## 4 Signal Processing Algorithms for RIS Channel Estimation
### Optimization-Based Algorithms for RIS Channel Estimation  
Optimization algorithms are also used in the RIS channel estimation process commonly. The generalized EM (GEM) algorithm is used to estimate the parameters in the  
\begin{table}
\begin{tabular}{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} & **Channel Model** & **Major Algorithm** & **Performance Analysis** \\ \hline [PERSON] et al. [77] & 2021 & MIMO/Passive RIS & Static channel method & AoA estimation and LS method & Improved estimation performance compared with the existing channel estimation algorithms (With low pilot overhead) \\ [PERSON] et al. [80] & 2022 & MIMO/Passive RIS & Static channel array geometry & Reduced the \(e_{\text{NMSE}}\) to \(-40\) dB (\(r_{\text{SNR}}=0\) dB) \\ [PERSON] et al. [81] & 2022 & SISO/Passive RIS & Time-varying and double selective channel & MMSE and the end-to-end system model & Reduced the required transmitted power across the range of locations spanning from 140 m to 800 m. \\ [PERSON] et al. [83] & 2021 & MIMO/Passive RIS & Static channel factorization & Maximum-margin matrix factorization & Performance achieved up to 6 dB enhancement as the sub-RIS size increases \\ [PERSON] et al. [84] & 2020 & MIMO/Passive RIS & Time-varying channel & RIS-elements grouping and partition & Achieved close rate performance to the case with phase shifts when the number of bits was 3 \\ [PERSON] et al. [85] & 2021 & MIMO/Passive RIS & Time-varying channel & MMSE, KF, and state-space model & Exhibited a better BER performance than the MMSE estimator \\ [PERSON] et al. [86] & 2021 & MIMO/Passive RIS & Time-varying channel & KF and codebook-based low complexity design and improved estimation accuracy \\ [PERSON] et al. [82] & 2023 & SISO/Passive RIS & High-dimensional and high-Doppler reflected fading channels & Reduced the error floor while achieving higher rates in high channel coefficient & mobility systems \\ [PERSON] et al. [87] & 2023 & MIMO/Passive RIS & Static channel & Two-phase and anchor-aided channel estimation & Reduced polit overhead and improved estimation accuracy \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of conventional algorithms for RIS channel estimation.  
absence of variables, the model and parameters are adjusted to obtain the optimal solution continuously. A spatially alternating GEM-based scheme for RIS-assisted channel measurements was presented in [88]. The scheme was estimated by multipath component tracking and maximum likelihood (ML) calculations.  
In [89], a new over-the-air (OTA) method to estimate the true phase shift of the RIS elements was modeled as a high-dimensional non-convex optimization problem. The ML estimates were obtained by iterative algorithm. The root MSE (RMSE) of the phase estimation was closed to the CRB, thus verifying the effectiveness of the proposed algorithm.  
To reduce the pilot overhead by exploiting the common-link variables and the user-specific variables, [PERSON] et al. [60] used an alternating optimization algorithm to achieve a locally optimal solution for a non-convex optimization problem, with a low computational complexity.  
Modelling the millimetre wave channel estimation problem as a non-convex optimisation problem, [PERSON] et al. [24] used alternating minimisation and manifold optimization algorithms to generate locally optimal solutions. To reduce the computational complexity, a multi-stage computer-based algorithm is further designed.  
The optimization-based channel estimation algorithms and performance comparison are described in Table 6. Table 6 describes the methods from four aspects: antenna/RIS architecture, channel model, major algorithm, and performance analysis.