  We develop an optimal version of a prior two-stage channel estimation
protocol for RIS-assisted channels. The new design uses a modified DFT matrix
(MDFT) for the training phases at the RIS and is shown to minimize the total
channel estimation error variance. In conjunction with interpolation
(estimating fewer RIS channels), the MDFT approach accelerates channel
estimation even when the channel from base station to RIS is line-of-sight. In
contrast, prior two-stage techniques required a full-rank channel for efficient
estimation. We investigate the resulting channel estimation errors by comparing
different training phase designs for a variety of propagation conditions using
a ray-based channel model. To examine the overall performance, we simulate the
spectral efficiency with MRC processing for a single-user RIS-assisted system
using an existing optimal design for the RIS transmission phases. Results
verify the optimality of MDFT while simulations and analysis show that the
performance is more dependent on the user-to-RIS channel correlation and the
coarseness of the interpolation used, rather than the training phase design.
For example, under a scenario with more highly correlated channels, the
procedure accelerates channel estimation by a factor of 16, while the
improvement is a factor of 5 in a less correlated case. The overall procedure
is extremely robust, with a maximum performance loss of 1.5bits/sec/Hz compared
to that with perfect channel state information for the considered channel
conditions.
