Fresnel-aware polynomial projectors for low-complexity subspace estimation in near-field XL–MIMO

Near-field extra-large multiple-input multiple-output (XL–MIMO) systems enable high-resolution range–angle focusing but require accurate estimation of high-dimensional channel subspaces. Conventional eigendecomposition-based subspace extraction provides near-oracle accuracy, yet its cubic complexity becomes prohibitive for large antenna arrays. This paper proposes a low-complexity Polynomial Projector (PolyProj) for near-field XL–MIMO subspace estimation. The proposed method approximates the dominant signal-space projector using a Jackson-damped Chebyshev polynomial filter, evaluated through stable Clenshaw recursion and combined with randomized range compression. This avoids full eigendecomposition of the large covariance matrix and requires only matrix–block multiplications followed by a reduced eigendecomposition of dimension s ≪ N a . Theoretical analysis shows that the projector error consists of a polynomial leakage term and a sketching term, with scaling governed by O ( k / ( L 2 γ 2 ) ) + O ( 1 / ( s − k + 1 ) ) , where L is the polynomial degree, s is the sketch dimension, k is the target rank, and γ is the signal-to-residual eigengap. Numerical results demonstrate that PolyProj achieves near-SVD NMSE and spectral-efficiency performance while substantially reducing runtime relative to full eigendecomposition. The proposed framework provides a scalable alternative for real-time subspace processing in near-field XL–MIMO systems.