# A Survey on Optimal Channel Estimation Methods for RIS-Aided Communication Systems
## 2 System Models for RIS-Assisted Systems
### MIMO Systems
#### 2.2.2 Cell Communication System  
Figure 3 shows an RIS-assisted downlink cell mMIMO communication system [3], where the BS has \(M\) antennas in communication with the \(K\) single-antenna non-cooperative UEs behind the obstacles. The RIS has \(N_{j}\) reflecting elements, where \(j=1\) and \(2\),..., \(N\) RIS is in the LoS of the BS to assist the contact with the UEs. The same scenario was considered  
where the BS and RIS are installed at a high altitude and with fixed locations. The size of each RIS element is \(dH\times dV\), where \(dV\) and \(dH\) stand for its vertical height and its horizontal width, accordingly. The suggested model also examines the potential presence of direct links between the BS and the UEs. * Channel Model  
A pseudo-static attenuation model with a larger bandwidth, rather than the channel bandwidth, is calculated. They generate a typical block attenuation model with each coherence interval/block included, \(\tau_{c}=B_{\mathbb{C}}T_{\mathbb{C}}\) for the channel usage, \(B_{\mathbb{C}}\) is the amplitude coherence (Hz) and \(T_{\mathbb{C}}\) is the time coherence (s). Inside the transmission of each relevance block and during the \(n\)-th time slot, let \(H_{1}=[h_{11},\dots,h_{1L}]\in\mathbb{C}^{M\times N_{j}}\) and \(h_{2k,j}\in\mathbb{C}^{N_{j}\times 1}\) be the LoS channel between the BS and the RIS and the channel between the RIS and \(k\)-thUE at the \(n\)-th time instant. Consider that \(h_{1i}\) for \(i=1,\dots,N_{j}\) expresses the \(i\)-th column vector of \(H_{1}\). \(g_{k,j}\in\mathbb{C}^{M\times 1}\) is the direct channel between the BS and \(k\)-th UE at the \(n\)-th moment. The greater part of existing studies e.g., [4, 5], assumed the independent Rayleigh model, but, compared fading appeared and changed the performance [83]. \(h_{2k,j}\) and \(g_{k,j}\) are depicted in terms of related Rayleigh fading distributions as  
\[h_{2k,j}=\sqrt{\beta_{h_{2k,j}}R_{RIS,k}}q_{2k,j} \tag{13}\]  
\[g_{k,j}=\sqrt{\beta_{g,k}R_{BS,k}}q_{k,j}, \tag{14}\]  
where \(R_{RIS,k}\in\mathbb{C}^{N_{j}\times N_{j}}\) and \(R_{BS,k}\in\mathbb{C}^{M\times M}\) stand for the deterministic Hermitian-symmetric positive semi-definite correlation matrices at the RIS and the BS accordingly, with \(tr\Big{(}R_{RIS,k}\Big{)}=N_{j1}\) and \(tr\Big{(}R_{BS,k}\Big{)}=M_{1}\) given. Furthermore, \(\beta_{h_{2k,j}}\) expresses the path-losses of the RIS \(k\)UE and \(\beta_{g,k}\) is defined as the path-losses of the BS \(k\)UE links. Particularly, \(\beta_{g,k}\) is supposed to be small because of the blockages between the BS and the UEs. Furthermore, \(q_{2k,j}\sim\mathcal{CN}\Big{(}0,I_{N_{j1}}\Big{)}\) and \(q_{k,n}\sim\mathcal{CN}(0,I_{M1})\) indicate the matching fast-fading vectors at the \(n\)-th moment. Notice that fast fading vectors shift within each coherence block, while the parallel matrices are believed to be continuous for many coherence blocks [3]. The high level LoS channel \(H_{1}\) is illustrated as  
\[[H_{1}]_{m,\hat{j}}=\sqrt{\beta_{1}}e^{(\frac{2\pi}{\hat{\lambda}}(m-1)d_{BS} \sin\theta_{1,N_{j}}\sin\varphi_{1,N_{j}}+(l-1)d_{RIS}\sin\theta_{2,m}\sin \varphi_{2,m})}, \tag{15}\]  
where \(\beta_{1}\) is the path-loss between the BS and RIS, while \(d_{BS}\) and \(d_{RIS}\) are the inter-antenna partition at the BS and inter-element partition at the RIS, accordingly [49]. Moreover, \(\theta_{1,N_{j}}\) and \(\varphi_{1,N_{j}}\) represent the elevation and azimuth line of sight (LoS) AoD at the BS regarding the RIS reflecting element \(N_{j}\), \(\theta_{2,m}\) and \(\varphi_{2,m}\) represent the elevation and azimuth LoS AoA at  
Figure 3: RIS-assisted cell mMIMO communication system. the RIS. It is useful to point out that \(H_{1}\) can be acquired similarly to the covariance matrices since the dependence of their statements on the distances and the angles is related [3].