## 6 Tensor-Based Design of Vehicle Combination Matrix  
We can extend the time dimension of the RIS-vehicle channel matrix in (34) by expressing the tensor form as follows:  
\[\begin{array}{l}\mathbf{\mathcal{G}}=\sqrt{\frac{N_{t}N_{BS}}{L_{bl}}}\sqrt{A_{ kl}}\sum\limits_{l=1}^{L_{bl}}\mathbf{a}_{lE}(\phi_{RL,r}^{(l)},\theta_{RL,r}^{(l)}) \circ\mathbf{a}_{RIS}^{c}(\phi_{RL,l}^{(l)},\theta_{RL,l}^{(l)})\circ\widetilde{ \mathbf{\beta}}_{l}\\ =\sqrt{\frac{N_{t}N_{BS}L_{bl}}{L_{bl}}}\cdot(\mathbf{\Sigma}\times_{1}\mathbf{A}_{l} \times_{2}\mathbf{A}_{R}^{c}\times_{3}\mathbf{\Sigma}_{\mathbf{\mathcal{G}}})\end{array} \tag{72}\]where \(\widetilde{\mathbf{\beta}}_{l}=[\beta_{l}(1),\ldots,\beta_{l}(T)]^{T}\in\mathbb{C}^{T \times 1}\) is the component vector of the gain under each path containing the Doppler shift; \(\mathbf{\mathcal{G}}\) is the third-order channel tensor of the RIS-vehicle; and its three factor matrices are defined as  
\[\mathbf{A}_{U}=[\mathbf{a}_{UE}((\varphi^{(1)}_{RU,r^{\prime}},\theta^{(l)}_{RU,r}), \ldots,\mathbf{a}_{UE}((\varphi^{(L_{RUl})}_{RU,r},\theta^{(L_{RUl})}_{RU,r})])\in \mathbb{C}^{N_{U}\times L_{RU}} \tag{73}\]  
\[\mathbf{A}_{R}=[\mathbf{a}_{RIS}(\varphi^{(1)}_{RU,r},\theta^{(1)}_{RU,l}),\ldots,\mathbf{ a}_{RIS}(\varphi^{(L_{RUl})}_{RU,l},\theta^{(L_{RUl})}_{RU,l})]\in\mathbb{C}^{N_{ RIS}\times L_{RUl}} \tag{74}\]  
\[\Xi\mathbf{\mathcal{G}}=[\widetilde{\mathbf{\beta}}_{1},\ldots,\widetilde{\mathbf{\beta}} _{L_{RUl}}]\in\mathbb{C}^{T\times L_{RUl}} \tag{75}\]  
Then, the received signal \(\mathbf{Y}(k)\) in (29) can be expressed as  
\[\mathbf{Y}(k)=\sqrt{\frac{P_{t}}{N_{s}}}\mathbf{W}^{H}_{all}\mathbf{H}_{RU}\widetilde\mathbf{{F}}_{all}\mathbf{s_{2}}+\mathbf{N} \tag{76}\]  
\[\widetilde{\mathbf{F}}_{all}=\mathbf{\Phi}\mathbf{H}_{\text{BR}}\mathbf{F}_{all}=\widetilde\mathbf{{F}}_{RF}\widetilde{\mathbf{F}}_{BB} \tag{77}\]  
Based on the above transformation, we can consider \(\widetilde{\mathbf{F}}_{all}\) as the equivalent beamforming matrix. \(\widetilde{\mathbf{F}}_{BB}\) and \(\widetilde{\mathbf{F}}_{RF}\) are the baseband equivalent digital beamformer and analogue equivalent beamformer, respectively. Then, the tensor form of the received signal \(\mathbf{Y}(k)\) can be derived from the channel tensor \(\mathbf{\mathcal{G}}\) of the RIS- vehicle, i.e.,  
\[\begin{array}{l}\overline{\mathbf{\mathcal{Y}}}=\sqrt{\frac{P_{t}}{N_{s}}}(\bm {\mathcal{G}}\times_{1}\mathbf{W}^{H}_{all}\times_{2}\widetilde{\mathbf{F}}^{T}_{all} \mathbf{s_{2}})+\overline{\mathbf{\mathcal{N}}}\\ =\sqrt{\frac{P_{t}}{N_{s}}}(\mathbf{\mathcal{I}}\times_{1}\mathbf{W}^{H}_{all}\mathbf{A}_ {U}\times_{2}\widetilde{\mathbf{F}}^{T}_{all}\mathbf{A}^{C}_{R}\mathbf{s_{2}}\times_{3} \mathbf{\Xi}_{\mathbf{\mathcal{G}}})+\overline{\mathbf{\mathcal{N}}}\end{array} \tag{78}\]  
When the beamforming matrix of BS \(\mathbf{F}_{all}\) and the phase shift matrix of RIS \(\mathbf{\Phi}\) are determined, \(\widetilde{\mathbf{F}}_{all}\) is a known quantity. At this moment, the digital combiner \(\mathbf{W}_{BB}\) by using the MMSE criterion is represented as  
\[(\mathbf{W}_{BB})^{H}=\mathbb{E}[\mathbf{s_{2}}\overline{\mathbf{y}}^{H}(n)]\mathbb{E}[ \overline{\mathbf{y}}(n)\overline{\mathbf{y}}^{H}(n)]^{-1} \tag{79}\]  
where \(\overline{\mathbf{y}}(n)\in\mathbb{C}^{M_{r}\times 1}\) is a column tensor of the tensor \(\overline{\mathbf{\mathcal{Y}}}\), \(\mathbb{E}[\mathbf{  
Finally, the combination matrix of vehicle can be expressed as  
\[\mathbf{W}=\mathbf{W}_{RF}(\widetilde{\mathbf{H}}_{RF}(n)\widetilde{\mathbf{F}}_{BB}\widetilde{ \mathbf{F}}_{BB}^{H}\widetilde{\mathbf{H}}_{RF}^{H}(n)+\frac{N_{s}\sigma^{2}}{P_{t}}\mathbf{W }_{RF}^{H}\mathbf{W}_{RF})^{-1}\widetilde{\mathbf{H}}_{RF}(n)\widetilde{\mathbf{F}}_{BB} \tag{83}\]