Deconstructing the Composite Channel for Beyond Diagonal RIS: Channel Estimation and Beamforming Design

As beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) gain increasing attention in high-frequency wireless communications, accurate and scalable channel-estimation methods become essential. This paper develops a parametric channel-estimation and beamforming framework that deconstructs the composite BD-RIS channel into its generating directional factors, revealing the tensor structure induced jointly by propagation geometry and beyond-diagonal scattering. We propose two tensor-based estimators: Fourth-Order Tucker Channel Estimation (FORTE), which models the partially structured channel as a fourth-order Tucker tensor, and Fourth-Order PARAFAC Channel Estimation (FORPE), which captures the fully structured channel through a fourth-order PARAFAC model. By exploiting partial and full channel geometry, the proposed methods achieve higher estimation accuracy than Least Squares and Block Tucker Kronecker Factorization benchmarks. In particular, FORTE outperforms FORPE due to its more compact representation, attaining an NMSE of about 10^{-4} at 5 dB SNR. In contrast, FORPE provides essentially unique estimates of the composite-channel factor matrices, whereas FORTE identifies their subspaces. The proposed deconstruction also provides a structured representation useful for sensing-oriented parameter extraction and tensor-structured system optimization. Finally, the Tensor Optimization Framework for Beamforming, Combining, and Scattering (TenFormer) achieves spectral efficiency comparable to the benchmark design while significantly reducing computational complexity through parallel tensor-structured optimization.

TenSIM: Tensor-Based Channel Estimation for MIMO Systems with Stacked Intelligent Metasurfaces

Stacked intelligent metasurfaces (SIMs) are emerging as a promising architecture for the sixth generation (6G) and beyond of wireless systems, enabling richer electromagnetic-wave manipulation than conventional single-layer metasurfaces. However, realizing these gains requires accurate and scalable channel estimation under the strong inter-layer coupling and multilinear parameter interactions introduced by the stacked programmable metasurface layers. This paper proposes TenSIM, a tensor-based channel-estimation framework for SIM-assisted multiple-input multiple-output (MIMO) systems. By exploiting a structured SIM training protocol, TenSIM derives two parity-dependent observation models: a PARAllel FACtor (PARAFAC) model for odd-layer SIMs and a Tucker model for even-layer SIMs. These formulations decouple the transmitter-SIM and SIM-receiver channels while explicitly accounting for inter-layer wave coupling. Based on the resulting tensor models, we develop alternating least squares estimators, establish identifiability conditions using the associated design matrices, and characterize practical sufficient conditions for full-column-rank training designs, including those involving scaling ambiguities. The proposed framework is validated through extensive numerical experiments and reveals the main operating trade-offs. We show that both TenSIM-PARAFAC and TenSIM-Tucker outperform unstructured least squares baselines by exploiting the tensor structure of the SIM cascade. Moreover, TenSIM-PARAFAC offers better scalability, lower computational complexity, and stronger robustness to inter-layer spacing, while TenSIM-Tucker can provide more accurate channel reconstruction when sufficient training and strong layer coupling are available. Finally, it is shown that the proposed TenSIM framework remains effective under imperfect or blind SIM training when additional pilot diversity is available.

Joint Channel and Symbol Estimation for RIS-Assisted Fluid Antenna Systems

This paper addresses joint channel and symbol estimation in reconfigurable intelligent surface (RIS)-aided multiuser uplink systems with fluid antennas (FAs) at the base station. We propose the Nested Tucker for Fluid Antenna Systems (NTFAS) protocol, in which FA port selection and user-dependent coding vary across blocks while the transmitted symbol matrix is shared across observations. This structure yields coupled Tucker models with common channel and data factors. A two-stage semi-blind bilinear alternating least squares (BALS) receiver is then developed to estimate the cascaded channel and symbols, and to separate the user-to-RIS and RIS-to-BS channels through the embedded PARAFAC structure. Simulations show that NTFAS improves cascaded-channel NMSE and spectral efficiency (SE) with respect to a competing semi-blind benchmark, while maintaining comparable BER performance.

Channel Estimation for Movable Intelligent Surface

This paper proposes a tensor-based channel estimation framework for an uplink MIMO system assisted by a movable intelligent surface. The considered architecture combines a fixed transmissive metasurface with a smaller movable layer, whose discrete positions create an additional structured training dimension. By jointly exploiting fixed-layer phase patterns and movable-layer positions, the received pilots are modeled as a fourth-order PARAFAC tensor. A trilinear alternating least-squares receiver is then derived to estimate the individual channels and the position-dependent response. Importantly, the proposed method does not require prior knowledge of the movable-layer phase response at the receiver, since this unknown factor is estimated from the tensor structure of the received signal. Simulation results show that increasing the training length improves the NMSE of the estimated factors and the reconstructed cascaded channel.

Channel Estimation for Flexible Intelligent Metasurface Aided MIMO Communications

Flexible Intelligent Metasurfaces (FIMs) enable wireless systems to adapt their three-dimensional geometry through morphing, thereby providing new spatial degrees of freedom. However, continuous deformation complicates the accurate acquisition of Channel State Information (CSI). This work proposes a multidimensional framework for MIMO systems with active FIM arrays at both the transmitter and receiver. A split single-time-scale training protocol sequentially introduces spatial variation by morphing the receiver, then the transmitter. The resulting signal model is formulated as a PARAFAC decomposition, and an alternating least squares (ALS) algorithm is employed to estimate steering matrices and path gains. Our numerical results show that the proposed channel estimation method yields accurate CSI recovery for different system setups.

Low-Complexity Tensor-Based Monostatic Sensing for IRS-Assisted Communication Systems

This paper proposes a tensor-based parameter estimation algorithm for sensing in an intelligent reflecting surface-assisted system. We present a higher-order singular value decomposition-based solution that exploits the tensor structure of the received echo signal to jointly estimate the target's delay, Doppler, and angular information. Our tensor-based solution can estimate the parameters individually at low complexity, benefiting from parallel computation. Complexity analysis is carried out in comparison with a baseline scheme that does not exploit the intrinsic multilinear structure of the sensed signal. Simulation results show that our proposed tensor-based method can achieve the same performance as the reference method while drastically reducing the computational complexity.

PARAFAC-Based Time-Varying Channel Estimation for IRS-Aided Communications

This paper proposes a tensor-based parametric channel estimation technique for IRS-assisted communication systems with time-varying channel parameters. We exploit the multidimensional structure of the received signal by developing a $3$rd-order PARAFAC tensor model that is solved by employing the iteratively ALS algorithm. Our simulation results show that the proposed approach provides enhanced performance in terms of NMSE of the concatenated channel compared to the competing solutions by capitalizing on the intrinsic tensor structure of the received signal without increasing the computational complexity of the channel estimation.

Dimming Space-Time Code (DSTC) for Visible Light Communication with Semi-Blind Detection

Visible light communication (VLC) provides a unified framework for wireless data transmission and illumination, but its practical deployment requires transmission schemes that jointly satisfy communication and lighting constraints. In color-shift keying (CSK) systems, dimming remains a challenging and underexplored problem because the average optical power must be controlled without altering the perceived chromaticity. This paper proposes a dimming space-time code (DSTC) for CSK-based VLC systems, where a structured dimming matrix introduces controlled temporal power variations while satisfying physical feasibility, color preservation, and identifiability conditions. Two receiver architectures are developed: a pilot-assisted zero-forcing (ZF) receiver and a tensor-based semi-blind PARAFAC receiver that jointly estimates the channel and transmitted symbols using only one training time slot. Simulation results show that the proposed DSTC provides diversity gains and substantial BER reductions with respect to conventional CSK, while the tensor-based receiver improves spectral efficiency by reducing training overhead, with particular benefits in large-scale MIMO configurations.

Decoupled Delay-Doppler and Angle Estimation in BD-RIS Sensing via Nested Tucker Decomposition

We study single-target localization in a group-connected beyond-diagonal reconfigurable intelligent surface (BD-RIS)-assisted monostatic network with K element groups. We propose a Nested Tensor Factorization and Estimation (NTFE) algorithm that models the received signal as a 3rd-order nested Tucker tensor, decoupling the delay-Doppler and angle domains. The resulting two-stage procedure estimates the target-bearing tensor factors and then extracts the other physical parameters using subspace and closed-form steps. We also analyze identifiability and uniqueness conditions. Simulations show that NTFE exploits the group-connected BD-RIS structure and outperforms state-of-the-art sensing benchmarks.

Time-Varying Parametric Channel Estimation With CP Decomposition Tensor Processing

Integrated sensing and communications (ISAC) is a key use case for sixth-generation (6G) wireless systems, where parametric channel estimation (PCE) plays a central role in enabling sensing, localization, and channel equalization in high-mobility scenarios. However, PCE is typically more computationally demanding than conventional channel estimation, which motivates the development of lower-complexity solutions. In this letter, we propose a fast PCE algorithm for time-varying and frequency-selective (TVFS) channels based on canonical polyadic (CP) decomposition and tensor processing, combined with ESPRIT-based initialization, component refinement, and exact line-search alternating coordinate descent. Two variants are presented: one for fully digital and another for hybrid receiver architectures. Numerical results show that the proposed method clearly outperforms a related CP-based baseline while achieving estimation performance close to a multiple-start SAGE benchmark at a substantially lower computational cost, with about one order of magnitude shorter execution time.