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Near-Field Channel Estimation for Extremely Large-Scale Reconfigurable Intelligent Surface (XL-RIS)-Aided Wideband mmWave Systems

Songjie Yang, Chenfei Xie, Wanting Lyu, Boyu Ning, Zhongpei Zhang, Chau Yuen

arXiv:2304.00440v1cs.ITeess.SPmath.IT

TL;DR

Near-field XL-RIS estimation is challenging because spherical propagation makes channel characterization depend on distance as well as angle. The paper designs a wideband spherical-domain dictionary and a multi-frequency recovery framework with correlation-coefficient atom matching. Its reported results show CC-MMPSR outperforming inner-product matching in NMSE and time complexity, while a 2D-OLS estimator supplies a lower-bound benchmark.

  • Problem

    Near-field channel characterization must account for spherical wavefronts and distance as well as angle, making estimation more demanding than in the far field.

    Source: §I · pp. 2–3

  • Method

    The paper designs a low-coherence wideband spherical-domain dictionary and uses MMPSR with correlation-coefficient atom matching for multi-frequency channel recovery.

    Source: B. Contributions and Novelty · p. 5, 1) CC-Based Atom Matching · p. 18

  • Results

    CC-MMPSR outperforms IN-MMPSR in NMSE and time complexity, and CC matching performs better than IN matching in low- and high-power scenarios.

    Source: A. Evaluation of Different Methods · p. 24, §VII · p. 27

  • Takeaways & Limitations

    The 2D-OLS estimator provides a lower-bound benchmark for 2D-CS estimation with perfectly known channel support.

    Source: §V · p. 20

Abstract

from arXiv · show

Near-field communications present new opportunities over near-field channels, however, the spherical wavefront propagation makes near-field signal processing challenging. In this context, this paper proposes efficient near-field channel estimation methods for wideband MIMO mmWave systems with the aid of extremely large-scale reconfigurable intelligent surfaces (XL-RIS). For the wideband signals reflected by the analog RIS, we characterize their near-field beam squint effect in both angle and distance domains. Based on the mathematical analysis of the near-field beam patterns over all frequencies, a wideband spherical-domain dictionary is constructed by minimizing the coherence of two arbitrary beams. In light of this, we formulate a two-dimensional compressive sensing problem to recover the channel parameter based on the spherical-domain sparsity of mmWave channels. To this end, we present a correlation coefficient-based atom matching method within our proposed multi-frequency parallelizable subspace recovery framework for efficient solutions. Additionally, we propose a two-dimensional oracle estimator as a benchmark and derive its lower bound across all subcarriers. Our findings emphasize the significance of system hyperparameters and the sensing matrix of each subcarrier in determining the accuracy of the estimation. Finally, numerical results show that our proposed method achieves considerable performance compared with the lower bound and has a time complexity linear to the number of RIS elements.

I. INTRODUCTION

Near-field XL-RIS channels require estimation that accounts for spherical wavefronts, distance, and wideband beam squint. The paper develops a spherical-domain dictionary, a multi-frequency recovery framework, and a multi-subcarrier lower-bound benchmark for this setting.

I. INTRODUCTION

  • XL-array near-field channels depend on both angle and distance, making estimation and modeling more demanding than in the far field.Near-field XL-RIS techniques also seek to optimize signal quality at a specific location.Source: §I · pp. 2–3

A. State-of-the-Art Works

  • Earlier near-field XL-RIS beam-squint studies considered ULAs, while UPAs exhibit spherical-domain beam squint and pose dictionary-design challenges.Wideband signals commonly used in OFDM also exhibit beam squint, complicating near-field XL-RIS processing.Source: A. State-of-the-Art Works · p. 4

B. Contributions and Novelty

  • The paper analyzes angle- and distance-domain near-field beam squint and uses its frequency relationship to design a wideband spherical-domain dictionary minimizing atom coherence.The beam relationship is captured using an error function, and the analysis visualizes beam trajectories across frequencies.Source: B. Contributions and Novelty · p. 5
  • MMPSR converts 2D compressive sensing into multiple sparse-vector recovery problems with joint multi-frequency processing, using correlation coefficient matching to assess phase relation.The framework is presented as more suitable than Kronecker CS for multi-parameter, large-scale dictionaries.Source: B. Contributions and Novelty · p. 5
  • The proposed 2D-OLS estimator provides a benchmark and an NMSE lower bound across multi-frequency channels, with performance affected by system hyperparameters and each subcarrier’s sensing matrix.The stated hyperparameters include antenna, subcarrier, and channel-path counts.Source: B. Contributions and Novelty · p. 5

II. NEAR- AND FAR-FIELD WIDEBAND SYSTEM MODEL

  • The system model considers a TDD K-subcarrier MIMO-OFDM system assisted by XL-RIS and uses a one-sided near-field model for the RIS-user channel when the RIS is much larger than the user.The model considers an arbitrary user for clarity, citing parallel uplink estimation enabled by orthogonal pilots.Source: §II · p. 6, A. Channel Model · p. 7

B. Uplink Training

The section describes uplink pilot training and channel estimation, then motivates sparse dictionary-based recovery as an alternative to conventional least squares. The near-field RIS dictionary must account for spherical-wave effects.

B. Uplink Training

  • Uplink training uses Q subframes, each with a fixed RIS phase matrix, while the user transmits N_X pilot sequences per subframe.This requires QN_X symbol durations for single-user channel estimation.Source: B. Uplink Training · p. 8
  • With suitable BS–RIS deployment, the BS–RIS channel is approximated as line-of-sight, leaving the RIS–user channel to estimate.The BS combiner can be set toward the BS angle of arrival.Source: B. Uplink Training · pp. 8–9

III. NEAR-FIELD DICTIONARY DESIGN WITH BEAM SQUINT

  • 2D-LS requires pilot overhead at least equal to the number of elements, is noise-susceptible, and cannot jointly estimate all subcarriers.The section presents compressive sensing as a way to overcome these limitations using sparse channel representations.Source: §III · p. 10
  • The channel is represented sparsely with RIS and user dictionaries because the number of channel paths is limited.The RIS dictionary is harder to design because it must account for near-field effects.Source: §III · p. 10

A. Near-Field Spherical-Domain Beam Squint

The section characterizes near-field beam squint as a frequency-dependent spatial shift and analyzes the spherical array response using its phase terms. The elevation and azimuth angles are interdependent through a shared distance parameter, so the initial equations do not yield a unique solution.

A. Near-Field Spherical-Domain Beam Squint

  • Near-field beams at frequencies other than the central frequency shift spatially, motivating a mathematical description of the beam displacement.The section models the spherical array response using a Fresnel approximation that neglects terms above quadratic order.Source: A. Near-Field Spherical-Domain Beam… · p. 11
  • The beam pattern is defined over elevation, azimuth, and distance for a beam specified at a reference subcarrier.The spherical-domain response is parameterized by the desired beam and evaluated on subcarrier k.Source: A. Near-Field Spherical-Domain Beam… · p. 11
  • The equations do not yield a unique solution because elevation and azimuth are interdependent through a shared distance parameter.The authors address the mismatch by analyzing the near-field phase components sequentially.Source: A. Near-Field Spherical-Domain Beam… · pp. 11–12

1) Linear Term

The analysis separates linear and quadratic phase terms to characterize how near-field beams shift across frequencies. A spherical-domain trajectory map illustrates the resulting change in angle and distance.

1) Linear Term

  • Because linear terms dominate phase in both near- and far-field regions, the analysis first ignores nonlinear terms to find optimal subcarrier angles.The section then turns to quadratic terms, which make wideband near-field focusing more challenging.Source: 1) Linear Term · p. 12, 2) Quadratic Term · p. 12

2) Quadratic Term

  • The quadratic phase term complicates near-field wideband focusing because it depends on distance, a challenge aggravated by using uniform planar arrays.The distance solution is obtained numerically because the analytic expression involves a product of two error functions.Source: 2) Quadratic Term · pp. 12–13
  • The trajectory map’s start and end beam points are (40.56, 55.69, 17.65) and (48.6, 38.57, 22.28), versus the central-frequency point (45, 45, 20).The text interprets these points as beam shifts at other frequencies relative to the central frequency.Source: 2) Quadratic Term · p. 13, Figure 3 · p. 13

B. Wideband Spherical-Domain Dictionary Design

The dictionary design seeks low coherence between distinct beams while sampling spatial angle and distance. Its wideband form uses frequency-specific response vectors and can be generated offline for later channel estimation.

B. Wideband Spherical-Domain Dictionary Design

  • Angle sampling uses a 2D-DFT matrix, while distance sampling is considered after fixing each angle direction.The reciprocal distance can be sampled linearly based on the numerical solution and a minimum acceptable coherence.Source: B. Wideband Spherical-Domain… · pp. 13–15
  • The design aims to minimize coherence between arbitrary distinct dictionary atoms, approximating it to zero.The section distinguishes this subcarrier-independent coherence objective from the beam-pattern analysis across subcarriers.Source: B. Wideband Spherical-Domain… · p. 14
  • The distance-sampling procedure constructs narrowband dictionary atoms for each angle direction using the allowed minimum communication distance and coherence threshold.The resulting dictionary is formed from the response vectors.Source: B. Wideband Spherical-Domain… · p. 15
  • The wideband dictionary uses response vectors at different frequencies and can be generated offline for online channel estimation.The inverse function used in its construction requires numerical computation.Source: B. Wideband Spherical-Domain… · pp. 15–16

IV. NEAR- AND FAR-FIELD CHANNEL PARAMETER RECOVERY

The section uses common channel sparsity across subcarriers to recover channel parameters while avoiding the costly Kronecker representation. Its MMPSR framework exploits the channel’s 2D structure and supports parallel recovery.

IV. NEAR- AND FAR-FIELD CHANNEL PARAMETER RECOVERY

  • The KCS approach is computationally expensive for XL systems because vectorization creates a lengthy signal and large-scale dictionary.Parameter coupling also makes high-resolution recovery challenging in this framework.Source: §IV · p. 16
  • The multi-frequency recovery problem uses a common sparsity support across subcarriers to jointly recover channel parameters.The sensing matrix still depends on the subcarrier, distinguishing this problem from the standard multiple-measurement-vector setup.Source: §IV · pp. 16–17
  • The proposed recovery uses estimated atom supports to reconstruct the channel through a 2D least-squares estimator.Proposition 3 describes channel reconstruction from the estimated supports.Source: §IV · p. 17
  • MMPSR uses the 2D structure of the channel support to avoid Kronecker products and solve recovery subproblems in parallel.The framework also uses smaller dictionaries for each subproblem, reducing complexity and facilitating superresolution or refinement.Source: §IV · pp. 17–18

1) CC-Based Atom Matching

CC-based atom matching selects atoms using phase difference rather than modulus-sensitive inner products, then uses the matched atoms to estimate parameters and reconstruct channels.

1) CC-Based Atom Matching

  • Correlation-coefficient atom matching uses phase difference alone, avoiding the influence of varying atom modulus on inner-product matching.The method targets accurate matching of the phase relation between the measured signal and sensing atom.Source: 1) CC-Based Atom Matching · p. 18
  • For each subspace component, CC vectors compare the signal vectors with dictionary columns to identify matching atoms.The resulting indices are used to estimate the channel parameters across subcarriers.Source: 1) CC-Based Atom Matching · pp. 18–19
  • The algorithm refines the estimated parameters and reconstructs the channel for each subcarrier from them.The stated output is the estimated channel sequence.Source: 1) CC-Based Atom Matching · p. 19

2) Joint Multi-Frequency Processing

The method selects shared-support atoms across subcarriers, then refines parameters to address estimation error from low-resolution dictionaries.

2) Joint Multi-Frequency Processing

  • The best atom for each component is selected across all subcarriers, using the shared support to guide joint multi-frequency processing.The selected atom indices represent the best atoms for each component over the subcarriers.Source: 2) Joint Multi-Frequency Processing · p. 19

3) Parameter Refinement

  • Low-resolution dictionaries reduce computation and storage but can increase channel estimation error, making support refinement important.The passage identifies this concern particularly for multi-parameter recovery problems.Source: 3) Parameter Refinement · pp. 19–20
  • Refinement samples angle and distance parameters more finely, then repeats atom matching and joint multi-frequency processing.The refined CC vectors are evaluated using the new sampling points.Source: 3) Parameter Refinement · p. 20

V. LOWER BOUND DERIVATION

The section derives a mean-NMSE lower bound for the 2D-OLS estimator, relating it to system hyperparameters and each subcarrier’s sensing matrix.

V. LOWER BOUND DERIVATION

  • The proposed 2D-OLS estimator has a derived lower bound on its mean NMSE.The estimator uses known response vectors for the RIS and user channels.Source: §V · p. 20
  • The lower bound depends on system hyperparameters and the sensing matrix used at each subcarrier.The discussion identifies measurement-matrix optimization as a way to support a semi-orthogonal sensing-matrix assumption and potentially reduce the bound; this is left for future work.Source: §V · p. 22

VI. SIMULATION RESULTS

The simulations assess the proposed channel-estimation methods under a wideband mmWave XL-RIS setup. The setup specifies channel, measurement, and refinement parameters.

VI. SIMULATION RESULTS

  • The system uses a 28 GHz carrier, 2 GHz bandwidth, and −90 dBm noise power.The BS–RIS distance is 45 meters, and users are distributed 5–20 meters from RIS 2.Source: §VI · p. 23
  • The channel setting has three paths, including one LoS path, with users distributed uniformly in elevation and azimuth coverage.Path gains follow the cited 28 GHz mmWave channel setting; the passage gives separate path-loss parameters for LoS and NLoS paths.Source: §VI · p. 23
  • Measurement matrices are generated with complex Gaussian elements of modulus 1, and refinement step sizes are set to 0.005.The matrices include RIS phase shifts and the user training matrix.Source: §VI · p. 23
  • The passages describe 2D-LS via Prop. 1, K-OMP via solving problem (P1) with OMP, CC-MMPSR with CC-based atom matching, IN-MMPSR under the proposed MMPSR framework, and 2D-OLS.Source: §VI · p. 23

A. Evaluation of Different Methods

The simulations compare estimation methods, examine subcarrier and near-field effects, evaluate a lower bound, and analyze complexity. CC-MMPSR performs better than IN-MMPSR in the reported NMSE/MSE comparisons.

A. Evaluation of Different Methods

  • CC-MMPSR outperforms IN-MMPSR at low and high power, while 2D-LS performs poorly despite Q = NR pilot overhead; gains plateau beyond a measurement threshold.The authors attribute CC-MMPSR’s advantage to unequal atom norms interfering with phase matching for IN-based matching.Source: Figure 4 · p. 24, A. Evaluation of Different Methods · p. 24

B. Impact of the Number of Subcarriers K

C. Impact of the Near-Field Effect

  • With accurate angles but distance ignored, gain decreases as RIS size grows at fixed distance and increases as user distance grows at fixed RIS size.The passage attributes these trends to stronger near-field effects with larger aperture and planar-wave validity farther away.Source: C. Impact of the Near-Field Effect · p. 25, Figure 7 · p. 26

D. Lower Bound Analysis

  • In Fig. 8, the proposed lower bound achieves lower NMSE than 2D-OLS across uplink-power and measurement settings, including Q = 8–72 and, for K = 32, Q = 40–72.Fig. 8(a) uses uplink power 15–30 dB and Q = 8–72; Fig. 8(b) uses K = 32, Q = 40–72, and uplink power 5–30 dB.Source: D. Lower Bound Analysis · pp. 25–26

E. Time Complexity Analysis

  • Under the stated XL-RIS condition, CC-MMPSR and IN-MMPSR have total complexity O(KG_RQ), dominated by atom matching.The MMPSR steps are SVD, atom matching, and parameter refinement; the refinement can be designed to be no more complex than atom matching.Source: E. Time Complexity Analysis · pp. 26–27

VII. CONCLUSIONS

  • The conclusion describes MMPSR as more efficient than KCS and says CC-MMPSR outperforms IN-MMPSR in NMSE and time complexity.It presents 2D-OLS as a benchmark that cannot be surpassed by 2D channel reconstruction methods and derives a multi-frequency lower bound; optimizing training matrices to reduce pilot overhead remains challenging under near-field beam squint.Source: §VII · p. 27

APPENDIX A

The supplied appendix passages describe joint recovery through common sparsity support and refer to proof steps for later equations.

APPENDIX B

  • The appendix links each subspace to one channel support and describes joint recovery of the subspace signal through common sparsity support.The passage states that a single parameter is recovered in each subspace.Source: Appendix B · p. 28

APPENDIX C

  • For a simplified form, the passages define enq1,n1 ≜ [enq1]n1 and enq2,n2 ≜ [enq2]n2, and conclude that equation (d) can be proved at q1 = q2.Source: Appendix C · p. 28
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