In MIMO channel estimation, the linear minimum mean squares estimator (LMMSE) yields better performance than the least-squares (LS) estimator. However, it requires knowing the channel covariance matrix (which constitutes prior information). In practice, the channel covariance has to be estimated based on previous channel estimations, but how are these previous channels estimated? With LS? With MMSE using an identity matrix as covariance? Anything else?
You can find a survey of a few different covariance estimation methods in "Toward Massive MIMO 2.0: Understanding Spatial Correlation, Interference Suppression, and Pilot Contamination" (https://arxiv.org/pdf/1904.03406.pdf)
The basic approach is to modify the pilot transmission to obtain observations that can be used to form sample covariance matrices. It can be done by adding extra pilots for that purpose or switch between different pilot assignments. Regularization of the sample covariance matrices (scaling down the off-diagonal elements) is also useful when the covariance matrix estimates are rough.
Thanks a lot for the answer. Reading the corresponding part in your paper, I understand that LS channel estimates are used to form sample covariances (for the R direct method). Have you heard of a method using direction of arrival (DoA) estimates (obtained for example using sparse recovery methods) to constrain channel covariance estimation? (in order to reduce the number of coherence blocks needed to obtain a good covariance estimate)
You can check out the work by Saeid Haghighatshoar and Giuseppe Caire, such as https://arxiv.org/pdf/2110.03324
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