  Emerging technologies, such as holographic multiple-input multiple-output
(HMIMO) and stacked intelligent metasurface (SIM), are driving the development
of wireless communication systems. Specifically, the SIM is physically
constructed by stacking multiple layers of metasurfaces and has an architecture
similar to an artificial neural network (ANN), which can flexibly manipulate
the electromagnetic waves that propagate through it at the speed of light. This
architecture enables the SIM to achieve HMIMO precoding and combining in the
wave domain, thus significantly reducing the hardware cost and energy
consumption. In this letter, we investigate the channel estimation problem in
SIM-assisted multi-user HMIMO communication systems. Since the number of
antennas at the base station (BS) is much smaller than the number of meta-atoms
per layer of the SIM, it is challenging to acquire the channel state
information (CSI) in SIM-assisted multi-user systems. To address this issue, we
collect multiple copies of the uplink pilot signals that propagate through the
SIM. Furthermore, we leverage the array geometry to identify the subspace that
spans arbitrary spatial correlation matrices. Based on partial CSI about the
channel statistics, a pair of subspace-based channel estimators are proposed.
Additionally, we compute the mean square error (MSE) of the proposed channel
estimators and optimize the phase shifts of the SIM to minimize the MSE.
Numerical results are illustrated to analyze the effectiveness of the proposed
channel estimation schemes.
