Abstract: To estimate the true (unknown) directions which may not exactly fall on the preselected grid, a novel direction-of-arrival (DOA) estimation method based on the sparse spatial covariance model and the off-grid representation of the steering vector with Taylor expansion is presented. Utilizing the spatial sparse property of incident signals, this paper formulates the DOA estimation problem as an array covariance matrix sparse representation model in a discretized grid, and relaxes the model as a convex problem. Thus, an alternating iterative estimator with grid matching (AIEGM) is proposed. Because of the limitations of grid-based model, the estimation performance of conventional methods based on sparse signal reconstruction can be highly deteriorated if the true directions of arrival are not on the preselected discretized grid. The proposed algorithm solves a series of basis pursuit denoising (BPDN) problems on a coarse grid for that problem, and revises the DOA estimation results to achieve higher estimation accuracy and has lower computational complexity than the existing off-grid DOA estimation methods. Simulation results confirm the efficacy of AIEGM.