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.

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.

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.

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.

Low-Complexity Tensor Beamforming for RIS-Aided Multiuser Multistream MIMO Systems

We address joint active and passive beamforming for uplink RIS-assisted multi-user multi-stream MIMO systems with joint detection. The coupled design of the receive combiner, block-diagonal user precoders, and RIS phase vector is formulated through a third-order composite channel tensor. Exploiting this multilinear structure, we propose a multi-stream tensor alternating optimization method that updates the combiner, user precoders, and RIS coefficients via low-dimensional tensor projections. Simulations show that the proposed method approaches a multi-start alternating-optimization benchmark while reducing computational complexity and improving large-RIS scaling.

Multi-Target Estimation via Tensor Decomposition for Beyond Diagonal RIS-Aided Bistatic Sensing

We investigate the performance of beyond-diagonal reconfigurable intelligent surfaces (BD-RIS) for bistatic MIMO multi-target sensing using a two-stage tensor Doppler-delay-angle estimation (TenDAE). The first stage solves a Kronecker sum approximation (KSA) with a rank equal to the number of targets. The second stage employs a nested tensor factorization estimation (NTFE) that exploits the inherent multidimensional structure via two tensor decompositions that are solved in parallel. The first employs a PARAFAC decomposition to extract the targets' angles, and the second uses a nested PARAFAC decomposition to find the targets' delay and Doppler parameters. This two-stage approach decouples acquisition of the angles and delays/Dopplers using either alternating least squares or a higher-order singular value decomposition, followed by a high-resolution subspace technique, such as ESPRIT. We further compare the performance of a BD-RIS with a classical diagonal RIS. For the latter, we solve a Khatri-Rao sum approximation problem rather than the KSA due to the specific structure of the received signal. Notably, our NTFE framework remains blind to the underlying RIS architecture while simultaneously estimating all targets with minimal sensing resources. Additionally, we show that employing a nested-PARAFAC decomposition enables the decoupling of the delay-Doppler and angle domains. We also derive the Cramér-Rao lower bound to further assess the performance of the TenDAE framework. Finally, we numerically evaluate the solutions presented in this paper and demonstrate their efficiency in terms of RMSE compared with state-of-the-art approaches.

PARAFAC-Based Channel Estimation for Beyond Diagonal Reconfigurable Surfaces

Channel estimation is a central bottleneck in BD-RIS-assisted MIMO systems. The richer inter-element coupling that enables large performance gains also makes training and hardware control substantially harder than in diagonal RIS architectures. Existing estimators either target only cascaded channels or require block-by-block reconfiguration of the BD-RIS interconnections, which is costly and difficult to implement in practice. To overcome this limitation, we propose a pilot-assisted tensor framework for group-connected BD-RIS under a two-timescale protocol, where the scattering structure is designed as a low-rank PARAFAC model with fixed factor matrices. This design keeps the interconnection topology constant across blocks and updates only phase shifts, enabling practical operation without sacrificing estimation quality. Building on this structure, we develop a PARAFAC-based alternating least-squares (PALS) receiver that recovers the individual channels. Numerical results confirm that PALS delivers markedly lower composite-channel NMSE than conventional LS, matches the accuracy of state-of-the-art tensor receivers, and sharply reduces BD-RIS design complexity

Channel Estimation via Tensor Decomposition for Dynamic Metasurface Antennas with Known Mutual Coupling: Algorithms and Experiments

Dynamic metasurface antennas (DMAs) are an emerging hybrid-MIMO technology distinguished by an ultrathin form factor, low cost, and low power consumption. In real-world DMA prototypes, mutual coupling (MC) between meta-elements is generally non-negligible; some architectures even deliberately exploit strong MC to enhance wave-domain flexibility. In this paper, we address channel estimation (CE) for DMAs with known MC by formulating it as a tensor-decomposition problem. We develop a generalized block Tucker alternating least squares (BTALS) algorithm, together with specialized variants for cases with known direct and/or feed channel. We also develop a reciprocity-aware bilinear factorization method for the case with known direct channel. We experimentally validate our algorithms using an 18 GHz DMA prototype whose 7 feeds and 96 meta-elements are strongly coupled via a chaotic cavity. Our general BTALS algorithm reaches an accuracy of 43.1 dB, only 0.3 dB below the upper bound imposed by experimental noise. All proposed algorithms generally outperform the prior-art reference scheme thanks to the superior noise rejection enabled by the tensor-based framework. We further study the minimum number of required measurements as a function of the number of feeds and demonstrate the importance of MC awareness by comparison with an MC-unaware benchmark. Finally, we apply BTALS to a second setup enabling the prediction of the DMA's full dual-polarization 3D radiation diagram. We also measure the latter for DMA configurations optimized for channel-gain enhancement based on the estimated channels. Altogether, our work establishes the practical relevance of MC-aware tensor methods; beyond DMAs, it applies to all wireless systems with wave-domain programmability enabled by tunable lumped elements.

Performance Evaluation of Beyond Diagonal RIS under Hardware Impairments

Beyond diagonal reconfigurable intelligent surface (BD-RIS) improves the traditional reconfigurable intelligent surface (RIS) architecture functionality by interconnecting elements for advanced wave control. However, real-world implementations face hardware imperfections, such as impedance mismatches and varactor nonidealities, which can degrade overall system performance. In this paper, we propose three hardware impairment models that directly affect the BD-RIS scattering matrix structure and evaluate their impact on the channel estimation accuracy using the normalized mean square error (NMSE) as a performance metric. The proposed impairment models consider imperfections affecting self-impedances, mutual impedances, or both. Our results reveal how each impairment type degrades the system performance, allowing us to identify scenarios where the traditional RIS can outperform the BD-RIS.

A Decoupled Channel Estimation Method for Beyond Diagonal RIS

Beyond diagonal reconfigurable intelligent surface (BD-RIS) is a new architecture for RIS where elements are interconnected to provide more wave manipulation flexibility than traditional single connected RIS, enhancing data rate and coverage. However, channel estimation for BD-RIS is challenging due to the more complex multiple-connection structure involving their scattering elements. To address this issue, this paper proposes a decoupled channel estimation method for BD-RIS that yields separate estimates of the involved channels to enhance the accuracy of the overall combined channel by capitalizing on its Kronecker structure. Starting from a least squares estimate of the combined channel and by properly reshaping the resulting filtered signal, the proposed algorithm resorts to a Khatri-Rao Factorization (KRF) method that teases out the individual channels based on simple rank-one matrix approximation steps. Numerical results show that the proposed decoupled channel estimation yields more accurate channel estimates than the classical least squares scheme.