## 4 Signal Processing Algorithms for RIS Channel Estimation
### The Conventional Algorithms for RIS Channel Estimation  
In the generic channel estimation method, the transmitter transmits the pilot sequence and the receiver receives the signal sequence to recover the channel matrix in the constructed channel model.  
**LS/LMMSE:** LS estimation [77] and LMMSE are classical channel estimation methods for finding model solutions for channel estimation, which was widely used in RIS-aided systems for pilot-based channel estimation. The RIS channel estimation of conventional LS/LMMSE has a large channel estimation pilot overhead; to minimize the LS/LMMSE channel estimation error, the training reflection pattern and the transmit pilot sequence can be designed jointly [78; 79].  
The conventional LS estimation scheme had a high pilot overhead, which was reduced in [80]. Using a low-rank subspace, a reduced subspace LS estimation scheme and the RIS phase shift pattern was optimized to minimize the MSE of the channel estimation. [PERSON] et al. [81] applied minimum MSE (MMSE) channel estimation to unmanned aerial vehicle (UAV) scenarios with low-complexity optimization of the RIS phase shift matrix.  
At high Doppler frequencies, the existing RIS channel estimation techniques assuming block fading model pairs cause large errors. In [82], the characteristic of Doppler frequency shift of the time-varying channel leads to a large estimation overhead. To reduce the computational complexity, a correlation coefficient between direct line-of-sight and non-line-of-sight was established. The function of the channel was modeled and the MMSE estimation process was refined to improve the accuracy.  
**Grouping RIS:** To reduce the training overhead and make full use of the spatial correlation of the channel, a scheme of grouping RIS elements was proposed in [71]. The large RIS was divided into several small RIS groups and each group shared the same common reflection coefficients, which was widely used in the subsequent RIS channel estimation process.  
\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
For the channel estimation problem of the uplink channel of a RIS-assisted multi-user MIMO (MU-MIMO) system, the entire RIS surface was divided into sub-RISs and an algorithm for estimating the channel using the symmetric positive definite (SPD) matrix [83] was proposed.  
**Kalman filtering (KF):** The channel coefficient matrix was kept constant during the coherent time and the channel was estimated for the elements of the multiple time blocks [84]. [PERSON] et al. [85] used KF for mobile scenes and derived the best reflection coefficients for RIS elements according to the MMSE.  
For downlink communication system in a frequency division duplex MIMO system, [PERSON] et al. [86] proposed that the direct channel and the cascaded channel were estimated by two KF tracking. The reflection coefficients at the pilot and RIS were designed jointly and the optimal coefficient matrix was approximated by the columns of the discrete Fourier transform (DFT) matrix.  
**Multi-stages:** To further improve the estimation accuracy, the distributed multi-stage algorithm was used for RIS channel estimation widely. To obtain higher accuracy channel estimation with lower pilot overhead, in [87], an anchor-assisted multi-stage multiuser wideband cascaded channel estimation scheme was proposed to estimate the cascaded channels. In the first stage, [PERSON] et al. [87] obtained BS-RIS CSI based on cyclic prefix single carrier transmission and phase configuration at RIS. In the second stage, the cascade channel information was estimated in the frequency domain using partial BS-RIS channel states.  
The conventional-based channel estimation algorithms and performance comparison are described in Table 5. Table 5 describes the methods from four aspects: antenna/RIS architecture, channel model, major algorithm, and performance analysis.