Deep Learning for Beamforming in Multi-User Continuous Aperture Array (CAPA) Systems

A DeepCAPA (Deep Learning for Continuous Aperture Array (CAPA)) framework is proposed to learn beamforming in CAPA systems. The beamforming optimization problem is firstly formulated, and it is mathematically proved that the optimal beamforming lies in the subspace spanned by users' conjugate channel responses. Two challenges are encountered when directly applying deep neural networks (DNNs) for solving the formulated problem, i) both the input and output spaces are infinite-dimensional, which are not compatible with DNNs. The finite-dimensional representations of inputs and outputs are derived to address this challenge. ii) A closed-form loss function is unavailable for training the DNN. To tackle this challenge, two additional DNNs are trained to approximate the operations without closed-form expressions for expediting gradient back-propagation. To improve learning performance and reduce training complexity, the permutation equivariance properties of the mappings to be learned are mathematically proved. As a further advance, the DNNs are designed as graph neural networks to leverage the properties. Numerical results demonstrate that: i) the proposed DeepCAPA framework achieves higher spectral efficiency and lower inference complexity compared to match-filtering and state-of-art Fourier-based discretization method, and ii) DeepCAPA approaches the performance upper bound of optimizing beamforming in the spatially discrete array-based system as the number of antennas in a fixed-sized area tends toward infinity.