DOA estimation using sparse array with gain-phase error based on a novel atomic norm

• DOA estimation method based on gridless idea is provided for sparse linear array with gain-phase errors. • A novel atomic norm for the received signal of difference coarray is defined by introducing error parameters. • The process of DOA estimation is described as a dual formulation of constrained ANM approach. • A rigorous derivation and proof on how to convert dual problem into an SDP problem is given. Sparse arrays have attracted great attention in the field of direction-of-arrival (DOA) estimation due to the extended degrees of freedom (DOFs). Nevertheless, the traditional DOA estimation methods for sparse arrays suffer from degraded performance when sensor elements are uncalibrated. This paper presents a novel atomic norm-based algorithm for source localization with arbitrary sparse linear array (SLA) in the scenario with gain-phase uncertainties. Our proposed approach defines a new atomic norm for second order virtual signal by taking model errors into consideration. Then, the dual problem corresponding to original optimization problem is formulated to recover the DOAs by defining the dual atomic norm. We further present the corresponding semidefinite program characteristic that can be solved. The proposed method avoids iterations and restrictions on array configuration. It makes full use of all the DOFs provided by difference coarray of arbitrary SLA to estimate more sources and to provide high accuracy. Besides, compared with the existing coarray-based calibrated algorithms, the proposed algorithm does not need discretization on spatial domain. Computer simulations are carried out to demonstrate the superiority of the proposed algorithm.