# A Survey on Optimal Channel Estimation Methods for RIS-Aided Communication Systems
## 2 System Models for RIS-Assisted Systems
### MISO Systems
#### 2.1.1 System Model  
The system and channel models for the considered RIS-assisted MISO communication systems are similar to those in [74]. Specifically, Figure 1 depicts an RIS-assisted MU-MISO system with M antennas at the BS, communicating with K users, and the RIS is equipped with \(J\) antennas for signal transmission. Each \(j\)-th reflective element of the RIS has a reflective coefficient \(\xi_{j}\) and the reflectance matrix of the RIS panel is depicted as \(\Xi=\text{diag}(\xi_{1},\ldots,\xi_{j})\), where \(\xi_{j}=e^{j\varphi_{j}}\) with \(\varphi_{j}\) denoting the phase shift of the \(j\)-th element of the RIS. All channel models follow the Rician distribution [74]. \(h_{k,0}\in\mathbb{C}^{M\times 1}\) is the direct channel from BS to the \(k\)-th UE, \(H_{1}\in\mathbb{C}^{J\times M}\) that from the BS to RIS and \(h_{k,2}\in\mathbb{C}^{J\times 1}\) that from the RIS to the \(k\)-th UE. The corresponding effective channel from the transmitter to the \(k\)-th UE would be \(h_{k}^{T}\triangleq h_{k,0}^{T}+h_{k,2}^{T}\Xi\text{H}_{1}\)[74]. The channel matrix \(h_{k,2}\) between the RIS and \(k\)-th UE is given as [74; 75; 76]  
\[h_{k,2}=\sqrt{\frac{\xi_{k,2}\kappa_{k,2}}{1+\kappa_{k,2}}}\bar{h}_{k,2}+ \sqrt{\frac{\delta_{k,2}\kappa_{k,2}}{1+\kappa_{k,2}}}\widetilde{h}_{k,2}, \tag{1}\]  
where \(\sqrt{\delta_{k,2}}\) is the distance dependent path-loss factor, \(\kappa_{k,2}\) is the Rician factor between the RIS and the \(k\)-th UE. Furthermore, \(\bar{h}_{k,2}=a_{RIS}\left(\varphi_{k}^{(RIS)},\psi_{k}^{(RIS)}\right)\) where \(\varphi_{k}^{(RIS)},\psi_{k}^{(RIS)}\) is the azimuth and elevation angle of departure (AoD) from the RIS to the \(k\)-th UE, respectively, and \(a_{RIS}\left(\varphi_{k}^{(RIS)},\psi_{k}^{(RIS)}\right)\) is the array response vector at the RIS side. For \(a_{RIS}\left(\varphi_{k}^{(RIS)},\psi_{k}^{(RIS)}\right)\) is the assumed uniform planar arrays (UPAs) and is given by  
\[ \begin{split} a_{RIS}\left(\varphi_{k}^{(RIS)},\psi_{k}^{(RIS)} \right)=\\ \frac{1}{\sqrt{J_{H}}\text{I}V}\left[\begin{array}{c}1,\ldots,e^ {j\frac{2\pi}{\lambda}D(\text{isin}\,\varphi_{k}^{(RIS)}\sin(\varphi_{k}^{( RIS)})+\text{cos}\,\varphi_{k}^{(RIS)})},\ldots,\\ e^{j\frac{2\pi}{\lambda}D(\left(J_{H}\right)\sin\,\varphi_{k}^{(RIS)}\sin( \varphi_{k}^{(RIS)})+\left(J_{V-1}\right)\,\text{cos}\,\varphi_{k}^{(RIS)} \right)}\end{array}\right]^{T},\end{split}  \tag{2}\]  
where \(D\) is the distance between antenna elements and \(J=J_{H}J_{V}\), where \(J_{H}\) and \(J_{V}\) are the number of elements at the horizontal and vertical axes, accordingly [74]. Figure 1: Illustration of the RIS-assisted MU-MISO communication system. The channels between the BS and RIS, and the direct one from the BS to the \(k\)-th receiver are depicted as [75; 76; 77]  
\[H_{1}=\sqrt{\frac{\delta_{1}\kappa_{1}}{1+\kappa_{1}}}\widetilde{H_{1}}+\sqrt{ \frac{\delta_{1}}{1+\kappa_{1}}}\widetilde{H_{1}}, \tag{3}\]  
\[h_{\text{k},0}=\sqrt{\frac{\delta_{\text{k},0}\kappa_{\text{k},0}}{1+\kappa_{ \text{k},0}}}\widetilde{h_{\text{k},0}}+\sqrt{\frac{\delta_{\text{k},0}}{1+ \kappa_{\text{k},0}}}\widetilde{h_{\text{k},0}}, \tag{4}\]  
where \(\delta_{1}\) is the distance dependent path-loss, \(\kappa_{1}\) is the Rician factor of the BS-RIS link. \(\delta_{\text{u},0}\) and \(\kappa_{\text{u},0}\) stand for the distance dependent path-loss and the Rician factor between BS and the \(k\)-th UE link, correspondingly [74]. In addition, \(\widetilde{H_{1}}=a_{RIS}\left(\varphi^{(RIS)},\psi^{(RIS)}\right)a_{BS} \left(\varphi^{(BS)},\psi^{(BS)}\right)^{H}\), where \(\varphi^{(\text{RIS})}\) and \(\psi^{(\text{RIS})}\) are the azimuth and elevation angle of the angle of arrival (AoA) to the RIS and \(\varphi^{(\text{BS})}\) and \(\psi^{(\text{BS})}\) express the azimuth and elevation AoD from the BS to the RIS direction.