## 4 Signal Processing Algorithms for RIS Channel Estimation
### Compressed Sensing-Based Algorithms for RIS Channel Estimation  
The problem is formulated as an optimization algorithm to recover the original signal as accurately as possible from the sparse projection matrix, which is often used to deal with the channel estimation problem of RIS [90]. The greedy algorithm, a common algorithm for CS, has low computational complexity and is often used in the signal recovery process [91; 92]. Common CS techniques such as ANM [75], OMP algorithm [64], approximation messaging passing (AMP) [93], and tensor decomposition technique [37] were widely used for the channel estimation process. The CS algorithm constantly searches for the most suitable transform base to represent the channel matrix as sparse, which is an optimization problem. The channel matrix is computed accurately using a small amount of information, which reduces the pilot overhead during channel estimation significantly. **OMP algorithm:** The conventional OMP algorithm directly used for RIS channel estimation leads to large complexity, and how to reduce the training overhead becomes a topic worth discussing. The number of RIS components is large and the calculated pressure of the composite channel is high. For the composite channel models, the channel estimation problem was formulated as a sparse recovery problem for a set of independent dictionaries in the angular and temporal domains. The computational complexity of the multidimensional OMP strategy [94] was based on compressed channel estimation, which was greatly reduced by not computing on the large dictionary directly. The OMP algorithm was modified to exploit the joint corner \begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Ref.** & **Year** & \begin{tabular}{c} **Antenna/RIS** \\ **Architecture** \\ \end{tabular} & \begin{tabular}{c} **Channel** \\ **Model** \\ \end{tabular} & \begin{tabular}{c} **Major Algorithm** \\ \end{tabular} & \begin{tabular}{c} **Performance Analysis** \\ \end{tabular} \\ \hline [PERSON] et al. [88] & 2022 & \begin{tabular}{c} MIMO/Passive \\ RIS \\ \end{tabular} & Static channel & \
\begin{tabular}{c} Space-alternating GEM \\ and ML estimation \\ \end{tabular} &
\begin{tabular}{c} Showed better performance with RIS \\ designed by the ideal phase shift \\ Phase estimates were close to the \\ Cramer Rao Bounds (\(r_{\text{SNR}}=20\) dB) \\ Approximately 15 dB performance \\ gain over ON/OFF method \\ Performance \\ Performance improvements \\ achieved compared with several \\ state-of-the-art benchmark schemes \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of optimization-based algorithms for RIS channel estimation. domain sparse structure of the cascaded channels associated with different users. The matrices of the corner domain channels for different users had common non-zero rows. The joint estimation of a large number of common corner domain cascaded channels increased the number of training samples and reduced the effect of noise. **AMP algorithm:** The sparsity of the OMP algorithm is a fixed value, which is not compatible with the actual situation. The further proposed AMP algorithm has adaptive sparsity. A two-stage algorithm including matrix decomposition stage and matrix completion stage was proposed in [68]. In [93], in the first stage, conventional channel training methods were used to estimate the direct MIMO channel among UE terminals. In the second stage, the bilinear adaptive vector approximation messaging algorithm was used for dictionary matrix learning. The sparsity of the phase-shifted coefficient matrix was used to remove the permutation ambiguity from the channel during the training process. By using the statistical properties of the user-to-RIS and RIS-to-BS channels and the central limit theorem (CLT), it can verify the approximate Gaussian distribution when the channel propagates in the sub-6 GHz band. By using Gaussian approximation, the joint channel estimation and data detection technique of the BiG AMP algorithm [95] was proposed.