Near-Field Channel Estimation and Joint Angle-Range Recovery in XL-MIMO Systems: A Gridless Super-Resolution Approach

Existing near-field channel estimation methods for extremely large-scale MIMO (XL-MIMO) typically discretize angle and range parameters jointly, resulting in large polar-domain codebooks. This paper proposes a novel framework that formulates near-field channel estimation as a gridless super-resolution problem, eliminating the need for explicitly constructed codebooks. By employing a second-order approximation of spherical-wave steering vectors, the near-field channel is represented as a superposition of complex exponentials modulated by unknown waveforms. We demonstrate that these waveforms lie tightly in a common discrete chirp rate (DCR) subspace, with a dimension that scales as $Θ(\sqrt{N})$ for an $N$-element array. By leveraging this structure and applying a lifting technique, we reformulate the non-convex problem as a convex program using regularized atomic norm minimization, which admits an equivalent semidefinite program. From the solution to the convex program, we obtain gridless angle estimates and derive closed-form coarse range estimates, followed by refinement under the exact spherical model using gradient-based nonlinear least squares. The proposed method avoids basis mismatch and exhaustive two-dimensional grid searches while enabling accurate joint angle-range estimation with pilot budgets that scale sublinearly with array size in sparse multipath regimes. Simulations demonstrate accurate channel reconstruction and user localization across representative near-field scenarios.

💡 Research Summary

Paper Overview
The authors address the problem of channel estimation in extremely large‑scale MIMO (XL‑MIMO) systems when users lie in the radiating near‑field region, where the conventional far‑field planar‑wave assumption no longer holds. In this regime the channel depends jointly on an angle and a range for each propagation path, leading to a high‑dimensional parameter space. Existing near‑field estimators rely on a polar‑domain dictionary that discretizes both angle and range, which (i) creates an enormous codebook whose size grows roughly as N·S (N antennas, S range samples) and (ii) suffers from basis‑mismatch because the true parameters are continuous. The paper proposes a gridless super‑resolution framework that eliminates the need for any explicit dictionary.

Key Technical Contributions

1. Second‑order Taylor Approximation – By expanding the exact spherical‑wave steering vector up to second order, the authors rewrite the near‑field steering vector as a product of a conventional linear‑phase term (function of angle only) and a quadratic phase term that depends on both angle and range. The quadratic term can be expressed as a linear combination of a set of discrete chirp‑rate (DCR) atoms.

2. Low‑dimensional DCR Subspace – The authors prove that all unknown waveforms (the quadratic‑phase components) lie in a common subspace whose dimension scales as Θ(√N) despite the full waveform length being N. This dramatically reduces the degrees of freedom needed to describe the channel.

3. Lifting and Atomic Norm Formulation – By “lifting” the channel vector into a rank‑1 matrix formed by the outer product of the linear‑phase and the DCR‑subspace vectors, the estimation problem becomes one of recovering a structured low‑rank matrix that is a sparse sum of atomic elements. The authors formulate a regularized atomic‑norm minimization (ANM) problem, which is convex and can be recast as a semidefinite program (SDP).

4. Gridless Angle Recovery and Coarse Range Extraction – Solving the SDP yields the dual polynomial whose peaks give exact (gridless) angle estimates. Because the DCR subspace is known, the corresponding coefficients directly provide coarse range estimates without any exhaustive 2‑D search.

5. Non‑linear Refinement – To compensate for the approximation error introduced by the second‑order Taylor expansion, the authors perform a gradient‑based nonlinear least‑squares refinement on the exact spherical‑wave model, jointly updating angle and range for each path.

6. Theoretical Guarantees – The paper derives an error bound for the ANM solution under additive Gaussian noise, showing that the structured low‑rank matrix can be recovered with high probability as long as the number of measurements scales sublinearly with N in the sparse‑multipath regime.

7. Simulation Results – Extensive Monte‑Carlo simulations compare the proposed method against polar‑domain CS, DPSS‑based dictionaries, and conventional MUSIC/ML estimators. The gridless ANM approach achieves near‑optimal mean‑square error for both channel reconstruction and user localization, while requiring far fewer pilots (pilot budget grows like √N). It also demonstrates robustness to model mismatch and noise.

Implications
The work provides a principled, computationally tractable solution to near‑field channel estimation that avoids the curse of dimensionality associated with polar‑domain discretization. By exploiting the intrinsic low‑dimensional chirp structure, the method achieves super‑resolution in both angle and range, enabling accurate CSI acquisition and high‑precision localization for 6G XL‑MIMO systems. The convex SDP formulation ensures global optimality, and the subsequent refinement step guarantees practical performance close to the Cramér‑Rao bound. This framework can be extended to wideband, hybrid‑field, or RIS‑assisted scenarios, making it a valuable building block for future integrated sensing‑communication (ISAC) architectures.