A Fast Gridless Covariance Matrix Reconstruction Method for One- and Two-Dimensional Direction-of-Arrival Estimation

The covariance matrix of the array output is widely used for direction-of-arrival (DOA) estimation in array signal processing. However, the existing methods are only suitable for either 1-D DOA estimation or 2-D DOA estimation. In this paper, we present a computationally efficient method for both 1-D and 2-D DOA estimation referred to as a fast gridless maximum likelihood method. In particular, by exploiting the Hermitian-Toeplitz structure in the covariance matrix, we present a convex optimization problem for covariance matrix reconstruction and further derive a closed-form solution for the problem. The DOAs can then be efficiently retrieved by using the recovered covariance matrix according to root-MUSIC or Vandermonde decomposition theorem. The recovered covariance matrix can also be used for source number detection. Numerical experiments are provided to validate the proposed method in comparison with some of the existing methods for both 1-D and 2-D scenarios. Our extensive performance study shows that the proposed method has the following advantages: 1) it can be applied to common linear and rectangular array geometries with high estimation accuracy; 2) it is computationally efficient because of the available closed-form solution; 3) in 2-D DOA estimation, the azimuth and elevation angles are automatically paired, eliminating or, reducing the angle ambiguity effect; and 4) it is able to locate more sources than directly using the sample covariance matrix in some cases.