TTD-Based Hybrid Beamforming for Multi-User Near-Field Communications: A Riemannian Optimization Approach

Near-field wideband communication systems, empowered by extra-large scale antenna arrays (ELAAs), face a fundamental challenge: the spatial beam split effect due to frequency-flat analog beamformers. To mitigate this, true-time delay (TTD) based hybrid beamforming architectures have been proposed. Optimizing this architecture remains difficult due to non-convex constraints and frequency-selective channels. In this work, we propose a Riemannian optimization framework for hybrid analog-digital beamforming in fully-connected TTD-based architectures. We formulate the beamformer design as a constrained matrix approximation problem over product manifolds, incorporating unit-modulus constraints and realistic hardware limitations. A tailored Riemannian gradient descent algorithm is developed to efficiently approximate the fully-digital solution across all subcarriers. Through extensive numerical evaluations in near-field multi-user settings, the proposed method consistently delivers superior spectral efficiency and substantially reduced computational complexity relative to existing approaches across a broad frequency band.