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Channel Estimation for Extremely Large-Scale MIMO: Far-Field or Near-Field?
Mingyao Cui, Linglong Dai
TL;DR
Hybrid precoding in XL-MIMO requires accurate channel state information, but near-field spherical wavefronts spread each path across multiple angles and undermine angle-domain sparsity. The paper introduces polar-domain representations and on-grid P-SOMP and off-grid P-SIGW estimators, with simulations showing accurate channel recovery in both near- and far-field settings under low pilot overhead.
Problem
Near-field spherical wavefronts create an energy-spread effect across multiple angles, degrading angle-domain sparsity and existing far-field compressed-sensing channel estimators while hybrid precoding requires accurate channel state information.
Method
The paper represents near-field XL-MIMO channels in a polar domain that captures angle and distance, then develops on-grid P-SOMP and off-grid P-SIGW channel-estimation schemes.
Results
The proposed near-field schemes significantly outperform existing far-field algorithms at small distances and accurately recover channels in both near-field and far-field settings with low pilot overhead.
Takeaways & Limitations
Polar-domain estimation provides a channel-estimation approach that remains effective across near-field and far-field conditions while reducing pilot overhead for near-field XL-MIMO.
Abstract
from arXiv · showhide
Extremely large-scale multiple-input-multiple-output (XL-MIMO) with hybrid precoding is a promising technique to meet the high data rate requirements for future 6G communications. To realize efficient hybrid precoding, it is essential to obtain accurate channel state information. Existing channel estimation algorithms with low pilot overhead heavily rely on the channel sparsity in the angle domain, which is achieved by the classical far-field planar wavefront assumption. However, due to the non-negligible near-field spherical wavefront property in XL-MIMO systems, this channel sparsity in the angle domain is not available anymore, and thus existing far-field channel estimation schemes will suffer from severe performance loss. To address this problem, in this paper we study the near-field channel estimation by exploiting the polar-domain sparse representation of the near-field XL-MIMO channel. Specifically, unlike the classical angle-domain representation that only considers the angle information of the channel, we propose a new polar-domain representation, which simultaneously accounts for both the angle and distance information. In this way, the near-field channel also exhibits sparsity in the polar domain. By exploiting the channel sparsity in the polar domain, we propose the on-grid and off-grid near-field channel estimation schemes for XL-MIMO. Firstly, an on-grid polar-domain simultaneous orthogonal matching pursuit (P-SOMP) algorithm is proposed to efficiently estimate the near-field channel. Furthermore, to solve the resolution limitation of the on-grid P-SOMP algorithm, an off-grid polar-domain simultaneous iterative gridless weighted (P-SIGW) algorithm is proposed to improve the estimation accuracy, where the parameters of the near-field channel are directly estimated. Finally, numerical results are provided to verify the effectiveness of the proposed schemes.
I. INTRODUCTION
XL-MIMO hybrid precoding requires accurate channel state information despite limited RF chains and high pilot overhead. Near-field spherical wavefronts destroy angular-domain sparsity, motivating a polar-domain representation and corresponding estimation algorithms.
- Motivation: Accurate channel state information is essential for efficient hybrid precoding, but far fewer RF chains than antennas make simultaneous antenna observation impossible.This mismatch creates unacceptable pilot overhead when XL-MIMO arrays contain very many antennas.
- Problem: Existing compressed-sensing estimators rely on angular-domain sparsity under the far-field planar-wave assumption, but that sparsity may not hold in XL-MIMO near-field conditions.The near-field and far-field regions are separated by the Rayleigh distance, which depends on array dimensions and wavelength.
- Contributions: The paper introduces a polar-domain representation that jointly captures angle and distance, with uniform angle sampling and non-uniform distance sampling designed to reduce transform-matrix coherence.The representation preserves sparsity for both near-field and far-field channels, with the angular-domain representation as a special case.
- Contributions: The proposed estimators include on-grid P-SOMP and off-grid P-SIGW, with P-SIGW directly recovering path gains, angles, and distances.The paper positions these algorithms as near-field channel-estimation schemes exploiting polar-domain sparsity.
- Problem: A near-field path’s energy spreads across multiple angles, so the classical spatial Fourier representation is no longer sparse and far-field estimators can lose performance.The spherical-wave steering vector has nonlinear phase variation across antenna indices and cannot be represented by a single far-field Fourier vector.
III. NEAR-FIELD POLAR-DOMAIN REPRESENTATION
The paper proposes a polar-domain representation to address the energy-spread effect and reduce pilot overhead in near-field XL-MIMO channel estimation.
- III. NEAR-FIELD POLAR-DOMAIN REPRESENTATION: The polar-domain representation is proposed to enable efficient near-field channel estimation with reduced pilot overhead.It is introduced specifically to address the energy-spread effect that prevents useful angular-domain sparsity.
A. Polar-domain representation for the near-field channel
Near-field channels remain compressible but are not sparse in the angular domain because spherical-wave paths spread energy across multiple angles. The proposed polar-domain representation samples both angle and distance to recover sparsity and avoid this energy spread.
- Near-field channels are compressible because their number of paths remains limited, even though angular-domain sparsity is lost.
- A near-field path depends on both angle and distance, so the transform matrix W is built from steering vectors sampled across the angular-distance domain.
- The polar-domain representation makes near-field channels sparse by accounting simultaneously for angular and distance information.
- The polar-domain transform matrix W is generally wide, with Q sampled vectors and Q larger than N.
- Fresnel approximation supports the near-field model, while the nonlinear phase makes closed-form transform-matrix design difficult.
B. Angular sampling method
Angular sampling is derived by isolating the linear phase term, whose coherence depends only on angle when samples lie on the same distance ring. Accordingly, angles are sampled uniformly.
- On a distance ring, the distance-dependent phase term is removed, leaving coherence determined only by the two sampled angles.
- The resulting coherence matches that between far-field steering vectors, so angular sampling follows the existing angular-domain method.
- Angles are uniformly sampled on each distance ring according to the angular-domain sampling rule.
C. Distance sampling method
Distance sampling is derived from the quadratic phase term using Fresnel-function approximations. The resulting rule samples distances non-uniformly to control column coherence below a chosen threshold.
- For equal angles, coherence depends on distance-related terms because the angle-dependent linear phase vanishes.
- Fresnel functions approximate the coherence between same-angle steering vectors at different distances.
- The magnitude |G(β)| decreases overall as β increases, supporting distance spacing selected to reduce coherence.
- For a desired coherence threshold of 0.5, β0.5 ≈1.6, giving the approximate condition β ≥1.6.
- Distance differences are constrained through inverse-distance spacing and the threshold distance Z∆.
- Sampling distances according to (15) makes adjacent-distance coherence equal to ∆; angles are uniform while distances are non-uniform.
D. Design the polar-domain transform matrix
The polar-domain transform matrix is constructed by generating angle-distance submatrices on successive distance rings and concatenating them. In the far-field-only limit, this representation reduces to the angular-domain representation.
- Each submatrix Ws contains N near-field steering vectors sampled on one distance ring at different angles.
- The transform matrix W contains S distance-ring submatrices and therefore has Q = NS columns.
- Only rings farther than the minimum allowable distance ρmin are sampled, so ρmin determines the number of distance rings S.
- When ρmin is large enough to permit only far-field distances, the polar-domain representation becomes the angular-domain representation.
- The resulting polar-domain framework supports both on-grid P-SOMP and off-grid P-SIGW near-field channel estimation.
A. On-grid near-field channel estimation
The on-grid P-SOMP algorithm estimates near-field XL-MIMO channels using polar-domain sparsity across subcarriers, with preprocessing and iterative support selection. It can also operate in the far-field, but its accuracy is limited when physical angles and distances fall between grid points.
- A. On-grid near-field channel estimation: Pre-whitening converts the colored received-pilot noise into white noise before sparse recovery.The transformed covariance becomes σ^2I after applying D^-1.
- A. On-grid near-field channel estimation: Shared polar-domain support across subcarriers enables simultaneous estimation to improve accuracy.The algorithm aggregates correlation power across subcarriers when selecting path locations.
- A. On-grid near-field channel estimation: P-SOMP extends SOMP to the polar domain to recover near-field XL-MIMO channels.The polar-domain transform accounts for physical channel angle and distance.
- A. On-grid near-field channel estimation: P-SOMP iteratively selects path locations, estimates supported path gains by orthogonal least squares, updates residuals, and reconstructs the channel.The procedure repeats for the estimated number of paths.
- A. On-grid near-field channel estimation: P-SOMP also works in the far-field because its transform samples far-field distances, including s = 0.This property is identified for later verification through simulations.
- A. On-grid near-field channel estimation: On-grid P-SOMP has limited estimation accuracy because it assumes channel angles and distances coincide exactly with sampled polar-domain points.Actual angles and distances are continuously distributed off-grid.
B. Off-grid near-field channel estimation
The off-grid P-SIGW algorithm addresses on-grid resolution limits by refining near-field path angles, distances, and gains directly. It initializes with P-SOMP and uses likelihood-based alternating optimization with gradient updates and line search.
- B. Off-grid near-field channel estimation: P-SIGW refines path gains, angles, and distances simultaneously, unlike angular-domain SIGW, which refines only gains and angles.The refinement follows the maximum likelihood principle.
- B. Off-grid near-field channel estimation: P-SIGW uses P-SOMP to initialize detected path distances, angles, and complex path gains.The subsequent refinement stage starts from these estimated parameters.
- B. Off-grid near-field channel estimation: The recovered channel is reconstructed from steering vectors evaluated at the refined angles and distances, multiplied by refined path gains.The steering-vector matrix is formed by concatenating the detected near-field paths.
- B. Off-grid near-field channel estimation: For fixed angles and distances, P-SIGW obtains path gains optimally, then optimizes the non-convex objective by alternating minimization.The reformulated objective depends on the refined angles and distances.
- B. Off-grid near-field channel estimation: Angles are updated by iterative gradient descent, while distances are updated through gradients with respect to inverse distance because sampling is non-uniform in distance.Armijo backtracking selects step lengths and ensures the objective is non-increasing.
C. Complexity and convergence analysis
The proposed optimization converges through monotonic updates, while the algorithms’ computational costs are characterized by their iterative and initialization stages. The convergence analysis guarantees a feasible solution, but not strict local optimality.
- Convergence analysis: The objective function is lower-bounded, and alternating updates of G, θ, and r are monotonically non-increasing under Armijo backtracking.This establishes convergence of the alternating procedure.
- Complexity analysis: P-SOMP and P-SIGW computational complexities are summarized in Table I, with P-SIGW dominated by SOMP operations and refinement iterations.The refinement stage is evaluated over Niter iterations.
- Convergence analysis: The convergence analysis guarantees that P-SIGW reaches a feasible solution, while strict local optimality is not proved.Only the convergence rate is fully analyzed for establishing local optimality.
- Complexity analysis: P-SIGW has the same refinement complexity as far-field off-grid SS-SIGW-OLS, with the remaining complexity difference arising from initialization.The overall complexity includes both initialization and refinement contributions.
VI. CONCLUSIONS
The paper addresses near-field XL-MIMO channel estimation by replacing angle-domain sparsity with a polar-domain representation that captures both angle and distance. It proposes on-grid and off-grid estimation algorithms based on this representation.
- Near-field energy spreads across multiple angles, preventing angular-domain sparsity and degrading compressed-sensing channel estimation.
- The polar-domain transform jointly extracts angle and distance information, making both far-field and near-field channels sparse in that domain.
- The proposed on-grid P-SOMP and off-grid P-SIGW algorithms estimate near-field XL-MIMO channels using polar-domain sparsity.
APPENDIX A. PROOF OF LEMMA 1
Appendix A derives the approximation used to characterize column coherence in the polar-domain analysis. The derivation relates the coherence to functions of a parameter determined by the channel representation.
- The proof rewrites the column-coherence analysis through the function F(x) and related transformations.
- The resulting approximation expresses column coherence as f(θ, θ, rp, rq) ≈ |F(x)| ≈ |G(β)|.
- The appendix concludes the proof of Lemma 1 after establishing the coherence approximation.
APPENDIX B. DERIVATION OF THE GRADIENT
Appendix B derives gradients of the loss function with respect to the estimated angle and distance parameters. It uses matrix differentiation of the sensing-related quantities and applies the same procedure across parameters.
- The appendix derives the gradient of L(ˆθ, ˆr) with respect to each estimated angle and distance parameter.
- The derivation differentiates ˜Ψ through its dependence on the matrix product D−1A ˜W(ˆθ, ˆr).
- After combining the intermediate derivatives, the same procedure is applied to the remaining angle and distance parameters, with a corresponding modification for distance gradients.