# A Survey on Optimal Channel Estimation Methods for RIS-Aided Communication Systems
## 4 Results of the Proposed Algorithms
### MIMO Systems  
In [1], the authors use tensor algorithms for channel estimation in RIS-assisted MIMO systems, with trilinear and quadrilinear PARAFAC and it is referred to in Section 3. The results are that the proposed algorithms have a high-performance estimate for imperfections in the channel model and system settings. Furthermore, TALS-STI and HOSVD-STI algorithms have a similar performance. The TALS-STI algorithm is preferred for defect detection in the low SNR regime and when more flexible choices for the training parameters are required. Finally, HOSVD-STI is preferred when low processing latency is desired. In [80], the authors analyze an RIS-assisted cell-free mMIMO system, investigate the multi-BS cooperation and multi-UE joint estimation. They use the 3D-MLAOMP algorithm, and it is referred to in Section 3. In future work, the derivation of the results for time-scale channel estimation through multi-BS cooperation will be explored. In addition, non-orthogonal pilot sequences based on RIS-assisted cell-free channel estimation in high-mobility scenarios will be investigated. In [79], a closed-form analytical expression on the achievable sum of RIS-assisted cell-free systems is proposed, this evaluates the effect of key parameters of the system performance. To gain more information, a special case nLoS element was investigatedwith a power gain in the order of O(M). The proposed low-complexity algorithm uses a two-time-scale transmission protocol, so that the joint beamformers in the BS and RIS are optimized to increase the achievable weighted sum. In addition, the RIS beamformers were improved based on the penalty double decomposition (PDD) method exploiting S-CSI, while the BS beamformers were designed by the primal double degradation (PDS) method dependent on I-CSI. The influence of key system parameters, such as the number of RIS elements, CSI settings and the Rician factor was tested. Finally, the advantages of adopting the cell-free paradigm and developing the exploitation of RISs were demonstrated. In [2], a joint channel estimation and data detection (JCEDD) scheme for a HRIS-assisted mmWave orthogonal time frequency space (OTFS) system was proposed. They suggested a transmission structure, where the OTFS blocks were escorted by several pilot sequences. In the duration of the pilot sequences, partial HRIS elements were alternatively initiated in passive mode, and the impinging signal was entirely absorbed. The time domain channel model was investigated. In addition, the received signal model at both the HRIS and the BS was presented. Because CSI between the UE and the HRIS is obtained by the pilot sequences. HRIS beamforming design strategy improves the received signal strength at the BS. For the OTFS transmission, a JCEDD scheme was proposed. In this scheme, they resorted to a probabilistic graphical model, and designed a message passing (MP) algorithm to concurrently recover the data symbols and assess the equivalent channel gain. Moreover, an expectation maximization (EM) algorithm was employed to acquire channel parameters, i.e., the channel sparsity, the channel covariance, and the Doppler frequency shift. By iteratively processing between the MP and EM algorithms, the delay-Doppler domain channel and the transmitted data symbols can be acquired at the same time. Simulation results were required to validate the proposed JCEDD scheme and its robustness to the user velocity. In [3], the effect of channel aging on RIS-assisted mMIMO systems is studied and the effects of spatial correlation and I-CSI are considered. Correlation of channel aging and the Rayleigh attenuation in the data transmission phase and uplink training phase is introduced, the channel efficiency is estimated, and the DE attainable closed-form normalized zero-pressure RZF and sum SE is presented. Channel growth regarding the RIS phase shifts and power budget constraints is presented, applying the S-CSI-based alternating optimization (AO) algorithm to reduce computational complexity and feedback overhead.