The beamforming optimization in continuous aperture array (CAPA)-based multi-user communications is studied. In contrast to conventional spatially discrete antenna arrays, CAPAs can exploit the full spatial degrees of freedom (DoFs) by emitting information-bearing electromagnetic (EM) waves through continuous source current distributed across the aperture. Nevertheless, such an operation renders the beamforming optimization problem as a non-convex integral-based functional programming problem, which is challenging for conventional discrete optimization methods. A couple of low-complexity approaches are proposed to solve the functional programming problem. 1) Calculus of variations (CoV)-based approach: Closed-form structure of the optimal continuous source patterns are derived based on CoV, inspiring a low-complexity integral-free iterative algorithm for solving the functional programming problem. 2) Correlation-based zero-forcing (Corr-ZF) approach: Closed-form ZF source current patterns that completely eliminate the inter-user interference are derived based on the channel correlations. By using these patterns, the original functional programming problem is transformed to a simple power allocation problem, which can be solved using the classical water-filling approach with reduced complexity. Our numerical results validate the effectiveness of the proposed designs and reveal that: i) compared to the state-of-the-art Fourier-based discretization approach, the proposed CoV-based approach not only improves communication performance but also reduces computational complexity by up to hundreds of times for large CAPA apertures and high frequencies, and ii) the proposed Corr-ZF approach achieves asymptotically optimal performance compared to the CoV-based approach.

翻译：本文研究了基于连续孔径阵列（CAPA）的多用户通信中的波束成形优化问题。与传统的空间离散天线阵列不同，CAPA通过分布在孔径上的连续源电流发射承载信息的电磁波，从而能够利用全部空间自由度。然而，这种操作使得波束成形优化问题成为一个非凸的基于积分的泛函规划问题，这对传统的离散优化方法构成了挑战。本文提出了两种低复杂度方法来解决该泛函规划问题。1）基于变分法（CoV）的方法：利用变分法推导出最优连续源分布模式的闭式结构，并由此启发了一种低复杂度、无需积分的迭代算法来求解该泛函规划问题。2）基于相关性的迫零（Corr-ZF）方法：基于信道相关性，推导出能够完全消除用户间干扰的闭式迫零源电流分布模式。利用这些模式，原始的泛函规划问题被转化为一个简单的功率分配问题，该问题可采用经典注水法以较低复杂度求解。数值结果验证了所提设计的有效性，并表明：i）与最先进的基于傅里叶的离散化方法相比，所提基于变分法的方法不仅提升了通信性能，而且对于大孔径CAPA和高频场景，计算复杂度可降低高达数百倍；ii）与基于变分法的方法相比，所提基于相关性的迫零方法能够实现渐近最优的性能。