A Unified Dual Framework for Sparse-Array Near-Field Beam Focusing With Spatial Interference Suppression
2026-07-12 • Information Theory
Information Theory
AI summaryⓘ
The authors study how to focus signals in sparse arrays, like those used in satellite groups, while reducing interference. They provide a mathematical explanation showing how the best signal focusing methods relate to known techniques, offering a clearer understanding of their performance limits. They also find a way to certify when an algorithm has converged and show that array size mainly determines signal quality, not the specific optimization method. Lastly, they develop a practical algorithm that nearly reaches the theoretical performance.
beamformingsparse arrayssignal-to-interference ratio (SIR)Lagrangian dualityMVDRLCMVsecond-order cone programming (SOCP)cutting-plane methodsRiemannian conjugate-gradientconstant-modulus beamforming
Authors
Changhao He, Xiaojuan Zhang, Francois Chin Po Shin
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.