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Near-Field Channel Estimation in Mixed LoS/NLoS Environments for Extremely Large-Scale MIMO Systems

Yu Lu, Linglong Dai

arXiv:2205.03615v2cs.ITeess.SP

TL;DR

XL-MIMO’s enlarged apertures create near-field propagation that existing models do not accurately represent when LoS and NLoS components coexist. The paper proposes a mixed model, derives MIMO-RD and MIMO-ARD, and develops separate LoS/NLoS estimation; simulations show improved NMSE over existing schemes, including on QuaDRiGa.

  • Problem

    Existing near-field XL-MIMO models and estimation schemes do not accurately handle mixed LoS/NLoS propagation, particularly the LoS component.

  • Method

    The paper models LoS paths with geometric free-space propagation, NLoS paths with near-field array responses, derives MIMO-RD and MIMO-ARD, and estimates the components in two stages.

  • Results

    The proposed scheme achieves better NMSE than far-field and near-field codebook-based OMP methods, including about 4 dB improvement at 5 dB SNR and r = 60 m over near-field codebook-based OMP.

  • Takeaways & Limitations

    The mixed model and two-stage estimator provide an effective approach for practical near-field XL-MIMO in both the theoretical model and QuaDRiGa emulation.

Abstract

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Accurate channel model and channel estimation are essential to empower extremely large-scale MIMO (XL-MIMO) in 6G networks with ultra-high spectral efficiency. With the sharp increase in the antenna array aperture of the XL-MIMO scenario, the electromagnetic propagation field will change from far-field to near-field. Unfortunately, due to the near-field effect, most of the existing XL-MIMO channel models fail to describe mixed line-of-sight (LoS) and non-line-of-sight (NLoS) path components simultaneously. In this paper, a mixed LoS/NLoS near-field XL-MIMO channel model is proposed to match the practical near-field XL-MIMO scenario, where the LoS path component is modeled by the geometric free space propagation assumption while NLoS path components are modeled by the near-field array response vectors. Then, to define the range of near-field for XL-MIMO, the MIMO Rayleigh distance (MIMO-RD) and MIMO advanced RD (MIMO-ARD) is derived. Next, a two stage channel estimation algorithm is proposed, where the LoS path component and NLoS path components are estimated separately. Moreover, the Cramer-Rao lower bound (CRLB) of the proposed algorithm is derived in this paper. Numerical simulation results demonstrate that, the proposed two stage scheme is able to outperform the existing methods in both the theoretical channel model and the QuaDRiGa channel emulation platform.

I. INTRODUCTION

XL-MIMO’s enlarged apertures extend near-field propagation and expose limitations in existing channel models and estimation methods. The paper addresses this with a mixed LoS/NLoS model and a two-stage estimation scheme, supported by distance definitions, CRLB analysis, and simulations.

  • Motivation: XL-MIMO enlarges the near-field region because its antenna aperture increases sharply, making conventional far-field assumptions less suitable.The Rayleigh distance scales with the square of aperture and inversely with wavelength.
  • Motivation: Near-field XL-MIMO requires accurate channel modeling and low-overhead CSI estimation because far-field models and estimators can suffer serious performance loss.CSI is needed before beamforming, while practical hybrid systems require pilot overhead below the number of transmit antennas.
  • Research gap: Existing near-field XL-MIMO models describe NLoS components accurately but mismatch practical LoS components, especially when transmitter and receiver both use ELAAs.The resulting estimation problem was identified as lacking a corresponding study in the cited literature.
  • Contributions: The paper proposes a mixed LoS/NLoS near-field channel model and a two-stage estimator that separately handles the LoS and NLoS components.The contributions also include MIMO-RD, MIMO-ARD, CRLB, complexity analysis, and evaluation on the proposed model and QuaDRiGa.
  • Paper organization: The proposed framework is organized around a signal model, mixed channel model, near-field distance definitions, channel estimation, CRLB analysis, and simulations.The paper’s notation includes vector and matrix conventions, Frobenius norm, conjugate transpose, Gaussian and uniform distributions, Kronecker products, and the identity matrix.

B. Existing Near-Field XL-MISO Channel Model

Existing near-field XL-MISO modeling represents propagation with spherical-wave array responses and transforms the channel into a sparse polar-domain representation. The formulation accounts for antenna-specific distance differences and practical antenna-spacing considerations.

  • Near-field model: The near-field XL-MISO channel uses spherical-wave array response vectors whose phase depends on both scatterer angle and distance.The response is parameterized by the scatterer’s distance from the array center and its physical angle.
  • Practical assumption: The baseline formulation assumes half-wavelength antenna spacing to avoid coupling, while smaller spacing requires measuring and modeling coupling separately.The coupling is represented by a coefficient matrix with tunable loads.
  • Near-field model: The antenna-specific distance difference is approximated using a Taylor expansion to simplify the spherical-wave response.The approximation is applied to the difference between the scatterer distance from an antenna and from the array center.
  • Polar-domain representation: A polar-domain transform matrix is constructed from near-field response vectors sampled over angle and distance grids.The total grid size depends on the sampled distances associated with the angular grid points.
  • Estimation: The polar-domain channel exhibits sparsity, enabling compressed-sensing-based near-field channel estimation.The cited method uses the transformed representation to solve the near-field estimation problem.

C. Existing Near-Field XL-MIMO Channel Models

Existing near-field XL-MIMO models use near-field array-response vectors and polar-domain transforms, but their LoS representation mismatches the practical near-field LoS path.

  • Existing near-field XL-MIMO models replace far-field array-response vectors with near-field vectors under spherical-wave propagation.
  • The polar-domain transform matrices contain near-field response vectors sampled over angles and distances.
  • The resulting polar-domain XL-MIMO channel is sparse.
  • These models represent LoS and NLoS components identically, although the LoS representation mismatches near-field XL-MIMO behavior.

III. THE PROPOSED MIXED LOS/NLOS NEAR-FIELD XL-MIMO CHANNEL MODEL

The proposed model treats near-field LoS and NLoS components differently: LoS uses geometric free-space propagation per antenna pair, while NLoS uses near-field responses.

  • A. LoS Path Component: The LoS component is modeled with geometric free-space propagation for each transmitter–receiver antenna pair.The model uses the pairwise distance r_n2,n1 and normalizes each pair’s free-space path loss as 1/r_n2,n1.
  • A. LoS Path Component: Unlike NLoS components, the LoS component cannot be decoupled into near-field array-response vectors or represented by polar-domain transform matrices.
  • B. Proposed Mixed LoS/NLoS Near-Field XL-MIMO Channel: The mixed channel model combines the separately modeled LoS and NLoS path components.
  • B. Proposed Mixed LoS/NLoS Near-Field XL-MIMO Channel: As distance increases, the proposed model transitions toward the existing near-field representation and eventually toward a far-field MIMO model.
  • B. Proposed Mixed LoS/NLoS Near-Field XL-MIMO Channel: MIMO-RD and MIMO-ARD define boundaries between the mixed model, far-field MIMO, and existing near-field XL-MIMO models.

A. MIMO Rayleigh Distance (MIMO-RD)

MIMO-RD extends the Rayleigh-distance concept to XL-MIMO by accounting for both transmitter and receiver apertures and bounding planar–spherical phase discrepancy.

  • The phase discrepancy is determined primarily by the second-order Taylor-expansion term of the antenna-pair distance.
  • The maximum discrepancy occurs at the specified antenna geometry, which determines the required distance boundary.
  • The MIMO-RD condition limits the largest phase discrepancy between far-field planar and near-field spherical wavefronts to π/8.

B. MIMO Advanced Rayleigh Distance (MIMO-ARD)

MIMO-ARD bounds when the proposed mixed LoS model and the existing near-field response-vector model become effectively equivalent, separating three distance regimes.

  • MIMO-ARD is derived from the phase discrepancy between the true near-field LoS channel and the existing response-vector model.
  • The derived MIMO-ARD expression is proportional to 4D1D2/λ.
  • Above MIMO-RD, the phase discrepancy from the far-field model can be ignored, allowing far-field array-response modeling.
  • For distances between MIMO-ARD and MIMO-RD, the proposed and existing near-field LoS representations have negligible phase discrepancy.
  • Below MIMO-ARD, the phase discrepancy between the two near-field LoS representations cannot be ignored.

V. PROPOSED TWO STAGE CHANNEL ESTIMATION ALGORITHM

The proposed estimator separates LoS and NLoS estimation into two stages, using parameter estimation for LoS and OMP-based sparse recovery for NLoS components.

  • Overview: The algorithm estimates the LoS path component with parameter estimation and the NLoS path components with an OMP-based algorithm.The procedure also analyzes the CRLB and computational complexity.
  • Stage 1: LoS Path Component Estimation: The estimated LoS component is calculated from the optimized parameters and removed from the received pilots before NLoS estimation.This ordering exploits the usually dominant energy of the LoS path component.
  • Stage 1: LoS Path Component Estimation: LoS estimation searches a grid of distance, relative angle, and AoD values to obtain coarse parameter estimates.The grid is defined by lower and upper bounds with specified step sizes for each parameter.
  • Stage 1: LoS Path Component Estimation: The three LoS parameters are refined through iterative optimization after the initial on-grid search.The updates use gradient-based steps for distance and angles, with stopping based on an iteration limit or normalized parameter difference.
  • Stage 2: NLoS Path Components Estimation: The NLoS estimator uses iterative support selection, least-squares coefficient estimation, reshaping in the polar domain, and residual updates.After L iterations, the recovered coefficients are transformed into the estimated NLoS channel component.

B. Stage 2: NLoS Path Components Estimation

After removing the estimated LoS contribution, the NLoS channel is represented as a polar-domain sparse recovery problem and solved with a low-complexity OMP procedure.

  • NLoS sparse recovery: The received pilots are first adjusted to remove the estimated LoS path component, leaving a signal used for NLoS estimation.The resulting signal is expressed using the polar-domain representation.
  • NLoS sparse recovery: Because the NLoS channel is sparse in the polar domain, its estimation is reformulated as a sparse recovery problem.Transmitter and receiver sensing matrices are formed for the recovery process.
  • OMP procedure: The OMP procedure performs L iterations to identify supports in the transmitter and receiver sensing matrices.Each iteration evaluates correlations with the residual and updates the selected supports.
  • OMP procedure: Least-squares estimation obtains the current NLoS coefficients, which are reshaped into a polar-domain matrix before residual subtraction.After all iterations, the estimated NLoS path components are reconstructed.

C. Cram´er-Rao Lower Bound

The CRLB analysis reformulates the complex channel model into real and imaginary parts and derives a theoretical MSE bound for unbiased channel estimation.

  • CRLB formulation: The channel model is vectorized into a linear observation model involving the received signal, channel vector, noise, and sensing matrix.The sensing matrix is defined from the pilot and combing matrices.
  • CRLB formulation: The complex estimation problem is split into real and imaginary parts, whose CRLB contributions are analyzed separately.The estimated channel is reconstructed from the two estimated components.
  • CRLB derivation: The real-part CRLB is derived from the Gaussian noise model and its Fisher information matrix.The corresponding derivation applies to the imaginary part because both parts have the same form.
  • CRLB condition: The CRLB equality condition requires orthogonality of the pilot and combing matrix columns.This condition corresponds to equal eigenvalues in the relevant Gram matrices.
  • Practical limitation: In practice, the estimator cannot achieve the theoretical CRLB because the pilot and combing matrix columns are not orthogonal.The paper states that this gap is verified in the simulation results.

D. Computational Complexity Analysis

The proposed estimator combines coarse on-grid and refinement steps for LoS estimation with OMP-based NLoS estimation, and is evaluated against far-field and near-field codebook OMP methods using NMSE.

  • Computational complexity: LoS estimation combines coarse on-grid parameter search with a gradient-based refinement process.The coarse search uses a parameter collection of size SLoS, while refinement incurs gradient-calculation cost.
  • Computational complexity: The LoS-stage complexity is O((SLoS + MI)N_RF^r), while NLoS estimation has complexity O(N1N2(S1 + S2)L).The NLoS complexity follows the OMP-based estimation procedure.
  • Simulation setup: The simulations use N1 = 256, N2 = 128, f = 50 GHz, and distances generated from 50 m to 500 m.The evaluation uses NMSE, with CRLB used as a performance bound; pilot and combining matrices are randomly generated.
  • Simulation results: Across the distance range, the proposed algorithm maintains the lowest NMSE, while competing codebook OMP methods degrade as distance decreases.At distances larger than MIMO-ARD, the phase discrepancy between the proposed and existing models vanishes.
  • Simulation results: The proposed scheme achieves the best pilot-size NMSE performance and outperforms all competing schemes at smaller distance.At larger distance, the proposed and competing schemes have similar NMSE; at SNR 5 dB and r = 60 m, the proposed method gives about 4 dB improvement over near-field codebook OMP.
  • Simulation results: On the QuaDRiGa channel dataset, the proposed method outperforms both codebook OMP baselines.The results support the proposed mixed LoS/NLoS model and two-stage estimator in the practical near-field XL-MIMO setting.

VII. CONCLUSIONS

The paper proposes a mixed LoS/NLoS near-field XL-MIMO model, derives MIMO-RD and MIMO-ARD, and introduces a two-stage estimator. Simulations show better NMSE than codebook-based alternatives in theoretical and practical channel models.

  • Contributions: The mixed LoS/NLoS near-field XL-MIMO model uses geometric free-space propagation for LoS and near-field response vectors for NLoS components.The model generalizes both the far-field channel model and the existing near-field MIMO channel model.
  • Contributions: MIMO-RD and MIMO-ARD are derived to characterize the near-field region of XL-MIMO.These distances distinguish channel regimes according to the transmitter-receiver separation and LoS phase behavior.
  • Contributions: The proposed two-stage channel estimation scheme estimates LoS and NLoS path components separately.The paper evaluates the scheme against far-field and near-field codebook-based channel estimation methods.
  • Conclusions: The proposed scheme achieves better NMSE than far-field and near-field codebook-based estimators in theoretical and practical channel models.The practical evaluation uses a QuaDRiGa channel model, while future work is identified for UPA-based 3D near-field XL-MIMO modeling.

APPENDIX A OPTIMIZATION OF G(i) IN WITH REGARD TO r(i), θ(i), ϕ(i)

The appendix derives gradients of G with respect to distance and angular parameters by differentiating the channel-matrix elements used in the optimization.

  • Derivative setup: The appendix provides derivatives of G with respect to r(i), θ(i), and ϕ(i).The derivation suppresses the superscript (i) to simplify notation.
  • Channel-element derivatives: The derivation starts from the (n2, n1)-th channel element Hn2,n1 and its distance-dependent propagation expression.The element is written using rn2,n1 and the phase term e−j2πrn2,n1/λ.
  • Channel-element derivatives: The resulting expressions give explicit gradients with respect to r, θ, and ϕ for optimization of G.The gradients include geometric terms involving d1, d2, θ, ϕ, r, and the element-specific propagation factor Γ.
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