Channel Estimation for Movable Intelligent Surface

Technical Summary: Channel Estimation for Movable Intelligent Surface

Problem Statement
The paper addresses the challenge of channel estimation in uplink MIMO systems assisted by a Movable Intelligent Surface (MIS). While Reconfigurable Intelligent Surfaces (RIS) offer programmable radio environments, conventional single-layer RISs rely on accurate Channel State Information (CSI) and passive beamforming. Existing architectures like Active RISs or Stacked Intelligent Metasurfaces (SIMs) introduce complexity or power consumption. The MIS architecture, proposed in prior work [7], offers a compromise by combining a fixed transmissive layer (MS1) with a smaller movable layer (MS2). Mechanical displacement of MS2 relative to MS1 creates distinct effective phase profiles without requiring fast electronic tuning of individual elements.

However, this architecture introduces a significant estimation bottleneck. Similar to conventional RISs, the passive nature of the surface prevents direct baseband observation. Furthermore, the movable layer introduces a position-dependent response that must be estimated jointly with the wireless channels. Crucially, in practical scenarios, the effective phase response of the movable layer may be unknown or imperfectly calibrated at the receiver. Existing tensor-based channel estimation methods for RIS do not directly address this additional mechanical degree of freedom or the unknown nature of the movable-layer phase matrix.

Methodology
The authors propose a tensor-based framework that models the received pilot signals as a fourth-order PARAFAC (Parallel Factor) tensor. The system operates by jointly exploiting two dimensions of training:

1. Fixed-layer phase patterns: The fixed layer (MS1) applies $S$ predefined phase-shift patterns.
2. Movable-layer positions: The movable layer (MS2) explores $U$ discrete mechanical positions over the fixed grid.

The received signal at the Base Station (BS) is filtered and structured into a tensor $\mathcal{Y} \in \mathbb{C}^{N_r \times N_t \times S \times U}$, where $N_r$ and $N_t$ are the number of BS and user terminal (UT) antennas, respectively. The tensor follows the model:
$$\mathcal{Y} = \mathcal{I}_{4,M} \times_1 G \times_2 H \times_3 \Phi_f \times_4 \Phi_m$$
Here, $G$ and $H$ represent the MIS-BS and UT-MIS channels, $\Phi_f$ represents the known fixed-layer phase shifts, and $\Phi_m$ represents the unknown position-dependent phase response of the movable layer.

To estimate the unknown factors, the paper derives a Trilinear Alternating Least Squares (TALS) receiver. Unlike conventional approaches that assume full knowledge of the training matrix, this receiver treats the movable-layer phase matrix ($D$, corresponding to $\Phi_m$) as an unknown factor to be estimated. The algorithm iteratively updates estimates of the channel matrices ($G$, $H^T$) and the movable-layer response ($D$) by solving conditional least-squares subproblems based on the $n$-mode unfoldings of the received tensor.

Key Contributions
The paper outlines three primary contributions:

1. Training Protocol and Tensor Modeling: The authors propose a two-dimensional training protocol combining fixed-layer phase shifts and movable-layer positions, modeling the received signal as a fourth-order PARAFAC tensor.
2. TALS-PARAFAC Receiver: A receiver is derived that estimates the UT-MIS channel, the MIS-BS channel, and the position-dependent movable-layer response simultaneously. A critical feature is that the receiver operates without prior knowledge of the movable-layer phase matrix, estimating it directly from the tensor structure.
3. Analysis of Identifiability and Complexity: The paper discusses the conditions required for the identifiability of the channel factors (specifically regarding the rank of Khatri-Rao products involving training slots $S$, positions $U$, and antenna counts) and analyzes the computational complexity, which scales linearly with the number of iterations and positions.

Simulation Results
The performance was evaluated using the Normalized Mean Square Error (NMSE) of the estimated factors and the reconstructed cascaded channel.

• Comparison with BALS: The proposed TALS receiver was compared against a conventional Bilinear Alternating Least Squares (BALS) benchmark. The BALS benchmark achieved lower NMSE but relied on the assumption of perfect knowledge of the movable-layer training matrix. The TALS receiver, operating under the more challenging assumption of unknown movable-layer phases, demonstrated successful convergence and accurate estimation.
• Impact of Training Length: Simulation results indicated that increasing the number of training slots ($S$) significantly improves the NMSE. A transition was observed where insufficient training length led to high NMSE due to poor tensor factorization conditioning, while sufficient length ($S=64$) yielded a sharp decrease in error.
• Joint Estimation: The results confirmed that increasing the training length improves the estimation accuracy of both the movable-layer response ($D$) and the cascaded channel, validating the model's ability to recover the unknown phase matrix from the received data.

Significance and Claims
The paper claims to be the first work to address the channel estimation problem for MIS-assisted MIMO architectures in the literature. The primary significance lies in the practical advantage of the proposed formulation: it eliminates the need for prior calibration or knowledge of the movable-layer phase response. By treating the unknown mechanical response as an estimable tensor factor, the method enables flexible channel estimation in scenarios where the physical state of the movable surface is not perfectly known to the receiver. The authors conclude that the proposed framework successfully exploits the additional structured dimension provided by mechanical displacement to decouple and estimate the cascaded channels and the movable-layer response jointly.