Low Angle Estimation: Models, Methods, and Bounds

Abstract

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 di erent a 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-re ection are related via simple geometry 3 the sea re ection coe cient is known as a function of grazing angle.A compact expression for the Cram er-Rao lower bound is derived, including all special cases of the key assumptions. We prove that the Cram er-Rao bound is highly dependent on the multipath model, while it is the same for the di erent signal parameterizations and that it is independent of the noise covariance. However, the Cram er-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 Cram er-Rao bound. Simulations show that the Maximum Likelihood methods are statistically e cient 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 in uence of the number of sensors and the number of snapshots on the error variariance 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.

Use of a Multipath Model in the Meter-wave Radar Height-finding Applications

“…(7). (8) illustrate that the indirect signal has a product item pej(p-a) compared with the direct signal, named as the product factor herein. Two components of the product factor, i.e., the phase difference a and the specular reflection coefficient pej' will be analyzed in the following.…”

“…From the above analyses, the phase difference a and the specular reflection coefficient pej' are the function of the grazing angle which can be expressed by the signal arrival angles including the direct angle and indirect angle. Therefore, the product factor is the function of the signal arrival angles [8].…”

Altitude measurement of low‐angle target in complex terrain for very high‐frequency radar

“…In this paper, the complex reflection is modelled as the perturbation of the steering vector of multipath signal, of which the perturbation parameter Γ is introduced. To simplify the derivation and put emphasis on the perturbation parameter, the flat‐earth model is assumed, and we suppose that the reflecting surface is approximately horizontal (no slope) and the height of the antenna is much less than the range and height of the target, leading to the inference that the incident angle is approximately equal to the reflection angle [21]. We adopt a flat‐earth model for simplicity, however, our model can be extended directly to spherical‐earth model and sloped surface model as in [22, 23].…”

Beamspace approach for detection of the number of coherent sources

“…Readers can examine [2] and [3] for a summary of the recent works on this problem. It should be noted that for low altitude targets at large distances, the multipath echo is not separable in time or in space, that is both the main and specular component impinge on the array lined in the elevation direction at the same time and have an angular separation that is in the order of a fraction of beamwidth, [3].…”