A Unified Dual Framework for Sparse-Array Near-Field Beam Focusing With Spatial Interference Suppression

Jul 16, 2026

Sparse-array near-field beam focusing with spatial interference suppression is investigated for coherent satellite formations and other distributed non-terrestrial arrays. Existing numerical solutions rely on SOCP with cutting-plane refinement, but the achievable signal-to-interference ratio (SIR) and its connection to classical adaptive beamforming lacked analytic characterization. A Lagrangian-dual analysis provides three results: generalized matched-filter structure, finite-support dual measure with convergence certificates, and an asymptotic logarithmic SIR scaling law in array size under near-collinear geometry. A Riemannian conjugate-gradient method is developed for constant-modulus implementation, achieving performance near the derived limit.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages12

Reading Time30 min

I. INTRODUCTION

A. Three Principal Challenges

B. Related Work and Motivation

This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. 1 A Unified Dual Framework for Sparse-Array Near-Field Beam Focusing With Spatial Interference Suppression Changhao He, Xiaojuan Zhang, Senior Member, IEEE, Francois Chin Po Shin arXiv :2607 . 10764v 1 [ cs .IT] 12 Jul 2026 Abstract—We study sparse-array near-field beam focusing with spatial interference suppression, a problem arising in coherent satellite formations and other distributed non-terrestrial arrays. State-of-the-art designs solve it numerically through second-order cone programming (SOCP) with cutting-plane refinement, yet the achievable signal-to-interference ratio (SIR) and its link to classical adaptive beamforming have remained without an analytical characterization. We supply this characterization via a Lagrangian-dual analysis, obtaining three results. First, every optimal beamformer is a generalized matched filter against an effective spatial covariance induced by an optimal dual measure; this closed form recovers MVDR, LCMV, and SOCP-based focusing as special cases. Second, the dual measure has finite support of cardinality at most M 2 in general, sharpening to M for uniform linear arrays (M the number of array elements), which yields a finite-dimensional convergence certificate for cutting-plane methods. Third, a closed-form upper bound on the mean-SIR admits an asymptotic logarithmic scaling law in M under near-collinear geometry, identifying array order, rather than the optimization algorithm, as the dominant performance factor. A Riemannian conjugate-gradient algorithm on the unit-torus manifold is developed for practical constant-modulus beamforming, and numerical results demonstrate that it closely approaches the derived performance limit. Index Terms—Sparse arrays, near-field beamforming, MVDR, LCMV, manifold optimization, semi-infinite programming, Lagrangian duality, constant modulus. I. INTRODUCTION THE migration toward regenerative-payload non-terrestrial networks (NTN) lets low-Earth-orbit (LEO) satellites act as on-orbit base stations rather than transparent relays [1] . Adownlink toward an airborne terminal then shares spectrum with terrestrial users beneath the footprint, and co-channel emissions at ground level can exceed the interference margins of the underlying cell. The mitigation is to focus radiated energy at the airborne target while constraining the field over a designated terrestrial protection region; realized through a coherent multi-satellite formation, this is the spatiallyconstrained near-field beam focusing problem studied here. The same structure: a sparse set of distributed coherent elements focusing a converging wavefront at a target while attenuating the field over a continuum of constrained locations also arises in ultra-large-scale arrays [2] and distributed MIMO [3]; C. H. He is with King Abdullah University of Science and Technology, Saudi Arabia ([changhao.he@kaust.edu.sa](changhao.he@kaust.edu.sa)). X. J. Zhang and C. P. Shin are with the Institute for Infocomm Research, Agency for Science, Technology and Research, Singapore ({xzhang, Francois [Chin](Chin}@a-star.edu.sg)[}](Chin}@a-star.edu.sg)[@a-star.edu.sg](Chin}@a-star.edu.sg)). Corresponding author: X. J. Zhang. the analysis below is therefore stated for a general distributed array and instantiated on the LEO formation. A. Three Principal Challenges Designing a coherent satellite formation that focuses energy at an airborne target while protecting a continuous spatial region raises three principal challenges and addressed by the framework developed in this paper. (C1) Semi-infinite constraints. A continuous protection region produces infinitely many constraints, demanding cuttingplane methods [4], [5] whose convergence and inter-sample leakage need careful analysis. (C2) Geometric degeneracy. Because the target altitud

FAQs

What problem does the paper study in satellite formations?

It studies spatially constrained near-field beam focusing that targets an airborne receiver while suppressing spatial interference over a continuous protection region in coherent multi-satellite formations and distributed non-terrestrial arrays.

How does the proposed framework relate optimal beamformers to classical adaptive beamforming?

Through a Lagrangian-dual analysis, it shows every optimal beamformer has a generalized matched-filter form against an effective spatial covariance, recovering MVDR, LCMV, and SOCP-based focusing as special cases.

Why is constant-modulus beamforming a key challenge, and how is it addressed?

Saturated power amplifiers impose a per-element constant-modulus constraint, making the design non-convex; the paper develops a Riemannian conjugate-gradient algorithm on the unit-torus manifold and demonstrates numerical performance close to the derived limit.

Jul 16, 2026

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