The optimal beamforming design for multi-user continuous aperture array (CAPA) systems is proposed. In contrast to conventional spatially discrete array (SPDA), the beamformer for CAPA is a continuous function rather than a discrete vector or matrix, rendering beamforming optimization a non-convex integral-based functional programming. To address this challenging issue, we first derive the closed-form optimal structure of the CAPA beamformer for maximizing generic system utility functions, by using the Lagrangian duality and the calculus of variations. The derived optimal structure is a linear combination of the continuous channel responses for CAPA, with the linear weights determined by the channel correlations. As a further advance, a monotonic optimization method is proposed for obtaining globally optimal CAPA beamforming based on the derived optimal structure. More particularly, a closed-form fixed-point iteration is proposed to obtain the globally optimal solution to the power minimization problem for CAPA beamforming. Furthermore, based on the optimal structure, the low-complexity maximum ratio transmission (MRT), zero-forcing (ZF), and minimum mean-squared error (MMSE) designs for CAPA beamforming are derived. It is theoretically proved that: 1) the MRT and ZF designs are asymptotically optimal in low and high signal-to-noise ratio (SNR) regimes, respectively, and 2) the MMSE design is optimal for signal-to-leakage-plus-noise ratio (SLNR) maximization. Our numerical results validate the effectiveness of the proposed designs and reveal that: i) CAPA achieves significant communication performance gain over SPDA, and ii) the MMSE design achieves nearly optimal performance in most cases, while the MRT and ZF designs achieve nearly optimal performance in specific cases.

翻译：本文提出了多用户连续孔径阵列系统的最优波束成形设计方案。与传统空间离散阵列相比，连续孔径阵列的波束成形器是一个连续函数而非离散向量或矩阵，这使得波束成形优化问题转化为一类基于积分的非凸泛函规划问题。为应对这一挑战，我们首先利用拉格朗日对偶和变分法，推导出最大化通用系统效用函数的连续孔径阵列波束成形器的闭式最优结构。该最优结构表现为连续孔径阵列连续信道响应的线性组合，其线性权重由信道相关性决定。在此基础上，我们进一步提出基于该最优结构的单调优化方法以获得全局最优的连续孔径阵列波束成形。特别地，针对功率最小化问题，提出了闭式定点迭代算法以获得连续孔径阵列波束成形的全局最优解。此外，基于最优结构推导了连续孔径阵列波束成形的低复杂度最大比传输、迫零和最小均方误差设计方案。理论分析证明：1）最大比传输与迫零设计分别在低信噪比与高信噪比区域渐近最优；2）最小均方误差设计在最大化信漏噪比意义下具有最优性。数值结果验证了所提方案的有效性，并表明：i）连续孔径阵列较空间离散阵列可获得显著的通信性能增益；ii）最小均方误差设计在多数情况下接近最优性能，而最大比传输与迫零设计在特定场景下接近最优。