A Cramér Rao bounds based analysis of 3D antenna array geometries made from ULA branches

Abstract

In the context of passive sources localization using antenna array, the estimation accuracy of elevation, and azimuth are related not only to the kind of estimator which is used, but also to the geometry of the considered antenna array. Although there are several available results on the linear array, and also for planar arrays, other geometries existing in the literature, such as 3D arrays, have been less studied. In this paper, we study the impact of the geometry of a family of 3D models of antenna array on the estimation performance of elevation, and azimuth. The Cramér-Rao Bound (CRB), which is widely spread in signal processing to characterize the estimation performance will be used here as a useful tool to find the optimal configuration. In particular, we give closed-form expressions of CRB for a 3D antenna array under both conditional, and unconditional observation models. Thanks to these explicit expressions, the impact of the third dimension to the estimation performance is analyzed. Particularly, we give criterions to design an isotropic 3D array depending on the considered observation model. Several 3D particular geometry antennas made from uniform linear array (ULA) are analyzed, and compared with 2D antenna arrays. The isotropy condition of such arrays is analyzed. The presented framework can be used for further studies of other types of arrays.

Aperture Maximization with Half-Wavelength Spacing, via a 2-Circle Concentric Array Geometry that is Uniform but Sparse

“…where, s(m) is the incident signal at time instant m and n(m) is additive complex-valued spatiotemporal white Gaussian noise with a mean of zero and a variance of σ 2 n which are both prior known [33,34,13,11,35,8,36,37,38,39,40,41,1,42,5].…”

Estimation and Minimization of the Cramer-Rao lower bound for radio direction-finding on the azimuth and elevation of planar antenna arrays

“…Based on the simplified CRLB equation the metric function is defined and minimized. A few papers consider only certain array shapes [13] like 3D particular geometry antennas made from uniform linear array (ULA). As it can be seen from the above, these approaches complicate the implementation of the search for the solution of cost functions, because it is necessary to use genetic algorithms.…”

“…Several different performance and design criteria have been introduced to be used in obtaining optimal arrays [13][14][15][16][17], we can say the array with the highest bound is optimum in the sense that array is constructed using prespecified performance levels, in our case Cramer-Rao Bounds on error variance and minimum and maximum coordinates in the XY plane. The calculations can be executed manually by means of the presented in the paper simple relationships.…”

Computationally Efficient 2D DOA Estimation with Uniform Rectangular Array in Low-Grazing Angle

“…As described in Section 2, according to [24], the CRB can be obtained as $\begin{array}{c}\hfill CRB=\frac{{\sigma }^2}{2L}{\left(\mathrm{Re}\left({\mathrm{boldD}}^H{\mathsf{\Pi }}_{\mathrm{boldA}}^{\perp }\mathrm{boldD}\oplus {\hat{\mathbf{R}}}_s^T\right)\right)}^{-1},\end{array}$ where $\begin{array}{c}\hfill \mathbf{D}=\left[\frac{\partial {\mathrm{bolda}}_1}{\partial {\theta }_1},\cdots ,\frac{\partial {\mathrm{bolda}}_K}{\partial {\theta }_K},\frac{\partial {\mathrm{bolda}}_1}{\partial {\varphi }_1},\cdots ,\frac{\partial {\mathrm{bolda}}_K}{\partial {\varphi }_K}\right]\end{array}$ and ${\stackrel{false^}{\mathrm{boldR}}}_s=\left(\begin{array}{cc}{\mathbf{R}}_s& {\mathbf{R}}_s\\ {\mathbf{R}}_s& {\mathbf{R}}_s\end{array}\right)$, ${\mathsf{\Pi }}_{\mathrm{boldA}}^{\perp }={\mathbf{I}}_{MN}-\mathrm{boldA}{\left({\mathrm{boldA}}^H\mathrm{boldA}\right)}^{-1}{\mathrm{boldA}}^H$, ${\mathrm{bolda}}_k$ is the k th column of $\mathbf{A}$, k = 1, ⋯, 2 K .…”