Sparse source localization using perturbed arrays via bi-affine modeling

Abstract

Non-uniform spatial sampling geometries, such as nested and coprime arrays, are provably capable of localizing O(M2) sources using only M sensors. However, such guarantees require the physical locations of the sensors to satisfy certain constraints, as dictated by the corresponding array geometries. In this paper, we consider the scenario when these constraints may be violated, leading to unknown perturbations on the locations of sensors. Such perturbations can have detrimental effect on the performance of virtual array based direction-of-arrival (DOA) estimation algorithms, since the perturbed virtual array will no longer be a uniform linear array (ULA). We propose a novel self-calibration approach for underdetermined DOA estimation with such arrays, that makes extensive use of the redundancies (or repeated elements) in the virtual array. Assuming small perturbations, and a sparse grid-based model for the DOAs, we extract a novel “bi-affine” model (affine in the perturbation variable, and linear in the source powers) from the covariance matrix of the received signals. The redundancies in the co-array are then exploited to eliminate the nuisance perturbation variable, and reduce the bi-affine problem to a linear underdetermined (sparse) problem in source powers, from which the DOAs can be exactly recovered under suitable conditions. This reduction is derived for both ULA and a newly-introduced robust version of coprime arrays, when the covariance matrix of the received signals is exactly known. Our approach is compared and contrasted with recently developed algorithms for blind gain and phase calibration (BGPC), whose signal model is fundamentally different from ours. We also provide an iterative algorithm to jointly solve for the DOAs and perturbation values when we can only estimate the covariance matrix using a finite number of snapshots.