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Convexity Meets Curvature: Lifted Near-Field Super-Resolution

Sajad Daei, Gábor Fodor, Mikael Skoglund

Abstract

Extra-large apertures, high carrier frequencies, and integrated sensing and communications (ISAC) are pushing array processing into the Fresnel region, where spherical wavefronts induce a range-dependent phase across the aperture. This curvature breaks the Fourier/Vandermonde structure behind classical subspace methods, and it is especially limiting with hybrid front-ends that provide only a small number of pilot measurements. Consequently, practical systems need continuous angle resolution and joint angle-range inference where many near-field approaches still rely on costly 2D gridding. We show that convexity can meet curvature via a lifted, gridless superresolution framework for near-field measurements. The key is a Bessel-Vandermonde factorization of the Fresnel-phase manifold that exposes a hidden Vandermonde structure in angle while isolating the range dependence into a compact coefficient map. Building on this, we introduce a lifting that maps each range bin and continuous angle to a structured rank-one atom, converting the nonlinear near-field model into a linear inverse problem over a row-sparse matrix. Recovery is posed as atomic-norm minimization and an explicit dual characterization via bounded trigonometric polynomials yields certificate-based localization that super-resolves off-grid angles and identifies active range bins. Simulations with strongly undersampled hybrid observations validate reliable joint angle-range recovery for next-generation wireless and ISAC systems.

Convexity Meets Curvature: Lifted Near-Field Super-Resolution

Abstract

Extra-large apertures, high carrier frequencies, and integrated sensing and communications (ISAC) are pushing array processing into the Fresnel region, where spherical wavefronts induce a range-dependent phase across the aperture. This curvature breaks the Fourier/Vandermonde structure behind classical subspace methods, and it is especially limiting with hybrid front-ends that provide only a small number of pilot measurements. Consequently, practical systems need continuous angle resolution and joint angle-range inference where many near-field approaches still rely on costly 2D gridding. We show that convexity can meet curvature via a lifted, gridless superresolution framework for near-field measurements. The key is a Bessel-Vandermonde factorization of the Fresnel-phase manifold that exposes a hidden Vandermonde structure in angle while isolating the range dependence into a compact coefficient map. Building on this, we introduce a lifting that maps each range bin and continuous angle to a structured rank-one atom, converting the nonlinear near-field model into a linear inverse problem over a row-sparse matrix. Recovery is posed as atomic-norm minimization and an explicit dual characterization via bounded trigonometric polynomials yields certificate-based localization that super-resolves off-grid angles and identifies active range bins. Simulations with strongly undersampled hybrid observations validate reliable joint angle-range recovery for next-generation wireless and ISAC systems.
Paper Structure (9 sections, 2 theorems, 29 equations, 3 figures)

This paper contains 9 sections, 2 theorems, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. System Model
  4. Uplink pilot measurements with hybrid combining
  5. Near-field channel model
  6. Bessel-Vandermonde Lifting and Linear Measurement Model
  7. Numerical Experiments
  8. Conclusion
  9. Proof of Theorem \ref{['thm:lifted']}

Key Result

Theorem 1

Consider a ULA with $N_r\ge 2$ antennas indexed by $n=0,\dots,N_r-1$ and spacing $d$. Let $(r,\theta)\in [r_{\min},r_{\max}]\times (0,\pi)$ be the polar coordinates of a communication scatterer or radar target. Assume $r_{\min}>0$ and define $z_1(n)\mathbin{\setstackgap{S}{0pt} = } k_\lambda nd, z_2 Let $p(\ell,q)$ be any bijection from $[-I_1,I_1]\times[-I_2,I_2]$ to $\{1,\dots,P\}$. Define $\mat

Figures (3)

  • Figure 1: Hybrid XL-MIMO geometry with Analog combiner: a large-aperture BS observes a superposition of spherical-wave paths from multiple single-antenna uplink users and nearby targets/scatterers.
  • Figure 2: The uplink channel is comprised of $L$ scatterers or targets. Each scatterer has a distinct range and angle parameter relative to the reference antenna with antenna element index $n=0$.
  • Figure 3: Gridless recovery outputs: (a) dual polynomial used for localization, (b) recovered amplitudes. The estimates $\widehat{r}_l,\widehat{\theta}_l$ and $\widehat{c}_l$ are precisely aligned with the ground-truth ones.

Theorems & Definitions (4)

  • Theorem 1
  • Remark 1
  • Remark 2: Far-field limit
  • Lemma 1