Stacked intelligent metasurface (SIM) is an emerging programmable metasurface
architecture that can implement signal processing directly in the
electromagnetic wave domain, thereby enabling efficient implementation of
ultra-massive multiple-input multiple-output (MIMO) transceivers with a limited
number of radio frequency (RF) chains. Channel estimation (CE) is challenging
for SIM-enabled communication systems due to the multi-layer architecture of
SIM, and because we need to estimate large dimensional channels between the SIM
and users with a limited number of RF chains. To efficiently solve this
problem, we develop a novel hybrid digital-wave domain channel estimator, in
which the received training symbols are first processed in the wave domain
within the SIM layers, and then processed in the digital domain. The wave
domain channel estimator, parametrized by the phase shifts applied by the
meta-atoms in all layers, is optimized to minimize the mean squared error (MSE)
using a gradient descent algorithm, within which the digital part is optimally
updated. For an SIM-enabled multi-user system equipped with 4 RF chains and a
6-layer SIM with 64 meta-atoms each, the proposed estimator yields an MSE that
is very close to that achieved by fully digital CE in a massive MIMO system
employing 64 RF chains. This high CE accuracy is achieved at the cost of a
training overhead that can be reduced by exploiting the potential low rank of
channel correlation matrices