Abstract An array geometry with three parallel nested arrays is designed and a corresponding two-dimensional (2D) grid-less angle estimation method is proposed. The array can use 4 M physical elements to achieve 4M2 virtual degrees of freedom (DOF) after the extended vectorization of two cross-covariance matrices (CCMs), which also transforms the 2D angle estimation problem into one-dimensional (1D) and one-snapshot sparse recovery. Thereafter, without any spatial smoothing, the Toeplitz covariance matrix of the virtual array can be recovered through a robust semi-definite programming (SDP) based on atomic norm minimization (ANM). Meanwhile, the denoising virtual output is obtained as a by-product. Finally, 1D angle is obtained from the Vandermonde decomposition of the Toeplitz matrix and the other 1D angle is obtained in succession from the denoising virtual output via total least squares (TLS). The proposed method is robust to grid mismatch problem, obtains automatically paired two-dimensional angle estimation and requires only one-level Toeplitz matrix recovery, which makes the maximum identifiable source number exceed the physical sensor number. Furthermore, the angle estimation performance of the proposed method outperforms other state-of-the-art methods using parallel arrays. Numerical simulations verify the effectiveness of our approach.

中文翻译：

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基于三个平行嵌套阵列的二维无网格角度估计

摘要 设计了具有三个平行嵌套阵列的阵列几何结构，并提出了相应的二维（2D）无网格角度估计方法。该阵列在对两个交叉协方差矩阵（CCM）进行扩展矢量化后，可以使用 4M 个物理元素实现 4M2 个虚拟自由度（DOF），这也将 2D 角度估计问题转化为一维（1D）和一维快照稀疏恢复。此后，无需任何空间平滑，虚拟阵列的 Toeplitz 协方差矩阵可以通过基于原子范数最小化 (ANM) 的稳健半定规划 (SDP) 恢复。同时，作为副产品获得去噪虚拟输出。最后，一维角度是从托普利茨矩阵的范德蒙德分解获得的，另一个一维角度是通过总最小二乘法（TLS）从去噪虚拟输出中连续获得的。该方法对网格失配问题具有鲁棒性，自动获得成对的二维角度估计，只需要一级托普利茨矩阵恢复，使得最大可识别源数超过物理传感器数。此外，所提出方法的角度估计性能优于其他使用平行阵列的最新方法。数值模拟验证了我们方法的有效性。自动获得成对的二维角度估计，只需要一级托普利茨矩阵恢复，使得最大可识别源数超过物理传感器数。此外，所提出方法的角度估计性能优于其他使用平行阵列的最新方法。数值模拟验证了我们方法的有效性。自动获得成对的二维角度估计，只需要一级托普利茨矩阵恢复，使得最大可识别源数超过物理传感器数。此外，所提出方法的角度估计性能优于其他使用平行阵列的最新方法。数值模拟验证了我们方法的有效性。