Hybrid-Field Channel Estimation for XL-MIMO: A Proximal Gradient Algorithm on the Fixed-Rank Matrix Manifold

This paper investigates hybrid-field channel estimation for extremely large-scale MIMO (XL-MIMO) systems. Different from the previous algorithms based on sparsity, the proposed algorithm considers both the low-rank structure and the sparsity of the channel, leading to a more accurate estimation performance. Particularly, we formulate the estimation problem of the hybrid-field channel as a sparse matrix recovery problem with a fixed-rank constraint. However, this problem encounters two challenges: the non-smooth
$\ell_{1}-\mathbf{norm}$
term and the non-convex fixed-rank matrix constraint. To address these challenges, we employ the proximal operator to handle the non-smooth
$\ell_{1}-\mathbf{norm}$
term and consider the set of matrices with a fixed rank as the fixed-rank matrix manifold. Then a fixed-rank matrix manifold-based proximal gradient (FRM-PG) algorithm is proposed to smoothly search the solution on the fixed-rank matrix manifold, where the descent direction is derived in each iteration. Simulation results demonstrate that the proposed algorithm outperforms the classical channel estimation algorithms.