Fast and accurate two-dimensional direction-of-arrival estimation using a modified projected descent algorithm

Two-dimensional (2-D) direction-of-arrival (DOA) estimation is crucial in array signal processing, but existing methods often fail to balance estimation accuracy and computational efficiency, particularly for large-scale uniform or sparse planar arrays. Inspired by the recently proposed maximum likelihood estimation via sequential alternating direction method of multipliers (MESA), which offers high statistical efficiency and robustness to source correlations but incurs high computational cost, we extend MESA to 2-D DOA estimation and develop a computationally efficient algorithm. We formulate a structured low-rank positive-semidefinite matrix recovery problem using the Vandermonde decomposition theory of two-level Toeplitz (2LT) covariance matrices. We then design a modified projected descent (MPD) algorithm that alternates between a descent step with modified gradient updates and a projection step enforcing 2LT, positive-semidefinite, and low-rank constraints. The per-iteration computational complexity is reduced from O(N²K) to approximately O(NKmax {K, log N}) for sparse planar arrays, where N and K denote the numbers of sensors in the virtual uniform planar array and sources, respectively, by implementing large-scale matrix multiplications using the fast Fourier transform and by using small-scale eigenvalue decompositions to replace large-scale ones. Numerical experiments demonstrate that the proposed method achieves superior accuracy, efficiency, and robustness to source correlations compared with state-of-the-art approaches.