Gridless DOA Estimation and Root-MUSIC for Non-Uniform Linear Arrays

Gridless direction of arrival (DOA) estimation is addressed for a 1-D non-uniform array (NUA) of arbitrary geometry. Currently, gridless DOA estimation is solved via convex relaxation, and is applicable only to uniform linear arrays (ULA). The ULA sample covariance matrix has Toeplitz structure, and gridless DOA is based on the Vandermonde decomposition of this matrix. The Vandermonde decomposition breaks a Toeplitz matrix into its harmonic components, from which the DOAs are estimated. First, we present the irregular Toeplitz matrix and irregular Vandermonde decomposition (IVD), which generalizes the Vandermonde decomposition. It is shown that IVD is related to the MUSIC and root-MUSIC algorithms. Next, atomic norm minimization (ANM) for gridless DOA is generalized to NUA using the IVD. The resulting non-convex optimization is solved using alternating projections (AP). Simulations show the AP based solution for NUA/ULA has similar accuracy as convex relaxation of gridless DOA for ULA.