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Explicit Ziv-Zakai Lower
Bound for Bearing
Estimation Using
Planar Arrays

Kristine L. Bell, Yariv Ephraim, and Harry L. Van Trees
George Mason University
C3I Center, MS 4B5
341 Science and Technology Building II
Fairfax, VA 22030-4444
tel: (703) 993-1707
fax: (703) 993-1706
email: kbell@gmu.edu

Abstract Estimating the two-dimensional direction-of-arrival, or bearing, of a narrowband planewave signal using a planar array of sensors has applications in many fields. It is a highly nonlinear problem for which calculation of the exact estimation performance is intractable, and lower bounds on the mean square estimation error are used for evaluating performance. In this work, the theory of the Ziv-Zakai lower bound is extended to handle vector parameters with arbitrary continuous prior probability density functions, and applied to the bearing estimation problem. The resulting bound has a simple closed form expression which is a function of the signal wavelength, the signal-to-noise ratio (SNR), the number of data snapshots, the number of sensors in the array, and the array configuration. Analysis of the bound suggests that there are several regions of operation, and expressions for the thresholds separating the regions are provided. In the asymptotic region where the number of snapshots and/or SNR are large, the bound approaches the inverse Fisher information. In the a priori performance region where the number of snapshots or SNR is small, the bound approaches the a priori covariance. In the transition region, the bound varies smoothly between the two extremes. Results from simulation of the maximum likelihood estimator (MLE) demonstrate that the bound closely predicts the performance of the MLE in all regions.



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