  Covariance fitting (CF) is a comprehensive approach for direction of arrival
(DoA) estimation, consolidating many common solutions. Standard practice is to
use Euclidean criteria for CF, disregarding the intrinsic Hermitian
positive-definite (HPD) geometry of the spatial covariance matrices. We assert
that this oversight leads to inherent limitations. In this paper, as a remedy,
we present a comprehensive study of the use of various Riemannian metrics of
HPD matrices in CF. We focus on the advantages of the Affine-Invariant (AI) and
the Log-Euclidean (LE) Riemannian metrics. Consequently, we propose a new
practical beamformer based on the LE metric and derive analytically its spatial
characteristics, such as the beamwidth and sidelobe attenuation, under noisy
conditions. Comparing these features to classical beamformers shows significant
advantage. In addition, we demonstrate, both theoretically and experimentally,
the LE beamformer's robustness in scenarios with small sample sizes and in the
presence of noise, interference, and multipath channels.
