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.

中文翻译：

---

非均匀线性阵列的无网格DOA估计和Root-MUSIC

针对任意几何形状的一维非均匀阵列（NUA），解决了无网格到达方向（DOA）估计问题。当前，无网格DOA估计是通过凸松弛来解决的，并且仅适用于均匀线性阵列（ULA）。ULA样本协方差矩阵具有Toeplitz结构，无网格DOA基于该矩阵的Vandermonde分解。Vandermonde分解将Toeplitz矩阵分解为其谐波分量，据此可以估算DOA。首先，我们介绍了不规则Toeplitz矩阵和不规则范德蒙德分解（IVD），它概括了范德蒙德分解。结果表明，IVD与MUSIC和root-MUSIC算法有关。接下来，使用IVD将无网格DOA的原子规范最小化（ANM）推广到NUA。使用交替投影（AP）可以解决最终的非凸优化问题。仿真表明，针对NUA / ULA的基于AP的解决方案与用于ULA的无网格DOA的凸松弛具有相似的精度。