Source Resolvability of Spatial-Smoothing-Based Subspace Methods: A Hadamard Product Perspective

A major drawback of subspace methods for direction-of-arrival estimation is their poor performance in the presence of coherent sources. Spatial smoothing is a common solution that can be used to restore the performance of these methods in such a case at the cost of increased array size requirement. In this paper, a Hadamard product perspective of the source resolvability problem of spatial-smoothing-based subspace methods is presented. The array size that ensures resolvability is derived as a function of the source number, the rank of the source covariance matrix, and the source coherency structure. This new result improves upon previous ones and recovers them in special cases. It is obtained by answering a long-standing question first asked explicitly in 1973 as to when the Hadamard product of two singular positive-semidefinite matrices is strictly positive definite. The problem of source identifiability is discussed as an extension. Numerical results are provided that corroborate our theoretical findings.

I. Introduction

Direction-of-arrival (DOA) estimation is of major interest in array processing [2], [3]. For uniform linear arrays (ULAs), this important task can be formulated as a (spatial) spectral estimation problem with equispaced samples. As compared to common (temporal) spectral estimation problems, multiple snapshots of the array output can be acquired by doing temporal sampling simultaneously.