AI 中文总结
研究具有空间干扰抑制的稀疏阵列近场波束聚焦问题，通过拉格朗日对偶分析得出最优波束形成器闭式等三个结果，开发黎曼共轭梯度算法，证明阵列阶数是主导性能因素，算法接近性能极限。
AI 中文摘要
我们研究了具有空间干扰抑制的稀疏阵列近场波束聚焦问题，该问题出现在相干卫星编队和其他分布式非地面阵列中。现有设计通过带有割平面细化的二阶锥规划（SOCP）进行数值求解，但可实现的信干比（SIR）及其与经典自适应波束形成的联系仍缺乏解析表征。我们通过拉格朗日对偶分析提供了这种表征，得到三个结果。首先，每个最优波束形成器是针对由最优对偶测度诱导的有效空间协方差的广义匹配滤波器；这种闭式形式在特殊情况下恢复了MVDR、LCMV和基于SOCP的聚焦。其次，对偶测度通常具有至多为\(M^2\)的有限支撑，对于均匀线性阵列则锐化为\(M\)（\(M\)为阵列元素数量），这为割平面方法提供了有限维收敛证明。第三，在近共线几何下，平均SIR的闭式上界在\(M\)中呈现渐近对数缩放定律，确定阵列阶数而非优化算法是主导性能因素。我们开发了一种在单位环面流形上的黎曼共轭梯度算法用于实际的恒模波束形成，数值结果表明它紧密接近推导的性能极限。
英文摘要
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.
