  We consider a narrowband MIMO reconfigurable intelligent surface
(RIS)-assisted wireless communication system and use tensor signal modelling
techniques to individually estimate all communication channels including the
non-RIS channels (direct path) and decoupled RIS channels. We model the
received signal as a third-order tensor composed of two CANDECOMP/PARAFAC
decomposition terms for the non-RIS and the RIS-assisted links, respectively,
and we propose two channel estimation methods based on an iterative alternating
least squares (ALS) algorithm: The two-stage RIS OFF-ON method estimates each
of the non-RIS and RIS-assisted terms in two pilot training stages, whereas the
enhanced alternating least squares (E-ALS) method improves upon the ALS
algorithm to jointly estimate all channels over the full training duration. A
key benefit of both methods compared to the traditional least squares (LS)
solution is that they exploit the structure of the tensor model to obtain
decoupled estimates of all communication channels. We provide the computational
complexities to obtain each of the channel estimates for our two proposed
methods. Numerical simulations are used to evaluate the accuracy and verify the
computational complexities of the proposed two-stage RIS OFF-ON, and E-ALS, and
compare them to the traditional LS methods. Results show that E-ALS will obtain
the most accurate estimate while only having a slightly higher run-time than
the two-stage method.
