Low-angle estimation : Models, methods and bounds

In this work we study the performance of elevation estimators and lower bounds on the estimation error variance for a low angle target in a smooth sea scenario using an array antenna. The article is structured around some key assumptions on multipath knowledge, signal parameterization and noise covariance, giving the reader a framework in which Maximum Likelihood estimators exploiting different á priori information can be found. The crucial factor that determines the estimator accuracy is the multipath modeling, and there are three alternative levels of knowledge that can be used: 1) two unknown target locations 2) the target and its corresponding sea-reflection are related via simple geometry 3) the sea-reflection coefficient is known as a function of grazing angle. A compact expression for the Cramér–Rao lower bound is derived, including all special cases of the key assumptions. We prove that the Cramér–Rao bound is highly dependent on the multipath model, while it is the same for the different signal parameterizations and that it is independent of the noise covariance. However, the Cramér–Rao bound is sometimes too optimistic and not achievable. The tighter Barankin bound is derived to predict the threshold behavior seen at low SNR. At high SNR the Barankin bound coincides with the Cramér–Rao bound. Simulations show that the Maximum Likelihood methods are statistically efficient and achieve the theoretical lower bound on error variance, in case of high enough SNR. The bounds are also useful tools to design an improved array structure that can give better performance than the standard uniform linear array structure. The influence of the number of sensors and the number of snapshots on the error variance is also studied, showing the rate of improvement with more sensors or snapshots. Finally we discuss the use of multiple frequencies, which is mainly a tool for suppressing ambiguities. We show for which signal models it provides improved performance