  The problem of gridless direction of arrival (DOA) estimation is addressed in
the non-uniform array (NUA) case. Traditionally, gridless DOA estimation and
root-MUSIC are only applicable for measurements from a uniform linear array
(ULA). This is because the sample covariance matrix of ULA measurements has
Toeplitz structure, and both algorithms are based on the Vandermonde
decomposition of a Toeplitz 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 to apply to a more general set of matrices. It is shown that the
IVD is related to the MUSIC and root-MUSIC algorithms. Next, gridless DOA is
generalized to the NUA case using IVD. The resulting non-convex optimization
problem is solved using alternating projections (AP). A numerical analysis is
performed on the AP based solution which shows that the generalization to NUAs
has similar performance to traditional gridless DOA.
