Efficient DoA Estimation for Linear and Rectangular Arrays with Hybrid Architectures Using Compact DFT Codebooks

Hybrid Analog and Digital (HAD) architectures significantly reduce hardware overhead but introduce severe dimensionality compression, which strips the Spatial Covariance Matrix (SCM) of the degrees of freedom required for high-resolution Direction-of-Arrival (DoA) estimation. This challenge is further compounded by passive Butler-matrix implementations of Discrete Fourier Transform (DFT) analog beamforming, which avoid active phase shifters and amplifiers. In this paper, we propose a Generalized Least Squares (GLS) framework that exploits the Cauchy-like displacement structure that arises after DFT beamforming. By leveraging this structure, we develop a highly efficient numerical technique to recover the SCM for uniform linear arrays with a complexity of $\mathcal{O}(N_{\text{RF}}^2 N_x)$, where $N_x$ is the number of antennas and $N_{\text{RF}}$ the number of RF-chains. Simulations demonstrate that our estimator approaches the Cram\'er-Rao Bound (CRB) while outperforming state-of-the-art methods.