mathbf{w}_{k,n}^{H}\mathbf{H}_{k,n} \mathbf{f}_{k,n}\mathbf{s}_{k,n}(t)+n_{c}(t) \tag{17}\]  
where \(\alpha_{k,n}=\widetilde{\alpha}e^{j\frac{2\pi}{\lambda}d_{k,n}}\) denotes the channel coefficient during communication, in which \(\widetilde{\alpha}\) is path loss, \(\frac{2\pi}{\lambda}d_{k,n}\) is the phase of the channel. \(\mathbf{H}_{k,n}\) denotes the mmWave channel between RSU and vehicle. \(\mathbf{w}_{k,n}^{H}\) and \(\mathbf{f}_{k,n}\) denote beam combining and precoding vector, and are similarly defined as in (8) with \(N_{t}\) and M antennas, respectively. \(n_{c}(t)\) denotes the Gaussian noise which has a mean of zero and a variance of \(\sigma_{c}^{2}\). Specifically, path loss \(\widetilde{\alpha}\) refers to 3 GPP TR 38.901 [22], which presents a path loss model of rural macrocellular (RMa) scenario including LoS and Non-Line of Sight (NLoS) path. The path loss is calculated as follows  
\[PL=Pr_{LOS}PL_{RMa-LOS}+(1-Pr_{LOS})PL_{RMa-NLoS} \tag{18}\]  
where \(Pr_{LOS}\) denotes the probability of LoS path, \(PL_{RMa-LOS}\) and \(PL_{RMa-LOS}\) denote the loss of the LoS path and NLoS path, respectively. The specific calculations are detailed in Appendix A.1. It is worth noting that \(\mathbf{w}_{k,n}^{H}\) is formulated by AOA at the vehicle side which AOA is the same as AOD in linear mobility scenario, while AOA is different from AOD in curvilinear mobility scenario. Thus, we employ the classical MUltiple SIgnal Classification (MUSIC) algorithm which has superior performance in angular resolution to estimate AOA in curvilinear mobility scenarios at the vehicle side [23]. \(\mathbf{H}_{k,n}\) is a wideband eSV channel containing an LoS path, where Rice factor is introduced to control the power ratio between the LoS and NLoS paths. Moreover, the time-delay of each path is different, thus ensuring that the channel is wideband. The wideband eSV channel used in the article is given by  
\[ \begin{split}\mathbf{H}_{\mathbf{k},n}&=\sum\limits_ {j=1}^{N_{cl}}\sum\limits_{i=1}^{N_{ray}}a_{k,n,j,i}e^{j2\pi\alpha_{k,n,j,i}t} \sqrt{\frac{N_{cl}M}{N_{ny}+1}}\mathbf{b}\left(\theta_{k,n,j,i}^{\alpha x} \right)\mathbf{a}^{H}\left(\theta_{k,n,j,i}^{\alpha x}\right)\sqrt{\frac{1}{K _{rice}+1}}\\ &\quad+a_{k,n}^{LoS}e^{j2\pi e_{k,n}^{LoS}t}\sqrt{\frac{N_{cl}M}{ N_{ny}+1}}\mathbf{b}\left(\theta_{k,n}^{LoS,rx}\right)\mathbf{a}^{H}\left( \theta_{k,n}^{LoS,rx}\right)\sqrt{\frac{K_{rice}}{K_{rice}+1}}\end{split}  \tag{19}\]  
where \(N_{cl}\) is the number of clusters, \(N_{ny}\) is the number of paths in each cluster, and therefore the total number of paths of NLoS is \(N_{cl}N_{ny}\). \(a_{k,n,j,i}\) and \(\phi_{k,n,j,i}=\frac{v_{k,n}\cos\theta_{k,n,j,i}f_{c}}{c}\) denote complex gain and Doppler frequency shift of the \(k\)-th vehicle for the \(i\)-th path in the \(j\)-th cluster at the \(n\)-th slot, respectively. \(N_{t}\) is the number of transmitting antennas at the RSU, while \(M\) is the number of receiving antennas at the vehicle. \(\mathbf{a}(\theta)\) and \(\mathbf{b}(\theta)\) denote array antenna response at the RSU and the vehicle, and are similarly defined as in (6) and (7) with \(N_{t}\) and \(M\) antennas, respectively. \(\theta_{k,n,j,i}^{xx}\) and \(\theta_{k,n,j,i}^{xx}\) denote the AOD and AOA of the \(k\)-th vehicle for the \(i\)-th path in the \(j\)-th cluster at the \(n\)-th slot. \(K_{rice}\) is the Rice factor. Correspondingly, \(a_{k,n}^{LoS}\), \(\theta_{k,n}^{LoS}\), \(\theta_{k,n}^{LoS,rx}\), \(\theta_{k,n}^{LoS,tx}\) are complex gain, Doppler frequency shift, AOA, AOD of the LoS path.