PaperlayerView exact revision on arXiv

Source-linked AI summary

Channel Estimation for RIS-Aided mmWave MIMO Systems via Atomic Norm Minimization

Jiguang He, Henk Wymeersch, Markku Juntti

arXiv:2007.08158v2eess.SPcs.IT

TL;DR

Passive RIS-aided mmWave MIMO systems face a channel-estimation problem amid transmission-distance limits from high free-space path loss. The paper proposes a two-stage atomic norm minimization scheme for super-resolution channel parameter estimation, whose results confirm advantages over two-stage OMP across several estimation and performance measures.

  • Problem

    Passive RIS-aided mmWave MIMO systems require channel estimation, while transmission distance is limited by high free-space path loss.

  • Method

    The paper proposes an efficient two-stage atomic norm minimization scheme for super-resolution channel parameter estimation.

  • Results

    The proposed scheme shows advantages over two-stage OMP in angular-parameter and path-gain product MSE, average effective SE bound, and RIS gains for homogeneous paths.

  • Takeaways & Limitations

    The scheme achieves super-resolution estimation and better performance for parameters related to the path perspective.

  • Takeaways & Limitations

    The paper includes a realistic assumption regarding location awareness.

Abstract

from arXiv · show

A reconfigurable intelligent surface (RIS) can shape the radio propagation environment by virtue of changing the impinging electromagnetic waves towards any desired directions, thus, breaking the general Snell's reflection law. However, the optimal control of the RIS requires perfect channel state information (CSI) of the individual channels that link the base station (BS) and the mobile station (MS) to each other via the RIS. Thereby super-resolution channel (parameter) estimation needs to be efficiently conducted at the BS or MS with CSI feedback to the RIS controller. In this paper, we adopt a two-stage channel estimation scheme for RIS-aided millimeter wave (mmWave) MIMO systems without a direct BS-MS channel, using atomic norm minimization to sequentially estimate the channel parameters, i.e., angular parameters, angle differences, and products of propagation path gains. We evaluate the mean square error of the parameter estimates, the RIS gains, the average effective spectrum efficiency bound, and average squared distance between the designed beamforming and combining vectors and the optimal ones. The results demonstrate that the proposed scheme achieves super-resolution estimation compared to the existing benchmark schemes, thus offering promising performance in the subsequent data transmission phase.

I. INTRODUCTION

The paper addresses channel estimation for passive RIS-aided mmWave MIMO systems when the direct BS–MS channel is blocked and both RIS links contain multiple paths. It proposes a two-stage atomic-norm approach that estimates channel parameters efficiently and supports RIS and beamforming design.

  • Motivation: Blocked direct BS–MS links motivate RIS deployment for maintaining connectivity in sparse mmWave MIMO systems.RIS elements passively steer beams by controlling their phases, without baseband processing in the discrete architecture.
  • Related limitations: Existing RIS channel-estimation methods face trade-offs involving active-element power, complexity, cost, restricted RIS states, or simplified channel assumptions.Prior work includes active RIS elements, on/off training, single-antenna MS models, and LoS-only BS–RIS and RIS–MS channels.
  • Problem formulation: The paper studies passive RIS-aided mmWave MIMO with an obstructed direct channel, multiple paths on both links, and no data-sharing backhaul between BS and RIS.A low-rate control link is assumed sufficient for coordination.
  • Proposed approach: The proposed estimator divides channel estimation into two compressed-sensing subproblems and sequentially applies atomic norm minimization to angular and gain-related parameters.The estimated quantities include angular parameters, angle differences, and products of propagation path gains.
  • Evaluation and findings: The evaluation covers parameter MSE, RIS gains, effective spectrum-efficiency bounds, and squared distance from optimal beamforming and combining vectors.Reported simulations compare the proposed scheme with a two-stage OMP counterpart and show approximation to perfect-CSI effective spectrum efficiency at low SNR with limited training.
  • Proposed approach: First-stage estimates of BS–RIS AoDs and RIS–MS AoAs are reused to design subsequent training sequences, reducing training overhead.The method exploits mmWave channel sparsity rather than estimating full individual or cascaded channel matrices.

II. CHANNEL MODEL

The channel model represents the BS–RIS and RIS–MS links geometrically using array responses, angular parameters, and path gains. Their RIS-mediated composite channel depends on a diagonal phase-control matrix whose design affects the effective communication channel.

  • System model: The considered system contains one multi-antenna BS, one multi-antenna MS, and one multi-element RIS, with a blocked direct BS–MS channel.The RIS provides the reflected connection between the endpoints.
  • Individual channels: Each individual link follows a geometric channel model parameterized by AoDs, AoAs, propagation path gains, and array-response matrices.The BS–RIS channel uses resolvable paths whose path gains distinguish LoS and NLoS components.
  • Composite channel: The RIS-mediated composite channel combines the two individual channels through the RIS phase-control matrix.For a discrete RIS, the matrix is diagonal with unit-modulus entries exp(jω), while practical reflection coefficients can include amplitude loss.
  • Model assumptions: The model assumes an ideal RIS with reflection coefficient a = 1, while fabrication can cause coefficients to vary across RIS elements.The idealization is used to focus on channel estimation, and the received power then serves as a theoretical upper bound under optimal phase control.
  • Channel estimation and design: The effective channel factor depends on RIS phase control and therefore influences achievable rate and the design of RIS phases, BS beamforming, and MS combining.The first estimation stage targets selected angles, while the second estimates remaining gains and angular parameters using training designed from those estimates.

III. SOUNDING PROCEDURE

The sounding procedure divides each quasi-static coherence interval into channel-estimation and data-transmission subintervals, with RIS phase control varying across sounding blocks.

  • The wireless channels are modeled as quasi-static block fading, with channel parameters unchanged during the coherence time.
  • Each CE subinterval contains T + 1 blocks, with the RIS phase matrix varying between blocks.
  • In the example, the phase matrix remains fixed for the first 9 symbol times and changes every 6 symbol times during stage 2 sounding.
  • Each coherence interval is divided into a CE subinterval followed by a data-transmission subinterval.
  • The RIS phase matrix can switch rapidly because nFET turn-on and turn-off times are about 300 ps, below a mmWave symbol duration.

A. Stage 1 Sounding

Stage 1 uses random training, RIS control, and combining matrices to obtain received signals for recovering the initial angular parameters under hybrid-architecture constraints.

  • The BS sends a random training matrix X0, which is reflected by a random RIS phase matrix and received at the MS as Y0.
  • The MS uses a hybrid analog-digital architecture and accesses at most an NRF-dimensional signal vector per symbol time.
  • The BS can explore only one transmitted beam per symbol duration, regardless of its number of RF chains.
  • When NRF < M0, each training beam requires ⌈M0/NRF⌉ transmissions, increasing the first-stage training overhead.

B. Stage 2 Sounding

Stage 2 reuses fixed training and combining matrices while changing RIS configurations, using the stage 1 angular estimates to reduce later-block training overhead.

  • Atomic norm minimization applied to Y0 recovers θB,R and φR,M, which guide the sequential training and combining matrices.
  • For t = 1, · · · , T, the scheme fixes X1 = · · · = XT and W1 = · · · = WT while changing Ωt.
  • The design chooses N0 ≫ LB,R and M0 ≫ LR,M to obtain accurate first-stage estimates.
  • Training overhead can be greatly reduced for blocks t = 1, · · · , T compared with the first block.
  • The second stage uses received signals {Y1, · · · , YT} to estimate the remaining channel parameters with atomic norm minimization.
  • Gradual refinement could increase computational complexity, while the longer first block already targets super-resolution estimates.

1) 1D Signal:

The paper formulates channel-estimation subproblems using one- and two-dimensional atomic sets and atomic norms, solved through convex optimization and semidefinite programming.

  • 1) 1D Signal:: For DoA or line-spectral estimation, the recovered one-dimensional signal has the form α(θ) ∈ C^Nu×1.
  • 1) 1D Signal:: The one-dimensional atomic set contains steering-vector atoms with continuously varying θ, so its cardinality is infinite.
  • 1) 1D Signal:: The atomic norm seeks a sparse representation using the fewest atoms from the predefined atomic set.
  • 1) 1D Signal:: The two-dimensional channel-estimation problem separates into two decoupled 2D signal-recovery subproblems under an MMV model.
  • 1) 1D Signal:: In the first stage, estimating φR,M from Y0 is formulated with an SDP whose regularization parameter µ controls sparsity versus data fitting.
  • 1) 1D Signal:: The number of significant paths is assumed known beforehand, though it may be identified through site-specific measurements or support-recovery algorithms.
  • 1) 1D Signal:: The estimate of φR,M is recovered from Toep(û̄1) using root finding or related methods such as MUSIC and ESPRIT.

2) Estimation of θB,R:

The first stage estimates BS–RIS and RIS–MS angular parameters, then uses these estimates to guide second-stage training and recovery of angle differences and gain products.

  • Two-stage design: The second stage forms a simplified observation model and applies SMV atomic norm minimization to separate observations.The recovered quantities are angle differences and products of propagation path gains.
  • Two-stage design: The first stage determines θB,R and φR,M, which guide the design of training and receiving beams for the second stage.The resulting beam matrices use LB,R and LR,M columns, generally far fewer than the first-stage training sequences.
  • Performance conditions: Super-resolution estimation is achievable in the high-SNR regime with reasonable training overhead.Performance depends on SNR, first-stage training-sequence count, and the first-stage combining-matrix size.
  • Estimation conditions: The first stage loses the ordering of entries in θB,R and φR,M, but this changes only the row order of Y and the order of parameter-pair estimation.The passages state that this loss does not negatively affect estimation accuracy.

NBNM[ ¯G]T

The second stage recovers each gain–angle-difference pair from rows of the observation matrix using separate atomic norm minimization problems.

  • SMV recovery: LB,RLR,M atomic norm minimization problems are formulated, one for each pair {ρi, ˜θi}.The gain–angle vector hi is recovered first, followed by estimation of ˜θi and ρi.
  • Optimization: The regularization parameter controls the trade-off between sparsity and data fitting in the atomic norm formulation.The first-stage formulation sets η proportionally to a problem-dependent quantity, while the second-stage parameter is νi.
  • Pipeline: The proposed two-stage channel-estimation approach is summarized in Fig. 3, with angular directions determined first and gain products and angle differences estimated second.The staged design reduces the dimensions of the subsequent training and receiving beams.
  • Identifiability: The loss of angular ordering changes the row order of Y but not the estimation accuracy, while coupling prevents separate recovery of individual channel gains.These are structural properties of the proposed parameterization and recovery procedure.

V. RIS CONTROL AND BEAMFORMING & COMBINING DESIGN

The paper designs RIS phase control and BS/MS beamforming from estimated composite-channel parameters, using phase projection and dominant singular vectors.

  • RIS control: The RIS phase-control objective maximizes the power of the effective channel G under a unit-modulus constraint.The heuristic solution projects elementwise phases through ω⋆ = exp(−jphase([J]:,1)).
  • RIS control: The estimated gain–angle pairs form E, and SVD of EE^H selects the conjugate of its dominant singular-vector column for RIS control.The resulting vector is projected onto the unit-modulus vector space before constructing Ω⋆.
  • Limitation: The phase-control design is heuristic and is not guaranteed to maximize spectrum efficiency.The paper leaves potentially better RIS-phase criteria for future investigation.
  • Beamforming and combining: The reconstructed composite channel uses Ω⋆ and the estimated {ρi, ˜θi} pairs before BS and MS beamforming design.The reconstructed channel is converted from vector to matrix form and then decomposed by SVD.
  • Beamforming and combining: The BS beamformer and MS combiner align with singular vectors associated with the largest singular value of the reconstructed channel.Hybrid-precoding hardware constraints make the implemented vectors approximate the unconstrained singular vectors.

A. Benchmarks

The evaluation compares the proposed atomic-norm estimator with OMP and perfect-CSI benchmarks across parameter accuracy, effective spectrum efficiency, beam alignment, and RIS gain.

  • Benchmarks: The OMP benchmark uses overcomplete dictionaries and greedy atom selection in both stages, with quantized angular domains in the second stage.The proposed method is compared against this grid-based two-stage recovery approach.
  • Benchmarks: The perfect-CSI benchmarks include jointly optimized RIS, BS, and MS designs and a line-of-sight-angle alignment benchmark.The perfect-CSI setting may use exact BS, MS, and RIS location and environmental information.
  • Simulation setup: The simulations use NB = NM = 16, NR = 64, and NRF = 8, with angle separations exceeding 4/NB, 4/NR, and 4/NM.Propagation gains and noise are modeled as complex Gaussian variables, and results are averaged over 2000 channel realizations.
  • Metrics: Performance is assessed using parameter-estimation MSEs, the average effective SE bound, ASD of beamformers and combiners, and RIS gain.The parameter MSEs cover first-stage angles and second-stage angle differences and gain products.
  • Metrics: The average effective SE bound depends on estimation accuracy, training overhead, and the joint active/passive beamformer design.It is an asymptotic theoretical lower bound on subsequent data-transmission rate after designing the beamformers and RIS phase control.
  • Results presentation: Fig. 4 compares the proposed CE algorithm with [34] for channel-parameter estimation under the stated simulation setting.The supplied caption identifies the comparison and metric but does not state a plotted numerical outcome.

C. Results and Discussion

The proposed scheme improves channel-parameter estimation and downstream RIS-aided transmission performance, including under limited training overhead and inhomogeneous path gains.

  • 1) Single Path Scenario:: With Tt = 30, the proposed CE scheme outperforms the iterative reweighted benchmark in the single-path scenario.
  • 2) Effect of Training Overhead:: Increasing training overhead improves channel-parameter estimation at both stages, while the proposed method remains effective in the low-SNR regime.
  • 2) Effect of Training Overhead:: The OMP-based angular-estimation benchmark saturates near 10^-2, whereas the proposed scheme achieves better performance before mild saturation.
  • 3) Effect of Path Gain Profile:: For inhomogeneous paths, partial estimation provides beamforming and combining performance comparable to full estimation when one path dominates each individual channel.
  • 3) Effect of Path Gain Profile:: The proposed scheme significantly outperforms the OMP-based counterpart in the evaluated beamforming and combining comparison.
  • 3) Effect of Path Gain Profile:: Full estimation can reduce the average effective SE bound relative to partial estimation because weak-path gain products are poorly estimated and degrade RIS phase control.

VII. CONCLUSIONS AND FUTURE WORK

The paper proposes a two-stage atomic norm minimization approach for super-resolution channel parameter estimation in RIS-aided mmWave MIMO systems. Simulations report advantages over two-stage OMP and identify training, regularization, transmit-power, path-count, and location-awareness issues for future work.

  • A two-stage atomic norm minimization problem efficiently performs super-resolution channel parameter estimation.
  • The RIS phase control matrix is designed by power maximization, followed by joint BS beamforming and MS combining based on the reconstructed composite channel.
  • Simulation results outperform the two-stage OMP approach in angular-parameter and propagation-path-gain MSE, average effective SE bound, and RIS gains for homogeneous paths.
  • The inhomogeneous-path evaluation examines parameter-estimation performance from the path perspective, including stronger performance for parameters related to strong paths.
  • Location information provides benefits in the inhomogeneous-path scenario, under a stated assumption of location awareness.
  • Future work includes optimizing stage-one training and combining matrices, regularization, sounding transmit powers, path-count knowledge, and location-information modeling.
Loading 2007.08158v2…