Cramer-Rao Bounds for Holographic Positioning

Multiple antennas arrays combined with high carrier frequencies play a key role in wireless networks for communications but also for localization and sensing applications. To understand the fundamental limits of electromagnetically large antenna arrays for localization, this paper combines wave propagation theory with estimation theory, and computes the Cramer-Rao Bound (CRB) for the estimation of the source position on the basis of the three Cartesian components of the electric field, observed over a rectangular surface area. The problem is referred to as holographic positioning and it intrinsically depends on the radiation angular pattern of the transmitting source, which is typically ignored in standard signal processing models. We assume that the source is a Hertzian dipole, and address the holographic positioning problem in both cases, that is, with and without a priori knowledge of its orientation. To simplify the analysis and gain further insights, we also consider the case in which the dipole is located on the line perpendicular to the surface center. Numerical and asymptotic results are given to quantify the CRBs, and to quantify the effect of various system parameters on the ultimate estimation accuracy. It turns out that square surfaces with side comparable to the distance are needed to guarantee a centimeter-level accuracy in the mmWave bands. Moreover, we show that the CRBs with and without a priori knowledge of the source dipole orientation are numerically the same. The provided CRBs are also used to benchmark different maximum-likelihood estimators (MLEs) derived on the basis of a discrete representation of different models of the electric field. The analysis shows that, if the standard models are used (neglecting the radiation angular pattern), the MLE accuracy is far from the CRB. On the other hand, it approaches the CRB when the more detailed electromagnetic model is considered.