Efficient Solvers for Coupling-Aware Beamforming in Continuous Aperture Arrays

TL;DR

This paper introduces two efficient computational methods for solving the complex integral equations involved in coupling-aware beamforming for continuous aperture arrays, significantly improving accuracy and scalability.

Contribution

It develops two novel solvers that efficiently handle the integral equations in coupling-aware beamforming, enhancing computational performance and scalability.

Findings

The LU-based solver offers stable and scalable solutions for large systems.

The kernel approximation reduces discretization demands while preserving operator structure.

Numerical results show improved accuracy and reduced computation time.

Abstract

In continuous aperture arrays (CAPAs), careful consideration of the underlying physics is essential, among which electromagnetic (EM) mutual coupling plays a critical role in beamforming performance. Building on a physically consistent mutual coupling model, the beamforming design is formulated as a functional optimization whose optimality condition leads to a Fredholm integral equation. The incorporation of the coupling model, however, substantially increases computational complexity, necessitating efficient and accurate integral equation solvers. In this letter, we propose two efficient solvers: 1) a coordinate-transformation-based kernel approximation that preserves the operator structure while alleviating discretization demands, and 2) a direct lower-upper (LU)-based solver that stably handles the Nystr\"om-discretized system. Numerical results demonstrate improved accuracy and reduced computational overhead compared to conventional methods, with the LU-based solver emerging as an efficient and scalable solution for large-scale CAPA optimization via offline factorization.