In this paper, we develop a functional weighted minimum mean-squared error (WMMSE) algorithm for downlink beamforming in multiuser continuous aperture array (CAPA) systems where both the base station (BS) and users are equipped with CAPAs. We first present a closed-form expression for the achievable rate in multiuser CAPA systems, based on which the equivalence between maximizing the sum rate and minimizing the sum of weighted mean-squared errors (MSE) is established. We then employ the orthonormal basis expansion to transform the formulated functional optimization problem into a parameter optimization problem. By deriving the first-order optimality conditions of the parameter optimization problem and mapping them back to the functional domain, we obtain the update equations of the proposed functional WMMSE algorithm. Simulation results show that the proposed method outperforms discretization-based baselines in both sum rate and computational complexity.

6 pages,4 figures