Functional WMMSE Algorithm for Continuous Aperture Array Systems

Abstract: In this paper, we propose a functional extension of the weighted minimum mean-squared error (WMMSE) algorithm for downlink beamforming in multiuser multiple-input multiple-output (MU-MIMO) systems where both the base station (BS) and the users employ continuous-aperture arrays (CAPAs). The method lifts the matrices and vectors in the classical discrete WMMSE recursion to continuous functions by replacing matrix products and inner products with integrals over the apertures. In practice, we apply a Galerkin projection to map functions to coefficient matrices, solve the resulting discrete WMMSE problem via closed-form updates, and then lift these updates back to the functional domain. All integrals are implemented using Gauss-Legendre quadrature, which preserves the closed-form structure through weighted matrix products. Simulations show that the proposed method outperforms baselines in both spectral efficiency (SE) and computational complexity.