🤖 AI Summary
This study addresses the challenge of spatial interference suppression in near-field beam focusing for sparse arrays, particularly in non-terrestrial distributed scenarios such as coherent satellite formations. By leveraging Lagrangian duality theory, the authors establish a unified analytical framework that reveals a generalized matched-filter relationship between the optimal beamformer and the effective spatial covariance matrix, and for the first time provides an analytical characterization of near-field focused beamforming performance. Key contributions include proving the finite support property of the dual measure, which guarantees convergence of the cutting-plane method; uncovering an asymptotic logarithmic growth of the average signal-to-interference ratio (SIR) with the number of array elements; and developing a Riemannian conjugate gradient algorithm on the unit torus manifold to implement constant-modulus beamforming. Numerical experiments demonstrate that the proposed approach closely approaches the theoretical SIR upper bound, confirming that array geometry—not the optimization algorithm—primarily governs performance.
📝 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.