Abstract
Signal processing via subspace-based methods require subspace estimates taken from either the eigenvectors of the sample covariance matrix or the principal singular vectors of the received data matrix. The subspaces spanned by the singular vectors or eigenvectors are perturbed away from ground truth due to the additive noise in the received data. The perturbation reduces the accuracy of algorithms that make use of these estimates. A statistically optimal estimate of the unperturbed subspace in terms of the perturbed signal and orthogonal subspaces has been derived and is accurate up to the first-order terms in the additive noise matrix (Vaccaro, 2019). Here, the second-order optimal approximation for the unperturbed subspace is derived and applied to a uniform linear array (ULA) processing model for both full and sparse geometries. The approximation provides little improvement for a full ULA highlighting the importance of first-order terms for a fully sampled geometry. However, for a sparse array, the source estimation performance is much more clearly improved for a low signal-to-noise ratio (SNR) environment with few temporal samples.
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