## 3 System Model  
Figure 1 shows the system model of the RIS-assisted mmWave communication, where RIS is used to connect the mmWave BS with a single-antenna mmWave user equipment (UE) by routing around the blocker. RIS is equipped with a uniform planner array (UPA) of \(M\) antenna elements, while mmWave BS is equipped with a uniform linear array (ULA) of \(N\) antenna elements. An RIS controller is used to control the PSs of RIS antenna elements based on the selected PS vector. In addition, mmWave BS and RIS are connected through a dedicated communication link for controlling and information exchange. As a result, the received signal at the UE can be expressed as follows:  
\[x=\mathbf{h}_{RU}^{H}\mathbf{\Phi}_{i}\mathbf{H}_{BR}f_{j}s+n, \tag{1}\]In (1), \(s\) is the transmitted symbol, and \(x\) is the received one where \(\mathbb{E}\big{[}ss^{H}\big{]}=P\), and \(P\) is the TX power. \(\left(.\right)^{H}\) means Hermitian transpose and \(n\sim\mathcal{CN}\big{(}0,\sigma_{0}^{2}\big{)}\) is the complex additive white Gaussian noise (AWGN) with zero mean and variance of \(\sigma_{0}^{2}\). \(\mathbf{f}_{j}\in\mathbb{C}^{N\times 1}\) is the analog precoder vector of size \(N\times 1\) applied at the mmWave BS, and \(\mathbf{\Phi}_{i}\in\mathbb{C}^{M\times M}\) is a diagonal matrix of size \(M\times M\) containing the PSs of the RIS antenna elements in its diagonal. \(\left\{i,j\colon 1\leq i\leq|\mathcal{R}|,\ 1\leq j\leq|\mathcal{F}|\right\}\) are the indices of the used \(\mathbf{\Phi}\) and \(\mathbf{f}\), where \(\mathcal{R}\) and \(\mathcal{F}\) are their finite sets. \(\mathbf{H}_{BR}\in\mathbb{C}^{M\times N}\) is the channel matrix of size \(M\times N\) between BS and RIS, while \(\mathbf{\mathbf{h}}_{RU}\in\mathbb{C}^{M\times 1}\) is the channel vector of size \(M\times 1\) between the RIS and UE. Following the geometric channel models with limited scatterers given in [30], \(\mathbf{\mathbf{H}}_{BR}\) and \(\mathbf{\mathbf{h}}_{RU}\) can be expressed as follows:  
\[\mathbf{\mathbf{H}}_{BR}=\sqrt{\frac{MN}{L_{BR}}}\sum_{l=1}^{L_{BR}}\xi_{l}\mathbf{ \mathbf{\Lambda}}_{R}\Big{(}\chi_{l}^{\left(AoA\right)},\delta_{l}^{\left(AoA \right)}\ \Big{)}\mathbf{\mathbf{\Lambda}}_{B}\Big{(}\chi_{l}^{\left(AoD \right)}\ \Big{)}, \tag{2}\]  
\[\mathbf{\mathbf{h}}_{RU}=\sqrt{\frac{M}{L_{RU}}}\sum_{l=1}^{L_{RU}}v_{l}\mathbf{ \mathbf{\Lambda}}_{R}\Big{(}\theta_{l}^{\left([PERSON]\right)},\phi_{l}^{\left([PERSON] \right)}\ \Big{)}, \tag{3}\]  
where \(L_{BR}\) and \(L_{RU}\) are the number of channel paths between BS and RIS and between RIS and UE, respectively. \(\xi_{l}\sim\mathcal{CN}\Big{(}0,\sigma_{\xi_{l}}^{2}\Big{)}\) and \(v_{l}\sim\mathcal{CN}\Big{(}0,\sigma_{\eta_{l}}^{2}\Big{)}\) are the complex path gains of the \(l\)-th path in \(L_{BR}\) and \(L_{RU}\), respectively. \(\mathbf{\mathbf{\Lambda}}_{R}\Big{(}\chi_{l}^{\left(AoA\right)},\delta_{l}^{ \left(AoA\right)}\ \Big{)}\), and \(\mathbf{\mathbf{\Lambda}}_{B}\Big{(}\chi_{l}^{\left(AoD\right)}\ \Big{)}\) are the response vectors of the \(l\)-th path array at the RIS and BS, where \(\chi_{l}^{\left(AoA\right)}\Big{(}\delta_{l}^{\left(AoA\right)}\Big{)}\) and \(\chi_{l}^{\left(AoD\right)}\ \) are the azimuth (elevation) angle of arrival (AoA) and angle of departure (AoD), respectively. In addition, \(\mathbf{\mathbf{\Lambda}}_{R}\Big{(}\theta_{l}^{\left(AoD\right)},\phi_{l}^{\left( AoD\right)}\ \Big{)}\) is the response vector of the \(l\)-th path at the RIS, where \(\theta_{l}^{\left([PERSON]\right)}\) and \(\phi_{l}^{\left([PERSON]\right)}\) are the corresponding azimuth and elevation AoD. Generally, for any \(\theta\) and \(\phi\), \(\mathbf{\mathbf{\Lambda}}_{R}(\theta,\phi\ )\) can be expressed as follows [30]:  
\[\mathbf{\mathbf{\Lambda}}_{R}(\theta,\phi\ )=\frac{1}{\sqrt{M}}\Big{[}1,\ldots,\ e^{j\frac{2\pi}{\lambda}d \left(p\sin\left(\theta\right)+q\cos\left(\phi\right)\right)},\ldots\Big{]}^{T}, \tag{4}\]  
where \(d\) is the antenna spacing and \(\lambda\) is the carrier wavelength and \(0\leq\left\{p,q\right\}\leq\left(\sqrt{M}-1\right)\).