An analytical study of sparse recovery algorithms in presence of an off-grid source

Abstract

Direction of arrival (DOA) estimation has been an active field of research for many decades. If the field is modeled as a superposition of a few planar wavefronts, the DOA estimation problem can be expressed as a sparse recovery problem and the Compressed Sensing (CS) framework can be applied. Many powerful CS-based DOA estimation algorithms have been proposed in recent years. However, they all face one common problem. Although, the model is sparse in a continuous angular domain, to apply the CS framework we need to construct a finite dictionary by sampling this domain with a predefined sampling grid. Therefore, the target locations are almost surely not located exactly on a subset of these grid points. Early solutions to this problem include adaptively refining the grid around the candidate targets found with an initial, mismatched grid [1]. Recent papers try to model the mismatch error explicitly and fit it to the observed data either statistically [2] or by interpolating between grid points [3]. In this paper we take an analytical approach to investigate the effect of recovering the spectrum of a source not contained in the dictionary. Unlike earlier works on the sensitivity of compressed sensing to basis mismatch [4] that have provided a quantitative analysis of the approximation error, we focus on the shape of the resulting spectrum, considering one target source for simplicity. We show that the recovered spectrum is not sparse but it can be well approximated by the closest two dictionary atoms on the grid and their coefficients can be exploited to estimate the grid offset.