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Channel Estimation for Reconfigurable Intelligent Surface Aided Multi-User mmWave MIMO Systems

Jie Chen, Ying-Chang Liang, Hei Victor Cheng, Wei Yu

arXiv:1912.03619v2eess.SP

TL;DR

The paper addresses channel estimation for RIS-aided multi-user mmWave MIMO, where passive RIS elements make training difficult. It proposes a CS-based protocol and two-step joint recovery method that exploits cascaded-channel sparsity and shared BS–RIS structure, enabling estimation with limited training overhead.

  • Problem

    Passive RIS elements cannot transmit or receive training sequences, making channel acquisition for RIS-involved links difficult.

  • Method

    The paper uses a novel CS-based protocol and two-step multi-user joint estimation that projects onto a common column subspace before recovering row-block sparse matrices.

  • Results

    The proposed estimator outperforms baseline schemes especially when the number of scatterers is small, while imperfect direct-link CSI degrades all methods.

  • Takeaways & Limitations

    Exploiting common BS–RIS-induced sparsity allows cascaded channels to be estimated with limited training overhead.

Abstract

from arXiv · show

Channel acquisition is one of the main challenges for the deployment of reconfigurable intelligent surface (RIS) aided communication systems. This is because an RIS has a large number of reflective elements, which are passive devices with no active transmitting/receiving abilities. In this paper, we study the channel estimation problem for the RIS aided multi-user millimeter-wave (mmWave) multi-input multi-output (MIMO) system. Specifically, we propose a novel channel estimation protocol for the above system to estimate the cascaded channels, which are the products of the channels from the base station (BS) to the RIS and from the RIS to the users. Further, since the cascaded channels are typically sparse, this allows us to formulate the channel estimation problem as a sparse recovery problem using compressive sensing (CS) techniques, thereby allowing the channels to be estimated with less training overhead. Moreover, the sparse channel matrices of the cascaded channels of all users have a common block sparsity structure due to the common channel between the BS and the RIS. To take advantage of the common sparsity pattern, we propose a two-step multi-user joint channel estimation procedure. In the first step, we make use of the common column-block sparsity and project the received signals onto the common column subspace. In the second step, we make use of the row-block sparsity of the projected signals and propose a multi-user joint sparse matrix recovery algorithm that takes into account the common channel between the BS and the RIS.

I. INTRODUCTION

RIS channel estimation is difficult because passive reflective elements cannot transmit or receive training signals, while multi-user mmWave cascaded channels have exploitable sparse and common-block structure. The paper proposes CS-based estimation and a two-step joint procedure that uses these structures.

  • Motivation: Passive RIS elements cannot transmit or receive training sequences, so channel estimation must occur at the BS or users.This makes channels involving the RIS difficult to estimate compared with systems using active devices.
  • Contribution: The paper proposes a novel protocol using compressive sensing to estimate cascaded channels in multi-user mmWave MIMO systems.Cascaded channels are products of the BS–RIS and RIS–user channels.
  • Sparse Representation: Cascaded channels exhibit row-column-block sparsity because limited propagation paths produce few AoA and AoD steering vectors.This two-dimensional structure differs from the row-block sparsity usually used for conventional mmWave MIMO channels.
  • Multi-User Structure: All users share common block sparsity because their cascaded channels contain the same BS–RIS channel.Estimating each user independently while ignoring this shared structure is suboptimal.
  • Proposed Procedure: The proposed two-step procedure first estimates and projects onto the common AoD subspace, then performs joint sparse matrix recovery using coupled scaling and row-block variables.The second-stage optimization uses a sparsity-promoting log-sum formulation.

B. Channel Estimation Protocol

The protocol estimates direct and cascaded channels in uplink TDD training, then uses sparse angular-domain representations to reduce cascaded-channel estimation overhead. It supports ULA and UPA arrays and assumes the direct channel is known when focusing on cascaded estimation.

  • Channel Estimation Protocol: The proposed TDD uplink protocol separates direct-link estimation from cascaded-channel estimation using two direct-link sub-frames and B cascaded-channel sub-frames.Orthogonal user pilots simplify recovery of each cascaded channel.
  • Direct-Link Estimation: Opposite RIS reflection configurations enable direct-link estimation, after which its estimated contribution is subtracted from the received signals.The direct-link estimation error is treated as additional noise; the analysis assumes perfect direct-link knowledge.
  • Cascaded-Channel Estimation: The conventional LS estimator recovers Gk from the pilot observations using the reflection matrix V and requires V to have full rank.Full rank implies B ≥ L, creating large training overhead when the RIS has many reflective elements.
  • Sparse Representation: The cascaded channel is represented in the virtual angular domain as Gk = AR Xk AH_T, where Xk stores path gains over cascaded AoA and BS AoD dictionaries.The dictionaries use angular resolutions Gr and Gt, and nonzero entries of Xk identify corresponding spatial paths.
  • Array Models: The proposed algorithms extend from ULAs to UPAs by replacing the RIS dictionary with a Kronecker-product structure.For a UPA, AR = ARv ⊗ ARh, while the subsequent algorithms apply directly.

B. Sparsity Structure Analysis and Conventional CS-based Techniques

mmWave propagation yields cascaded channels with a row-column-block sparse angular representation. Conventional CS estimators that ignore this structure require more training or finer grids, motivating structured recovery methods.

  • Sparsity Structure: High-frequency blockage and limited paths make the cascaded channel sparse in its angular representation.The sparse matrix Xk has only a few nonzero row and column vectors.
  • Sparsity Structure: Unlike conventional mmWave MIMO channels with mainly row-block sparsity, RIS cascaded channels exhibit two-dimensional row-column-block sparsity.The structure arises from limited paths on both the BS-RIS and RIS-user links.
  • Conventional CS-based Techniques: Channel estimation can therefore be formulated as recovery of the sparse channel matrix Xk using compressive sensing.The cascaded-channel model is substituted into the observations to obtain a sparse matrix recovery problem.
  • Conventional CS-based Techniques: Vectorizing the channel and ignoring block sparsity produces a conventional SMV problem solvable by methods such as OMP.This approach uses an observation vector of size B × M.
  • Conventional CS-based Techniques: Ignoring block sparsity increases training overhead and requires super-resolution AoA/AoD grids, raising computational complexity.A row-block-only alternative forms a new sparse matrix and a conventional MMV recovery problem.
  • Proposed Structured Estimation: The proposed two-step S-MJCE procedure first estimates a common AoD subspace, projects received signals onto it, and then performs multi-user joint recovery.The first step converts row-column-block sparsity into row-block sparsity; the second uses coupled variables and iterative sparse optimization.

C. Multi-User Common Sparsity Representation

The paper explicitly models the common block sparsity of all users’ cascaded channels by separating their shared BS-RIS channel from their distinct RIS-user channels.

  • C. Multi-User Common Sparsity Representation: A common-sparsity representation is developed because all users share the BS-RIS channel while having different RIS-user channels.This model supports joint estimation across users rather than independent recovery.

1) Common Column-Block Sparsity due to Common Scatterers between the BS and the RIS:

The common BS–RIS channel gives all users a shared column-block sparsity pattern, enabling estimation of a common AoD subspace before row-sparse recovery. Projecting onto this subspace avoids exact AoD-vector estimation, reduces noise components, and lowers subsequent recovery complexity.

  • The common BS–RIS channel makes the AoD support of all users’ cascaded channels identical, while their row-block sparsity patterns differ.
  • Scaling ambiguity prevents directly separating the RIS-user channel gains from the common BS–RIS channel in each cascaded channel.
  • The joint scaling property represents every user’s cascaded channel as a scaled version of one arbitrary cascaded channel.
  • The first step estimates the common column subspace and projects received signals onto it, transforming row-column-block sparsity into row-block sparsity.
  • The method estimates the subspace rather than exact AoD steering vectors, avoiding grid quantization and associated quantization and estimation errors.
  • Projection removes noise components in the common-subspace null space, increasing effective SNR and reducing the columns and complexity of later row-sparse recovery.

C. Second Step: Multi-User Joint Sparse Matrix Recovery

The second step jointly recovers the users’ sparse channel matrices after common-subspace projection. It couples a shared row-block sparse matrix with user-specific scaling gains and solves the resulting recovery problem efficiently.

  • The projected signals are rewritten using a combined common row-block sparse matrix derived from the joint scaling property.
  • The overall S-MJCE procedure estimates the common AoD subspace, projects the signals, solves the joint sparse recovery problem, and reconstructs each cascaded channel.
  • The recovery problem jointly estimates the shared sparse matrix and user-specific scaling gains under a combined-noise tolerance.
  • The proposed recovery problem is non-convex because the sparse matrix and scaling gains are coupled.

V. SOLUTION TO THE MULTI-USER JOINT SPARSE MATRIX RECOVERY PROBLEM

The recovery method replaces discontinuous sparsity with a log-sum surrogate and uses alternating optimization to decouple the coupled variables. Iterative updates continue until the objective converges.

  • The algorithm combines alternating optimization with iterative reweighting and includes convergence, initialization, and complexity analyses.
  • A log-sum function replaces the discontinuous l0-norm as the sparsity-promoting surrogate.
  • The penalty factor λk balances data fitting against solution sparsity and can be chosen according to received-signal power.
  • When users have different path losses, signal normalization can equalize their power levels so the method remains applicable.
  • Alternating optimization separates the coupled problem into one subproblem for scaling gains and another for the sparse matrix.

B. Least-Square Solution of αk

With the sparse matrix fixed, the scaling gains decompose into independent least-squares problems. The alternating algorithm then updates the sparse matrix through iterative reweighted optimization and is shown to converge.

  • For fixed sparse matrix X̄, the scaling-gain update decomposes into K independent optimization problems.
  • Vectorizing the L non-zero diagonal entries of αk reduces each scaling-gain subproblem to a standard least-squares problem.
  • The least-squares inverse requires rank( X̄ ) ≥ L so the relevant matrix has sufficient rank.
  • With scaling gains fixed, iterative reweighting upper-bounds the non-convex objective and produces a weighted least-squares update for the sparse matrix.
  • The proposed double-loop algorithm alternates sparse-matrix and scaling-gain updates, with an inner reweighted loop and an outer alternating loop.
  • The objective decreases monotonically and is bounded below, which guarantees convergence of the iterative algorithm.

2) Initialization Analysis:

Successful joint recovery depends on initializing α_k close to its actual value, while channel estimation imposes measurement and computational-complexity requirements.

  • Initialization: Proper initialization of α_k is crucial because joint recovery requires all cascaded channels to have similar sparse matrices.If α_k is not sufficiently close to its actual value, no matrix X̄ satisfies equality (21).
  • Initialization: α_k can be initialized by individually estimating each cascaded channel G_k with SMV, MMV, or a single-user version of (32).The estimates Ĝ_k are then used with the scaling property (20).
  • Training overhead: At least 2N_fN_hk measurements are required to form equations for the unknown supports and values in each sparse-matrix column.Stable recovery practically requires cN_fN_hk ln(eL/N_fN_hk) measurements, where c depends on the stability requirement.
  • Computational complexity: The total computational complexity of Algorithm 3 is O(I_inI_outΔ_1Δ_2).I_in and I_out denote the inner- and outer-loop iteration counts.

VI. TRAINING REFLECTION COEFFICIENTS OPTIMIZATION

The training reflection coefficients are optimized by reducing the mutual coherence of the equivalent dictionary while respecting RIS phase-only constraints. The method successively optimizes transformed coefficient vectors and projects them onto feasible reflection coefficients.

  • Optimization objective: The training reflection coefficients are designed to reduce the mutual coherence of the equivalent dictionary D = V^H A_R.Smaller mutual coherence corresponds to making the dictionary columns as orthogonal as possible.
  • Optimization objective: The target is to design V so that D^H D is as close as possible to a scaled identity matrix, with A_R fixed by angle quantization.A_R is a constant dictionary defined in (14), while B provides normalization.
  • Feasibility constraint: The unconstrained solution cannot be directly used because RIS reflection coefficients change phase without changing signal amplitudes.Therefore, V must satisfy the phase-only constraint in (52).
  • Constrained optimization: The constrained problem is solved by transforming the objective with A_R and A_R^H and then using an eigenvalue decomposition of A_R A_R^H.This modification adapts the method in to the RIS constraint.
  • Successive optimization: The optimized q_b is mapped to q̃_b, projected to obtain v_b, and the procedure repeats B times to produce V.The projection uses v_b = e^{j∠(U_R q̃_b)}.
  • Successive optimization: For each column, q_b is optimized from the eigenvector associated with the largest eigenvalue of E_b, eliminating the largest error for the given E_b.The eigenvalues are ordered by decreasing magnitude.

VII. SIMULATION RESULTS

Simulations evaluate NMSE across training overhead, propagation complexity, transmit power, reflection-coefficient design, CSI quality, penalty selection, and path-loss settings. The proposed estimator generally performs strongly, approaching the genie-aided lower bound, while its advantages depend on training overhead, SNR, scatterer count, CSI quality, and parameter selection.

  • Training overhead: The proposed estimator significantly outperforms baseline schemes as training overhead increases and achieves performance similar to the genie-aided lower bound.Subspace projection improves estimation performance particularly at larger B, while binary reflection performs worse because its training reflection power is lower.
  • Propagation complexity: All estimation schemes degrade as the number of BS-RIS scatterers grows, with the proposed method especially advantageous when scatterers are few.Performance degradation is strongest when the scatterer count equals the training overhead, as more unknown parameters must be estimated.
  • Transmit power: All schemes improve with transmit power, while matching-pursuit baselines become limited by the angular resolution of their dictionary matrices.The proposed algorithm performs relatively poorly at smaller training overhead or lower SNR but better at larger training overhead and higher SNR.
  • Training reflection coefficients: Optimizing RIS training reflection coefficients increasingly improves NMSE as pilot length grows by reducing equivalent-dictionary mutual coherence.The equivalent dictionary becomes higher-dimensional with longer pilots, allowing more orthogonal columns after optimization.
  • Direct-link CSI: Imperfect direct-link CSI worsens all schemes, but the proposed estimator’s gap to the lower-bound benchmark remains unaffected.Direct-link estimation errors act as extra noise and degrade subspace projection, while leaving the proposed algorithm’s relative gap unchanged.
  • Penalty factor: The proposed estimator’s NMSE first decreases and then increases with penalty factor λ because λ balances data fitting against solution sparsity.Both underfitting from excessive sparsity and loss of sparsity from excessive data fitting degrade recovery.

APPENDIX

The appendix derives a maximum-likelihood procedure for estimating the subspace associated with the AoD vectors without requiring their exact values. It reformulates the estimation using likelihood, eigenvalue, and unitary-matrix properties.

  • The appendix derives the likelihood function associated with the AoD vectors and presents maximum-likelihood subspace estimation.
  • The maximum-likelihood formulation treats A and P_b as deterministic unknown parameters and derives the likelihood of Y given them.
  • For a given A, the method first estimates each b, then maximizes the resulting log-likelihood with respect to A.
  • Because span(A) equals the span of W∥, the procedure estimates W∥ rather than requiring the exact AoD vectors A.
  • The reformulated optimization uses eigenvalue orderings and unitary-matrix constraints, with the resulting linear program having an optimal solution determined by τn,m = ∆nθm.
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