Ziv-Zakai Bound for 2D-DOAs Estimation

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 Cramér-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 Cramér-Rao bound.