Bounds for 2-D angle-of-arrival estimation with separate and joint processing

Cramer-Rao bounds for one- and two-dimensional angle-of-arrival estimation are reviewed for generalized 3-D array geometries. Assuming an elevated sensor array is used to locate sources on a ground plane, we give a simple procedure for drawing x-y location confidence ellipses from the Cramer-Rao covariance matrix. We modify the ordinary bounds for the case of "separate" 1-D estimates and numerically compare this with the full, joint bound. We prove that "separate" processing is optimal for a Uniform Cross Array with a single source, and that it is not optimal for the L-shaped array. A trade-off emerges between location accuracy and array height: for distant sources, increased height generally reduces error. When more than one source is present, significant gains are obtained from joint processing. We also show useful gains for distant sources by adding out-of-plane sensors in an "L + z" configuration with joint processing. These comparisons can aid system designers in deciding between separate and joint processing approaches.