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Channel Estimation for Stacked Intelligent Metasurfaces in Rician Fading Channels

Anastasios Papazafeiropoulos, Pandelis Kourtessis, Dimitra I. Kaklamani, Iakovos S. Venieris

TL;DR

This work tackles CSI acquisition for SIM-aided HMIMO in multi-user uplink under Rician fading, where the last SIM layer–to–user channel includes a LoS component modeled by $h_k = \sqrt{\frac{\beta_k}{1+\kappa}}(\sqrt{\kappa}\bar{h}_k + \tilde{h}_k)$. It introduces a fully wave-based MMSE estimator operating in a single phase and derives a NMSE expression $NMSE_k = 1 - S_k/D_k$, with $S_k$ and $D_k$ determined by the SIM design, correlation $R$, and path loss. An alternating optimization (AO) algorithm with closed-form gradients is developed to optimize per-layer phase shifts under unit-modulus constraints, demonstrating that increasing $L$ and $N$ reduces NMSE and can approach conventional MIMO performance for moderate sizes. The results support practical CSI procedures for SIM-HMIMO deployments and pave the way for subsequent rate analyses in uplink and downlink.

Abstract

The recent combination of the rising architectures, known as stacked intelligent metasurface (SIM) and holographic multiple-input multiple-output (HMIMO), drives toward breakthroughs for next-generation wireless communication systems. Given the fact that the number of elements per surface of the SIM is much larger than the base station (BS) antennas, the acquisition of the channel state information (CSI) in SIM-aided multi-user systems is challenging, especially when a line-of-sight (LoS) component is present. Thus, in this letter, we address the channel procedure under conditions of Rician fading by proposing a protocol in terms of a minimum mean square error (MMSE) estimator for wave-based design in a single phase. Moreover, we derive the normalized mean square error (NMSE) of the suggested estimator, and provide the optimal phase shifts minimising the NMSE. Numerical results illustrate the performance of the new channel estimation protocol.

Channel Estimation for Stacked Intelligent Metasurfaces in Rician Fading Channels

TL;DR

This work tackles CSI acquisition for SIM-aided HMIMO in multi-user uplink under Rician fading, where the last SIM layer–to–user channel includes a LoS component modeled by . It introduces a fully wave-based MMSE estimator operating in a single phase and derives a NMSE expression , with and determined by the SIM design, correlation , and path loss. An alternating optimization (AO) algorithm with closed-form gradients is developed to optimize per-layer phase shifts under unit-modulus constraints, demonstrating that increasing and reduces NMSE and can approach conventional MIMO performance for moderate sizes. The results support practical CSI procedures for SIM-HMIMO deployments and pave the way for subsequent rate analyses in uplink and downlink.

Abstract

The recent combination of the rising architectures, known as stacked intelligent metasurface (SIM) and holographic multiple-input multiple-output (HMIMO), drives toward breakthroughs for next-generation wireless communication systems. Given the fact that the number of elements per surface of the SIM is much larger than the base station (BS) antennas, the acquisition of the channel state information (CSI) in SIM-aided multi-user systems is challenging, especially when a line-of-sight (LoS) component is present. Thus, in this letter, we address the channel procedure under conditions of Rician fading by proposing a protocol in terms of a minimum mean square error (MMSE) estimator for wave-based design in a single phase. Moreover, we derive the normalized mean square error (NMSE) of the suggested estimator, and provide the optimal phase shifts minimising the NMSE. Numerical results illustrate the performance of the new channel estimation protocol.

Paper Structure

This paper contains 9 sections, 2 theorems, 33 equations, 4 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Channel Estimation
  4. Uplink Training
  5. NMSE Optimization
  6. Numerical Results
  7. Conclusion
  8. Proof of Theorem \ref{['Theorem1']}
  9. Proof of Proposition \ref{['propositionGradient']}

Key Result

Theorem 1

The MMSE estimate of the channel vector ${\mathbf{g}}_{k}$ is given by where Also, the covariance of $\hat{{\mathbf{g}}}_{k}$ is given by while the NMSE for user $k$ is obtained as where $S_{k}={{\kappa_{k}}}N\tr({{\mathbf{W}}^{1}}^{{\mathsf{H}}}{\mathbf{G}}^{{\mathsf{H}}}{\mathbf{G}}{\mathbf{W}}^{1})+q_{k}\tr( {\boldsymbol{\Psi}}_{k}{\mathbf{Q}}_{k}{\boldsymbol{\Psi}}_{k})$ and $D_{k}=\tr({{\

Figures (4)

  • Figure 1: Uplink of a multi-user SIM-aided MISO system.
  • Figure 2: NMSE versus the number of iterations.
  • Figure 3: NMSE versus the number of layers.
  • Figure 4: NMSE versus the training SNR.

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Remark 1
  • Proposition 1
  • proof