Group Sparsity Methods for Compressive Space-Frequency Channel Estimation and Spatial Equalization in Fluid Antenna System

Fluid Antenna Systems (FAS) have emerged as a transformative technology to overcome the spatial degrees-of-freedom limitations inherent in conventional static arrays. However, wideband FAS deployments face critical challenges in joint space-frequency channel estimation (CE) coupled and spatial equalization (SE), where unexplored frequency-space structure and exponential computational complexity impede practical implementation. To address this gap, this paper proposes an twostep framework combining: (1) A descending correlation group orthogonal matching pursuit (DC-GOMP) algorithm for CE, leveraging a leakage-aware group-sparse theory to exploit inherent channel correlations through dynamic hierarchical grouping, achieving near-order-of-magnitude efficiency gains over conventional OMP/GOMP; and (2) A mixed-integer linear programming (MILP) reformulation of max-min SE antenna positioning, solved globally via branch-and-bound, alongside the greedy row selection with isolated preselection (GRSIP) algorithm for real-time operation, reducing latency by two orders-of-magnitude while retaining $80 \%$ of optimal performance. Simulations demonstrate the DC-GOMP’s superiority in low-power recovery fidelity and convergence rate, alongside the SE framework’s scalable gains in deep-fading suppression and SNR enhancement (correlated with aperture size). This work establishes a foundational approach for efficient wideband FAS channel reconstruction and optimization.