# A Survey on Optimal Channel Estimation Methods for RIS-Aided Communication Systems
###### Abstract
In [24], the HBF scheme is used in a mmWave RIS-assisted broadband MIMO system with geometric mean beamforming decomposition in order to avoid complex bit/power allocation, while the authors in [23], designed a joint RIS phase shift matrix and HBF with a direct channel between the transmitter and receiver. To make use of the advantages promised by the RIS, accurate CSI is essential, although it is difficult to achieve because a complex signal processing capability of the RIS [25] is missing. Furthermore, the RIS consists of passive components, which do not process signals and do not evaluate the UE-RIS and RIS-BS channel separately, but sequentially (i.e., the cascaded end-to-end channel). When there are many BS antennas and RIS reflectors, the number of sequential channel coefficients will be large; and so will be the number of pilots [26]. This causes a problem for channel estimation in any communication system. Other parameters that affect the channel estimation are the distance of the users from the RIS, the type of communication system used, but mainly the algorithm used for the channel estimation. We cannot assume which algorithm is the best for channel estimation in every RIS-assisted system architecture because each system has different requirements. Hence, there has been great contributions on channel estimation for RIS-assisted communication systems, such as those in [1; 4; 17; 26; 27]. The purpose of the current research is to provide an inclusive and up-to-date survey of papers in RIS-assisted wireless communications, such as MIMO and MISO system communications. Emphasis is given on the practical challenges of BS, RIS and user channels to estimate their value using optimized algorithms with different channel models and system configurations. This paper presents several methods to optimally solve critical issues of channel estimation, such as pilot overhead reduction. The complexity of these methods is presented and the communication systems in which they can be used are selected according to their positive outcome. In addition, the challenges of the proposed methods are mentioned in order to be handled in future research. The rest of the paper is organized as follows: Section 2 presents system models for channel estimation in RIS-assisted systems with different system architectures. Section 3 presents optimal channel estimation algorithms using statistical CSI (S-CSI) or instantaneous CSI (I-CSI). Section 4 presents the results of the proposed optimal algorithms per communication system, and some concluding remarks are drawn in Section 5. Notations: \(x\) is a scalar, \(x\) is a vector, \(\mathbf{X}\) represents a matrix, and \(\mathcal{A}\) denotes a set. \(\left\|a\right\|_{\phi}\), \(\left\|a\right\|_{\mathbb{Z}}\), \(\left\|A\right\|_{l_{1}}\) and \(\left\|A\right\|_{F}\) denote \(\ell_{0}\), \(\ell_{2}\), \(\ell_{2}\) and the Frobenius norm, respectively. \(\bigotimes\) is the Kronecker product. \(\left[a\right]_{i}\), \(\left[A\right]_{i,j}\) and \(\left[A\right]_{i,j,k}\) are the \(i\)-th element of vector \(a\), the \((i,j)\)-th element of matrix \(A\) and the \((i,j,k)\)-th element of tensor \(A\), accordingly. The notations \(\left(\cdot\right)^{T}\), \(\left(\cdot\right)^{H}\) and \(tr(\cdot)\) each stand for the transpose, Hermitian transpose and trace operators. The \(\mathbb{E}[\cdot]\) is the expectation operator and \(diag(a)\) denotes an \(n\times n\) diagonal matrix that consists of the elements of vector \(a\).