In multi-source two-dimensional (2D) direction-of-arrival (DOA) estimation, the essential matching process between the estimated and the true DOAs in the mean square error (MSE) calculation is often based on minimum Euclidean distance criterion, which is substantially different from 1D DOA estimation that is based on simple ordering process. Hence, the ZZB for multi-source 2D DOA estimation is not the extension of that for 1D DOA estimation provided in existing work. Facing this problem, we analyze the effect of the minimum Euclidean distance criterion on the ZZB via the stochastic Euclidean bipartite matching problem, from which we derive a globally valid ZZB with closed-form solution for multi-source 2D DOA estimation. The derived ZZB outperforms the most commonly used Crame´\acute{\text{e}}r-Rao bound (CRB). In addition to the hybrid coherent/uncorrelated multi-source model, we consider the partially correlated multi-source model into the ZZB derivation and formulate the ZZB as an explicit function of the correlation coefficient matrix for the first time. Moreover, according to the matching process based on minimum Euclidean distance criterion, separately estimating azimuth/elevation from 2D DOA model is a new scenario that is different from 1D DOA estimation, such that we first derive the ZZB for azimuth/elevation estimation from 2D DOA model by adopting different weight vectors. Simulation results demonstrate the advantages of the derived ZZB over the widely-accepted Crame´\acute{\text{e}}r-Rao bound.

中文翻译：

---

Ziv'akai 束缚于 2D-DOA 估计


在多源二维（2D）波达方向（DOA）估计中，均方误差（MSE）计算中估计波达方向与真实DOA之间的基本匹配过程通常基于最小欧氏距离准则，即与基于简单排序过程的一维 DOA 估计有很大不同。因此，多源 2D DOA 估计的 ZZB 并不是现有工作中提供的 1D DOA 估计 ZZB 的扩展。面对这个问题，我们通过随机欧几里德二分匹配问题分析了最小欧几里德距离准则对ZZB的影响，由此推导出一个全局有效的ZZB，其具有闭式解，用于多源二维DOA估计。派生的 ZZB 优于最常用的 Crame´\acute{\text{e}}r-Rao 界限 (CRB)。除了混合相干/不相关多源模型之外，我们将部分相关多源模型纳入ZZB推导中，并首次将ZZB表示为相关系数矩阵的显式函数。此外，根据基于最小欧氏距离准则的匹配过程，从2D DOA模型单独估计方位角/仰角是一种不同于1D DOA估计的新场景，因此我们首先从2D DOA推导出用于方位角/仰角估计的ZZB采用不同的权重向量来构建模型。仿真结果证明了导出的 ZZB 相对于广泛接受的 Crame´\acute{\text{e}}r-Rao 界限的优势。