# A Survey on Optimal Channel Estimation Methods for RIS-Aided Communication Systems
## 2 System Models for RIS-Assisted Systems
### MIMO Systems
#### 2.2.1 Cell-Free Communication System  
* Scenario Caption and Signal Model  
Figure 2 illustrates an RIS-assisted single user cell-free communication system. Based on [80], they assume a time division duplex (TDD) mMIMO system, where \(M\) BSs with the assistance of \(J\) RISs serve \(K\) UE. The \(m\)-th BS is equipped with \(N_{m}\) antennas and each RIS has \(N_{j}\) reflective elements, respectively, where \(m=1,\,2,\ldots,\,M\) and \(j=1,\,2,\ldots,\,N\). In this model, the RISs function as antenna arrays far away from the BS to improve capacity and provide greater coverage at a low cost, to enhance cell-free communications. In addition, a central processing unit (CPU) facilitates the joint signal processing of multiple BSs. The RISs are managed by the wire to the BSs or CPU. Quasi-static block-fading channels are considered; the channel matrices from the \(m\)-th BS to the \(j\)-th RIS are represented by \(F_{mj}\in\mathbb{C}^{N_{j}\times N_{m}}\) and the channel matrices from the \(j\)-th RIS to the \(k\)-th UE are represented by \(h_{jk}^{H}\in\mathbb{C}^{1\times N_{m}}\). For the \(j\)-th RIS, the reflection coefficient matrix is \(V_{j}=diag(v_{j})\in\mathbb{C}^{N_{j}\times N_{j}}\), where \(v_{j}=\left[v_{j,1},v_{j,2},\ldots,v_{j,N}\right]^{T}\) and \(v_{j,l}=p_{j,l}e^{i\theta_{j,l}}\) is the reflection coefficient of the \(l\)-th element of the \(j\)-th RIS, \(\theta_{j,l}=1\) and \(\theta_{j,l}\) are the amplitude and phase, accordingly. The channel matrix from the \(m\)-th BS to the \(k\)-th UE affected by the \(j\)-th RIS is \(h_{jk}^{H}V_{j}F_{mj}\). Additionally, the RIS has the same number of reflection elements, i.e., \(L_{1}\) = \(L_{2}\) = \(\ldots\) = \(L_{N}\) = \(L\). The downlink CSI can be achieved by the uplink channel estimation due to TDD systems. To estimate the uplink channel, the orthogonal pilot sequences and reflection coefficients for the channel measurement should be designed. It also consists of subframes Q for RISs, and each subframe contains symbol durations \(T\), where \(T\geq K\)[80]. The reflection vector of the \(j\)-th RIS in the \(q\)-th sub-frame is equal to \(v_{j}^{t}\), the pilot sequence of the \(k\)-th UE is \(s_{k}^{H}=\left[s_{k,1},s_{k,2},\ldots,s_{k,T}\right]^{T}\in\mathbb{C}^{1 \times T}\) and \(s_{k}^{H}s_{k}=0,k\
eq l\), \(s_{k}^{H}s_{k}=0\). The \(j\)-th RIS is \(h_{jk}^{H}=\left[s_{k,1},s_{k,2},\ldots,s_{k,T}\right]^{T}\in\mathbb{C}^{1 \times T}\) and \(s_{  
\(\sigma_{\rho}^{2}T\), where \(\sigma_{\rho}^{2}\) is the transmit power of each UE. The channel that directly links the BS and the UE considers all RISs to be disabled and the direct channel estimation is not considered. In the \(q\)-th sub-frame, the received signal \(Y_{m,q}\in\mathbb{C}^{N_{m}\times T}\) at the \(m\)-th BS is written as  
\[Y_{m,q}=\sum_{k=1}^{K}\sum_{j=1}^{\rm N}F_{mj}^{H}V_{j}^{q}h_{jk}s_{k}^{H}+W_{m,q}, \tag{7}\]  
where \(W_{m,q}\in\mathbb{C}^{N_{m}\times T}\) represents the incoming Gaussian noise following \(\mathcal{CN}\Big{(}0,\sigma_{j}^{2}I_{N_{m}}\Big{)}\)[80]. In [80], since the orthogonal pilot sequences are employed and considering the low mobility scenarios for the \(k\)-th UE, we obtain  
\[\widetilde{y}_{m,q,k}=\frac{1}{\sigma_{\rho}^{2}T}Y_{m,q}s_{k}=\sum_{j=1}^{ \rm J}F_{mj}^{H}V_{j}^{q}h_{jk}+\widetilde{W}_{m,q}, \tag{8}\]  
where \(\widetilde{W}_{m,q}=\frac{1}{PT}W_{m,q}\). By collecting the signals of the Q sub-frames via raw stacking, the received signal \(\widetilde{Y}_{m,k}\in\mathbb{C}^{N_{m}\mathbb{Q}\times 1}\) is expressed as  
\[\widetilde{Y}_{m,k}= \begin{bmatrix}\sum\limits_{j=1}^{\rm N}F_{mj}^{H}V_{j}^{ 1}h_{jk}\\ \sum\limits_{j=1}^{\rm N}F_{mj}^{H}V_{j}^{2}h_{jk}\\ \vdots\\ \sum\limits_{j=1}^{\rm N}F_{mj}^{H}V_{j}^{Q}h_{jk}\end{bmatrix} + \begin{bmatrix} \widetilde{W}_{m,1}\\ \widetilde{W}_{m,2}\\ \vdots\\ \widetilde{W}_{m,Q}\end{bmatrix} .