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Structured Wavenumber-Domain Channel Estimation for LoS XL-MIMO

Nirosh Marthinu, Prathapasinghe Dharmawansa, Italo Atzeni

arXiv:2610.36609v1eess.SP

TL;DR

Near-field LoS XL-MIMO estimation faces growing channel dimensions, while the wavenumber-domain channel offers sparse and low-rank structure. The paper refines classical initial estimates with windowing and convex constraints, reporting improved accuracy, particularly at low SNR.

  • Problem

    Growing XL-MIMO antenna counts increase channel dimensions and make conventional estimation computationally demanding, while localized wavenumber-domain power has not been used to determine sparsity and rank bounds.

    Source: §I · p. 1

  • Method

    The framework applies windowing to an initial estimate to suppress out-of-band noise, then uses a convex constrained formulation with conservatively selected sparsity and rank bounds.

    Source: §I · p. 1, A. Constraint-Bound Selection · p. 3, C. Windowing-Based Refinement · p. 4

  • Results

    The proposed framework improves channel estimation accuracy, particularly at low SNR.

    Source: §VI · p. 5

  • Takeaways & Limitations

    The results support leveraging localized spatial-frequency power together with wavenumber-domain sparsity and low rank for LoS XL-MIMO estimation.

    Source: §I · p. 1, §VI · p. 5

Abstract

from arXiv · show

Near-field line-of-sight (LoS) channels in extremely large-scale multiple-input multiple-output (XL-MIMO) systems exhibit distinctive structural properties in the wavenumber domain. In this paper, we develop a structured wavenumber-domain channel estimation framework that first obtains an initial channel estimate using classical estimators and then refines it through a windowing-based refinement step that suppresses out-of-band noise by exploiting the localized spatial-frequency power distribution. The resulting estimate is subsequently enhanced through a convex constrained formulation that exploits the sparsity and low-rank structure of the wavenumber-domain channel. The corresponding sparsity and rank constraint bounds are selected using a conservative worst-case characterization over the relative position of the transmitter and receiver. Numerical results show that the proposed framework improves estimation accuracy, particularly at low signal-to-noise ratio, by effectively leveraging the structural properties of the wavenumber-domain channel.

I. INTRODUCTION

Near-field LoS channels are increasingly relevant in XL-MIMO, but growing channel dimensions make estimation computationally demanding unless channel structure is exploited. The paper addresses an underused wavenumber-domain structure with a framework that refines initial estimates and uses sparsity and low-rank constraints.

I. INTRODUCTION

  • Near-field LoS propagation matters for XL-MIMO because large apertures and high frequencies make spherical wavefronts and LoS components increasingly relevant.Conventional planar-wave modeling becomes inaccurate in this regime.Source: §I · p. 1
  • As antenna counts grow, channel dimensions make conventional estimation computationally demanding unless the underlying structure is exploited.Source: §I · p. 1
  • The Fourier plane-wave expansion yields a sparse wavenumber-domain representation, and near-field LoS channels also have low-rank structure, but localized power has not been used to set sparsity and rank bounds.Source: §I · p. 1
  • The proposed framework refines classical initial estimates with windowing to suppress out-of-band noise, then applies convex constraints with bounds conservatively selected over relative Tx–Rx positions.Numerical results show improved accuracy, particularly at low SNR.Source: §I · p. 1

II. SYSTEM MODEL

  • Section II introduces the considered LoS XL-MIMO system model and derives its wavenumber-domain representation.Source: §II · p. 1

A. LoS XL-MIMO System Model

The paper models a point-to-point LoS XL-MIMO link with uniform linear arrays at both ends. Its channel coefficients depend on the operating wavelength and the relative positions of the transmit and receive antennas.

A. LoS XL-MIMO System Model

B. Wavenumber-Domain Channel Representation

The channel is represented by approximating spherical propagation with a finite set of plane waves. This produces a wavenumber-domain matrix whose rows correspond to spatial-frequency components.

B. Wavenumber-Domain Channel Representation

  • The Fourier plane-wave series expansion approximates the spherical wave with a finite sum of plane waves to facilitate channel analysis and estimation.The approximation becomes asymptotically exact under the stated condition on the array transform.Source: B. Wavenumber-Domain Channel… · p. 2
  • The retained spatial-frequency indices form S, sampled at interval Δ, and L=|S| counts the discrete components.Source: B. Wavenumber-Domain Channel… · p. 2
  • The representation stacks array response vectors into Fw and expresses the channel as H≈FwHw, with wavenumber-domain entries given by component gains.Source: B. Wavenumber-Domain Channel… · p. 2
  • With receive-antenna spacing δr=λ/2, the codebook columns become asymptotically orthonormal as Nr grows.Source: B. Wavenumber-Domain Channel… · p. 2

III. SPARSE AND LOW-RANK CHANNEL ESTIMATION

The estimation method uses the wavenumber-domain channel’s localized sparsity and low rank. It refines an initial estimate under both constraints, then relaxes the resulting hard optimization into a convex problem.

III. SPARSE AND LOW-RANK CHANNEL ESTIMATION

  • Finite aperture localizes dominant spatial-frequency power, producing sparsity, while shared propagation geometry limits spatial degrees of freedom and produces low rank.Source: §III · p. 2
  • The method improves an initial estimate by finding a nearby channel matrix subject to sparsity and rank bounds.The initial estimate may come from least-squares or another low-complexity method.Source: §III · p. 2
  • Because the hard sparsity-and-rank problem is non-convex and NP-hard, the method replaces the ℓ0 and rank constraints with ℓ1 and nuclear-norm constraints.Source: §III · p. 2
  • The relaxed constrained problem is convex and can be solved with standard solvers such as CVX.Source: §III · p. 3

IV. CONSTRAINT-BOUND SELECTION, INITIAL CHANNEL ESTIMATION, AND WBR

The framework selects conservative constraint bounds, obtains an initial estimate with LS, LASSO, or IRL1, and applies WBR before constrained enhancement.

IV. CONSTRAINT-BOUND SELECTION, INITIAL CHANNEL ESTIMATION, AND WBR

  • The framework first selects α and β through a conservative worst-case approach.The problem also requires an initial channel estimate and these two constraint bounds.Source: §IV · p. 3
  • The initial channel estimate can be obtained using LS, LASSO, or IRL1.These are the three methods discussed for producing the estimate.Source: §IV · p. 3
  • WBR refines the initial channel estimate before it is used in the constrained formulation.WBR is presented as an intermediate post-processing step.Source: §IV · p. 3

A. Constraint-Bound Selection

The constraint bounds vary with array size and relative transmitter–receiver position, so the paper selects conservative upper bounds over a considered distance-angle region.

A. Constraint-Bound Selection

  • Because the exact transmitter-array position is unknown, α and β are selected as upper bounds over a considered distance-angle region G.This conservative choice makes the constraints accommodate possible transmitter-array placements.Source: A. Constraint-Bound Selection · p. 3
  • When prior information about θ or d is available, the bound-selection region can be restricted to that uncertainty region, yielding less conservative bounds.The bounds depend on array counts and the distance and angle between transmitter and receiver.Source: A. Constraint-Bound Selection · p. 3
  • For fixed Nr, α scales approximately with √Nt, while β scales approximately with Nt.The corresponding fitted models have coefficients that depend on Nr.Source: A. Constraint-Bound Selection · p. 3

B. Initial Channel Estimation

The initial wavenumber-domain estimate is formed from pilot observations using LS, LASSO, or IRL1; WBR can then refine any of these estimates before constrained enhancement.

B. Initial Channel Estimation

  • LS, LASSO, and IRL1 provide initial channel estimates with different computational-complexity and estimation-accuracy trade-offs.WBR can be applied as a post-processing step to any of these estimates.Source: B. Initial Channel Estimation · p. 3
  • The received pilot signal is modeled with additive white Gaussian noise, and the pilot matrix has orthogonal rows with τ ≥ Nt.The noise elements are i.i.d. CN(0, 1), and ρ denotes transmit SNR.Source: B. Initial Channel Estimation · p. 3
  • The estimate is obtained from the wavenumber-domain pilot observation by estimating the vectorized channel with LS, LASSO, or IRL1, then reshaping it as a matrix.LASSO promotes sparsity through ℓ1-norm regularization.Source: B. Initial Channel Estimation · p. 3
  • IRL1 iteratively solves weighted ℓ1-norm problems, penalizing small coefficients more heavily while preserving large ones.The parameter η is used to ensure numerical stability.Source: B. Initial Channel Estimation · pp. 3–4

C. Windowing-Based Refinement

Windowing-based refinement (WBR) refines an initial wavenumber-domain estimate by suppressing out-of-band noise while preserving dominant spatial-frequency components. Its window is centered on each transmitter antenna’s dominant component, with the length selected to retain most of the channel power.

C. Windowing-Based Refinement

  • WBR suppresses out-of-band noise while preserving dominant spatial-frequency components in the initial channel estimate.The initial estimators do not explicitly exploit the localized power distribution.Source: C. Windowing-Based Refinement · p. 4
  • The window for each transmitter antenna is centered on its dominant spatial-frequency component.The dominant index is determined from the initial wavenumber-domain estimate.Source: C. Windowing-Based Refinement · p. 4
  • The windowing parameter γ controls out-of-band suppression: γ = 0 gives hard thresholding, while γ > 0 retains residual contributions.The windowing matrix is applied element-wise to obtain the refined channel estimate.Source: C. Windowing-Based Refinement · p. 4
  • Across the considered Tx–Rx configurations, retaining 9 spatial-frequency components captures at least 95% of total power, so C⋆_sub is set to 9.The empirical selection uses ϵ = 0.05 and distances d ∈ [0.15dF, dF].Source: C. Windowing-Based Refinement · p. 4

V. NUMERICAL RESULTS

The numerical evaluation compares the proposed estimator with conventional estimators and examines WBR’s effect across SNR. The proposed framework significantly outperforms LS and OMP across the tested SNR range, while WBR improves MSE and its gains become more pronounced as SNR rises.

V. NUMERICAL RESULTS

  • The evaluation averages results over 100 independent channel and noise realizations for each SNR from 0 to 30 dB.The setup uses Nt = 8, Nr = 256, and pilot length τ = Nt.Source: §V · p. 4
  • WBR consistently improves MSE across the considered estimators and SNR range, with more pronounced gains as SNR increases.At low SNR, the gain is marginal because strong noise can prevent reliable identification of the dominant spatial-frequency component.Source: §V · p. 5, Figure 5 · p. 5
  • The proposed framework significantly outperforms LS and OMP across the considered SNR range, with the performance gap narrowing at high SNR.OMP uses a sparsity level of 9, matched to the empirical WBR window length.Source: §V · p. 5

VI. CONCLUSION

The conclusion presents a structured wavenumber-domain estimator that progressively refines an initial estimate using WBR and a convex constrained formulation. Numerical results show improved accuracy, particularly at low SNR; analytical constraint-bound selection and a more principled WBR design remain future work.

VI. CONCLUSION

  • The framework combines WBR with a convex constrained formulation that exploits wavenumber-domain sparsity and low-rank structure.Constraint bounds use a conservative worst-case characterization over relative Tx–Rx positions.Source: §VI · p. 5
  • Numerical results show that the proposed framework improves channel estimation accuracy, particularly at low SNR.Fig. 6 compares MSE versus SNR with conventional estimators.Source: §VI · p. 5, Figure 6 · p. 5
  • Analytical methods for selecting the sparsity and rank bounds, and a more principled WBR design, are left for future work.Source: §VI · p. 5
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