Continuous Aperture Array (CAPA) Beamforming: A Calculus of Variations Method

The beamforming optimization for maximizing weighted sum-rate (WSR) in continuous aperture array (CAPA)-based multi-user communications is studied. In particular, the transmit beamformers of CAPA are modelled as continuous source current patterns, rendering the beamforming optimization problem as a non-convex integral-based functional programming problem. In contrast to the state-of-the-art Fourier-based method that requires numerous Fourier basis functions to approximate the functional programming, a low-complexity calculus of variations (CoV)-based method is proposed to solve the functional programming problem for WSR maximization directly, where the optimal form of the continuous source patterns is derived. Based on this optimal form, a low-complexity integral-free iterative algorithm is developed. Our numerical results validate the effectiveness of the proposed designs. It is revealed that compared to the state-of-the-art Fourier-based method, the proposed CoV based method not only improves WSR performance but also reduces computational complexity by up to hundreds of times for large CAPA apertures and high frequencies.