The concept of the nested array has been used widely for underdetermined far-field source localization. In this case, the enhanced degrees-of-freedom (DOFs) is often achieved by using spatial smoothing because the difference co-array of a two-level nested array behaves like a uniformly-spaced linear array (ULA). This ingenious operation cannot be used for near-field localization because the ULA behavior of the co-array is violated by the near-field’s curved wavefronts and range-dependent attenuation. Thus, the application of nested arrays for near-field localization becomes challenging. In this letter, we propose to concatenate the vectorization of multiple fourth-order cumulant matrices to increase the DOFs offered by the nested arrays. We also show that when the sources are in the near-field, the maximum DOFs provided by a two-level nested array with $L$ physical sensors can mathematically be up to exactly $L^{2}$ . The well-posed identification conditions are discussed as well. Numerical examples are conducted to verify our analysis. Comparisons with existing ULA-based methods are also provided.

中文翻译：

---

两级嵌套阵列的近场源定位

嵌套阵列的概念已广泛用于欠定远场源定位。在这种情况下，增强的自由度 (DOF) 通常是通过使用空间平滑来实现的，因为两级嵌套阵列的差分共阵列的行为类似于均匀间隔的线性阵列 (ULA)。这种巧妙的操作不能用于近场定位，因为近场的弯曲波前和依赖于距离的衰减违反了共阵列的 ULA 行为。因此，嵌套阵列在近场定位中的应用变得具有挑战性。在这封信中，我们建议连接多个四阶累积矩阵的向量化，以增加嵌套数组提供的自由度。我们还表明，当源在近场时， $L$ 物理传感器在数学上可以精确到 $L^{2}$ . 还讨论了适定识别条件。进行了数值例子来验证我们的分析。还提供了与现有基于 ULA 的方法的比较。