An efficient beamforming design is proposed for continuous aperture array (CAPA)-based point-to-point multiple-input multiple-output (MIMO) systems. In contrast to conventional spatially discrete array (SPDA)-MIMO systems, whose optimal beamforming can be obtained using singular-value decomposition, CAPA-MIMO systems require solving the eigendecomposition of a Hermitian kernel operator, which is computationally prohibitive. To address this challenge, an explicit closed-form expression for the achievable rate of CAPA-MIMO systems is first derived as a function of the continuous transmit beamformer. Subsequently, an iterative weighted minimum mean-squared error (WMMSE) algorithm is proposed, directly addressing the CAPA-MIMO beamforming optimization without discretization approximation. Closed-form updates for each iteration of the WMMSE algorithm are derived via the calculus of variations (CoV) method. For low-complexity implementation, an equivalent matrix-based iterative solution is introduced using Gauss-Legendre quadrature. Our numerical results demonstrate that 1) CAPA-MIMO achieves substantial performance gain over the SPDA-MIMO, 2) the proposed WMMSE algorithm enhances performance while significantly reducing computational complexity compared to state-of-the-art Fourier-based approaches, and 3) the proposed WMMSE algorithm enables practical realization of parallel, non-interfering transmissions.

翻译：本文针对基于连续孔径阵列的点对点多输入多输出系统，提出了一种高效的波束成形设计方案。与传统的空间离散阵列多输入多输出系统（其最优波束成形可通过奇异值分解获得）不同，连续孔径阵列多输入多输出系统需要求解厄米核算子的特征分解，这在计算上是不可行的。为应对这一挑战，本文首先推导了连续孔径阵列多输入多输出系统可达速率的显式闭式表达式，该表达式是连续发射波束成形器的函数。随后，提出了一种迭代加权最小均方误差算法，直接处理连续孔径阵列多输入多输出系统的波束成形优化问题，而无需进行离散化近似。通过变分法推导了加权最小均方误差算法每次迭代的闭式更新公式。为实现低复杂度计算，利用高斯-勒让德积分引入了等效的基于矩阵的迭代解法。数值结果表明：1）连续孔径阵列多输入多输出系统相比空间离散阵列多输入多输出系统实现了显著的性能增益；2）与最先进的基于傅里叶的方法相比，所提出的加权最小均方误差算法在显著降低计算复杂度的同时提升了性能；3）所提出的加权最小均方误差算法能够实际实现并行且无干扰的传输。