Uplink Channel Estimation for Multi-User MISO Systems Assisted by a Fluid Reconfigurable Intelligent Surface

Technical Summary: Uplink Channel Estimation for Multi-User MISO Systems Assisted by a Fluid Reconfigurable Intelligent Surface

Problem Statement
Fluid Reconfigurable Intelligent Surfaces (FRISs) represent an emerging paradigm where reflecting elements can dynamically select their effective radiating positions from a dense preset grid, offering spatial diversity with fewer physical elements compared to conventional RISs. However, accurate Channel State Information (CSI) is critical for exploiting these gains. While channel estimation (CE) in conventional RIS systems is challenged by hardware nonidealities, FRIS-assisted systems face additional complexities due to the cascaded bilinear channel structure and, crucially, position uncertainties. These uncertainties arise from calibration inaccuracies or motion errors, leading to mismatches between commanded and realized element positions. To the best of the authors' knowledge, CE for FRIS-assisted systems under such position uncertainty has not been previously explored in the literature.

Methodology
The paper proposes a CE framework for a multi-user uplink MISO system where $Q$ single-antenna users communicate with a base station (BS) via an FRIS composed of $M$ fluid elements. The system operates under a two-time-scale FRIS configuration protocol:

1. Phase Shifts: Electronic phase shifts ($\phi$) remain fixed over $K$ blocks within a sub-frame but vary across sub-frames.
2. Element Positions: Controlled reflecting positions ($t$) vary across blocks but are repeated over sub-frames.

The received signal is modeled as a fourth-order tensor, which admits a Parallel Factor (PARAFAC) decomposition. The authors address two distinct scenarios:

• Scenario 1: Perfect Motion Realization (Idealized)
  Assuming the realized positions match the commanded ones, the goal is to estimate the individual channels ($G$ and $H$) given known motion ($T$) and phase ($\Phi$) matrices. The authors formulate a least-squares (LS) problem and utilize Khatri-Rao Factorization (KRF) to decouple the combined channel estimate ($\hat{\Theta}$) into individual channel components ($G$ and $H$). This approach leverages the denoising properties of KRF.

• Scenario 2: Motion Uncertainty (Realistic)
  When position mismatches occur due to motion errors, the motion matrix $T$ is treated as unknown. The authors propose a joint estimation framework to simultaneously recover the motion matrix $T$ and the channel matrices $G$ and $H$.

  • The problem is formulated as minimizing the error between the observed data and a model involving the Khatri-Rao product of $T$, $G^T$, and $H$.
  • The solution employs multidimensional LS Khatri–Rao factorization. Specifically, the columns of the estimated combined tensor are approximated as rank-one tensors.
  • A rank-one truncated Higher-Order Singular Value Decomposition (HOSVD) is applied to extract the dominant singular vectors, yielding the estimates for $\hat{T}$, $\hat{G}$, and $\hat{H}$.

Key Contributions

1. Novel Framework: This work presents the first known CE framework for FRIS-assisted systems that explicitly accounts for element-position uncertainty.
2. Tensor Modeling: The authors derive a closed-form algebraic solution based on tensor modeling (PARAFAC decomposition) that captures both reflection phase-shift and element-motion dynamics.
3. Joint Estimation Algorithm: A specific algorithm is proposed to jointly estimate the motion-induced phase coefficients and the individual channel matrices, addressing the challenge of unknown position deviations.
4. Two-Scenario Analysis: The paper systematically compares performance under idealized motion (where $T$ is known) versus realistic motion uncertainty (where $T$ is blind/unknown).

Performance Results
Numerical simulations evaluate the Normalized Mean Square Error (NMSE) of the combined channel under varying Signal-to-Noise Ratios (SNR) and position error magnitudes ($\sigma_{pos}$):

• Ideal Motion: In the absence of motion errors, the proposed KRF method outperforms standard LS methods and approaches the performance of classical static RIS systems, benefiting from the decoupling and denoising capabilities of KRF.
• Motion Uncertainty: When position errors are present (e.g., $\sigma_{pos} = \lambda/20$), methods that ignore these errors (using mismatched $T$) suffer significant performance degradation. The proposed joint estimation method remains robust, effectively mitigating the impact of unknown motion deviations and achieving performance levels comparable to the idealized motion case and static RIS benchmarks.
• Robustness: The proposed method demonstrates resilience against increasing levels of position error, whereas schemes that neglect motion estimation fail to maintain accuracy.

Significance and Claims
The paper claims that accurate channel estimation in FRIS systems is fundamentally hindered by position uncertainties, a factor previously unaddressed in literature. By introducing a two-time-scale protocol and a tensor-based joint estimation framework, the authors demonstrate that it is possible to achieve accurate CSI even when element positions deviate from commanded values. The work highlights that treating the motion matrix as an unknown variable to be jointly estimated is essential for maintaining system performance in realistic deployment scenarios where calibration and motion errors are inevitable. The authors note that the full version of the paper will further elaborate on identifiability conditions, computational complexity, and the impact of system parameters (number of elements and users).