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.

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