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 $\Theta(\sqrt{N})$ 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.