Array Design for Angle of Arrival Estimation using the Worst-Case Two-Target Cramér-Rao Bound

Sparse array design is used to help reduce computational, hardware, and power requirements compared to uniform arrays while maintaining acceptable performance. Although minimizing the Cramér-Rao bound has been adopted previously for sparse sensing, it did not consider multiple targets and unknown target directions. To handle the unknown target directions when optimizing the Cramér-Rao bound, we propose to use the worst-case Cramér-Rao bound of two uncorrelated equal power sources with arbitrary angles. This new worst-case two-target Cramér-Rao bound metric has some resemblance to the peak sidelobe level metric which is commonly used in unknown multi-target scenarios. We cast the sensor selection problem for 3-D arrays using the worst-case two-target Cramér-Rao bound as a convex semi-definite program and obtain the binary selection by randomized rounding. We illustrate the proposed method through numerical examples, comparing it to solutions obtained by minimizing the single-target Cramér-Rao bound, minimizing the Cramér-Rao bound for known target angles, the concentric rectangular array and the boundary array. We show that our method selects a combination of edge and center elements, which contrasts with solutions obtained by minimizing the single-target Cramér-Rao bound. The proposed selections also exhibit lower peak sidelobe levels without the need for sidelobe level constraints.