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Matrix-Calibration-Based Cascaded Channel Estimation for Reconfigurable Intelligent Surface Assisted Multiuser MIMO
Hang Liu, Xiaojun Yuan, Ying-Jun Angela Zhang
TL;DR
The paper targets the excessive training overhead of cascaded channel acquisition in fully passive RIS-assisted multiuser MIMO. It formulates estimation as matrix-calibration-based sparse factorization and develops a message-passing algorithm using slow-varying components and hidden sparsity. Simulations report reduced training overhead and accurate estimation, while a replica-method framework characterizes the large-system performance bound.
Problem
Fully passive RIS channel acquisition requires cascaded channel estimation, while state-of-the-art approaches use excessively long training sequences.
Method
The paper formulates cascaded estimation as matrix-calibration-based matrix factorization and uses Bayesian message passing with slow-varying channel information and hidden channel sparsity.
Results
The proposed algorithm requires about 30% of the training overhead of [18] and about 1% of [15] under the considered settings.
Takeaways & Limitations
The proposed estimator closely approaches the replica-method bound and improves estimation performance over the compared approaches in simulations.
Abstract
from arXiv · showhide
Reconfigurable intelligent surface (RIS) is envisioned to be an essential component of the paradigm for beyond 5G networks as it can potentially provide similar or higher array gains with much lower hardware cost and energy consumption compared with the massive multiple-input multiple-output (MIMO) technology. In this paper, we focus on one of the fundamental challenges, namely the channel acquisition, in an RIS-assisted multiuser MIMO system. The state-of-the-art channel acquisition approach in such a system with fully passive RIS elements estimates the cascaded transmitter-to-RIS and RIS-to-receiver channels by adopting excessively long training sequences. To estimate the cascaded channels with an affordable training overhead, we formulate the channel estimation problem in the RIS-assisted multiuser MIMO system as a matrix-calibration based matrix factorization task. By exploiting the information on the slow-varying channel components and the hidden channel sparsity, we propose a novel message-passing based algorithm to factorize the cascaded channels. Furthermore, we present an analytical framework to characterize the theoretical performance bound of the proposed estimator in the large-system limit. Finally, we conduct simulations to verify the high accuracy and efficiency of the proposed algorithm.
I. INTRODUCTION
The paper addresses channel acquisition in fully passive RIS-assisted multiuser MIMO, where accurate CSI is difficult and existing approaches can require excessive training. It exploits slow-varying channel information and sparsity to develop a lower-overhead matrix-factorization estimator.
- Motivation: RIS-assisted MIMO can provide similar or higher array gains than massive MIMO with lower hardware cost and energy consumption.RIS elements adjust independent phase shifts to combine reflected signals constructively at the receiver.
- Motivation: Accurate CSI is critical for configuring RIS parameters, but passive RIS elements make channel estimation more challenging than in conventional systems.Prior RIS communication studies often assume perfect CSI without addressing its acquisition difficulty.
- Research gap: Fully active or partially active RIS approaches ease channel estimation but sacrifice the extremely low hardware and deployment costs of purely passive elements.The paper therefore focuses on cascaded channel estimation with a fully passive RIS.
- Approach: The proposed formulation jointly calibrates the RIS-to-BS channel matrix and estimates RIS-to-user channel matrices using slow-varying components and channel sparsity.The slow-varying channel component is assumed estimable through long-term averaging before RIS channel estimation.
- Approach: Approximate message passing reduces the complexity of canonical message passing by updating message means and variances instead of performing high-dimensional integrations.The algorithm is developed for the large-system regime.
- System model: The system model uses K single-antenna users, an M-antenna BS, and an L-element passive RIS, with simultaneous user training over T time slots.The RIS elements are configured with a common phase during the described training interval.
B. Virtual Channel Representation
The channel model separates quasi-static and fast-varying RIS-to-BS components, then uses virtual angular representations to expose sparse coefficient matrices. With known angular bases and the averaged slow component, cascaded estimation becomes a matrix-calibration-based factorization problem.
- Channel decomposition: The slow-varying RIS-to-BS component is assumed static over intervals much longer than a coherence block and accurately estimated by long-term channel averaging.This component is available before the RIS channel estimation procedure.
- Virtual representation: Fast-varying RIS-to-BS paths are represented on discretized angular sampling grids using over-complete BS and RIS array-response bases.The bases cover BS angle of arrival and RIS angle-of-departure dimensions.
- Virtual representation: The angular coefficient matrix S is sparse because only a few entries correspond to the limited number of fast-varying propagation paths.Each nonzero entry identifies a path through its associated BS AoA and RIS AoD steering vectors.
- Virtual representation: User-to-RIS channels are represented by angular-domain coefficient vectors g_k, whose sparsity follows from limited scattering geometry.The sparsity of S and {g_k} is used in the channel-estimation design.
- Matrix formulation: The received-signal model can be rewritten as a matrix factorization involving S and G after applying the virtual channel representations.G collects the angular-domain user-channel vectors.
- Matrix formulation: Once the angular bases and averaged slow-varying component are known, the BS knows the sensing matrices and factorizes S and G from the training observations.The resulting task is termed matrix-calibration-based cascaded channel estimation because it resembles blind matrix calibration.
IV. MATRIX-CALIBRATION BASED CASCADED CHANNEL ESTIMATION ALGORITHM
The section derives Bayesian posterior-mean estimators for the sparse matrices S and G, then represents their factorization through a factor graph for canonical message passing.
- Bayesian inference: Bernoulli-Gaussian priors model the sparsity of S and G through Bernoulli activation parameters and nonzero-entry variances.The parameters are λS, λG, τS, and τG.
- Bayesian inference: The Bayesian framework estimates S and G using posterior means conditioned on the observation Y.Their MMSEs are characterized through posterior distributions and expectations over S, G, and Y.
- Message passing: Canonical sum-product message passing approximately computes the marginal posterior estimators, but its message integrations and normalizations are generally computationally intractable.The section therefore motivates approximate message updates.
- Factor-graph representation: The factor graph encodes variables S, G, W, Z, and Q together with their associated factorizable probability-density factors.Variable nodes represent matrix elements, while factor nodes represent priors and the relationships among W, Z, Q, and Y.
C. Approximations for Message Passing
The proposed message-passing algorithm uses large-system approximations to replace intractable high-dimensional message computations with tractable mean-and-variance updates.
- Approximate message passing: The message updates are simplified with approximate message passing in the large-system limit, where dimensions and noise variance grow with fixed ratios.The relevant ratios include M/K, M′/K, L/K, L′/K, T/K, and τN/K^2.
- Gaussian approximations: Second-order Taylor expansion and central-limit arguments approximate key message terms as Gaussian distributions.These approximations yield tractable means and variances for the message updates.
- Algorithm implementation: The algorithm outputs the estimated matrices S and G after completing the prescribed message-update loop.The stopping rule uses Imax or a tolerance ϵ for consecutive-iteration changes.
D. Computational Complexity
The section gives the proposed algorithm’s complexity and develops a replica-method framework whose fixed-point equations asymptotically describe the MMSEs under stated assumptions.
- Computational complexity: Each iteration computes updates for z, G, W, and S with component complexities O(MKT), O(MKL′), O(MKL′), and a matrix-dependent S-update cost.The overall expression is O(I(MKT + 2MKL′ + MM′(L′)^2)).
- Computational complexity: The proposed approximations substantially reduce complexity because conventional MMSE and canonical message-passing estimators require integrations involving S or G.The text contrasts this with component costs that grow exponentially with K^2.
- Asymptotic MSE analysis: The analysis assumes perfectly known prior distributions and uniformly covering sampling grids for the considered scenario.The stated assumptions include the specified large-system scaling and sampling-basis conditions.
- Asymptotic MSE analysis: The replica method derives fixed-point equations whose nontrivial solution gives the large-system limits of the MMSEs for S and G.The asymptotic result is obtained as K→∞ under the paper’s large-system assumptions.
- Asymptotic MSE analysis: The asymptotic MSEs can be computed efficiently by iteratively updating the fixed-point variables until convergence.The updates include MSES, MSEG, and auxiliary quantities associated with Z, W, S, and G.
B. Further Discussions
The simulations compare Algorithm 1 with analytical bounds and conventional MMSE estimators across channel bases, system settings, and training lengths. Algorithm 1 closely approaches the analytical bound while reducing computational and training overhead, although the replica bound becomes less tight for over-complete bases.
- Computational efficiency: Algorithm 1 runs much faster than conventional MMSE estimators, whose running time sharply increases as K becomes large.The approximations avoid the high-dimensional integrations required to compute the MMSE estimators directly.
- Normalized DFT bases: Algorithm 1 closely matches the replica-derived analytical bound under normalized DFT bases.The comparison uses MSE performance versus noise power for K = 40 and K = 100.
- Over-complete bases: Over-complete bases create a small analytical-to-simulation gap because the Gaussian approximations become less accurate.The resulting replica-method performance bound is less tight than under normalized DFT bases.
- Phase transitions: Algorithm 1 requires slightly larger parameters than the analytical result to achieve successful estimation, while remaining close to the theoretical bound.Success is defined by MSES < −20 dB and MSEG < −20 dB.
B. Simulation Results Under a More Realistic Channel Generation Model
Under a realistic channel model, simulations evaluate NMSE against noise power, sampling resolution, Rician factor, and training length. The proposed estimator generally outperforms the baselines, approaches the oracle bound, and achieves strong performance with substantially shorter training.
- Simulation setup: The simulations use K = 20 users, M = 60 antennas, T = 35, L = 16 RIS elements, κ = 9, and average results over 1500 Monte Carlo trials.The sampling-grid ratio is set to M′/M = L′_1/L_1 = L′_2/L_2 = 2 unless otherwise specified.
- Noise power: The proposed algorithm achieves an hUR,k NMSE close to the oracle bound and outperforms the baselines, especially at large noise power.The reported advantage is attributed to exploiting slow-varying channel components and hidden channel sparsity.
- Sampling resolution: As η increases, the proposed method’s NMSE decreases and substantially improves over baselines while closely approaching the oracle bound.Higher sampling-grid resolution yields higher angular resolution and sparser S and G; baselines that ignore angular sparsity remain invariant to η.
- Rician factor: As κ increases, the proposed method’s NMSE decreases because the known portion of HRB increases, while it generally outperforms the other channel-estimation algorithms.This experiment fixes τN = −95 dB and evaluates NMSE versus the Rician factor.
- Training length: T ≈ 35 is sufficient for the proposed algorithm, whereas baseline [20] achieves reasonably good performance only when T > 70.Increasing T initially gives sharp improvements, but very large T mainly adds redundant measurements and yields only slight gains.
- Overall outcome: The paper formulates cascaded estimation as matrix-calibration-based sparse matrix factorization and reports numerical performance improvements over the state-of-the-art approach.The estimator infers the cascaded BS-to-RIS and RIS-to-user channels using a message-passing algorithm.
APPENDIX A PROOF OF PROPOSITION 1
The appendix rewrites the received-signal relations using defined matrices and vectors to obtain a system model for subsequent analysis.
- Signal reformulation: The received signal is transformed by subtracting the contribution associated with the first user and defining y(t) accordingly.This manipulation is part of the derivation leading to the matrix-form model.
- Matrix representation: Definitions of X, Y, and N are used to rewrite the transformed observation in matrix form.The resulting expression introduces HUR as the matrix collecting the user-to-RIS channels.
- System model: Substituting the channel and basis representations into the rewritten model produces the system model used in the proposition’s proof.The passage identifies HUR = [hUR,1, · · ·, hUR,K].
APPENDIX B
The appendix derives approximate message updates for the proposed message-passing procedure, using Gaussian and central-limit approximations to obtain tractable mean and variance expressions.
- Message approximations: Fourier inversion is used to rewrite one message expression before deriving corresponding approximations for related updates.The rewritten form is matched to a previously established approximate-message-passing formulation.
- Update derivation: The appendix computes successive mean and variance updates by substituting intermediate expressions into the message equations.These substitutions cover updates associated with the variables and messages in the proposed algorithm.
- Gaussian approximation: Central-limit approximations model aggregate message terms as Gaussian random variables with specified means and variances.The derivation repeatedly substitutes these approximations into the message-update equations.
- Auxiliary variables: Auxiliary variables are introduced to express the approximated messages and complete the resulting update formulas.The auxiliary-variable definitions support the compact representation of the message updates.
APPENDIX C
The appendix applies the replica method and replica-symmetry assumptions to derive a large-system performance characterization, culminating in asymptotic MMSE expressions.
- Replica formulation: The replicate partition function is introduced, with replicated variables sharing the distributions of the corresponding original variables.The notation collects replicas of S, G, W, Z, and Q for the replica analysis.
- Covariance and saddle point: Covariance matrices are defined for the replicated variables and combined through a saddle-point expression for Fn.The appendix writes Fn as a normalized sum of the terms IQ, IZ, IW, IG, and IS.
- Replica symmetry: Replica symmetry imposes structured covariance forms, after which the explicit expression of Fn and stationary-point equations are obtained.The restrictions are applied to Co and eCo for o ∈ {S, G, W, Z}.
- Large-system approximation: Central-limit approximations are used for selected aggregate variables in the large-system limit.The appendix specifies Gaussian approximations for terms involving W and related variables.
- Asymptotic result: Evaluating the stationary points of the replica expression yields the MSEs defined earlier as asymptotic MMSEs.This provides the appendix’s final performance characterization.