Conical Localization From Angle Measurements: An Approximate Convex Solution

For several years, a substantial effort has been devoted to the study of 3-D source localization based on 2-D arrays by measuring the well-known azimuth and elevation angles. However, studies on 3-D source localization performed by 1-D arrays are still lacking. Perhaps, the most important drawback in the deployment of a 2-D array structure lies in the fact that it needs a planar space, which might not be available in some applications. This letter concentrates on the problem of 3-D source localization based on 1-D angle measurement provided by a linear array. Different from the traditional 2-D structures where each measurement induces a straight line, each measurement in the 1-D array results in a conic surface originating at the array location. The localization problem is formulated as a constrained weighted least squares optimization problem and the semidefinite relaxation technique has been utilized to recast it as a convex optimization problem. Numerical simulation indicates that the performance of the proposed method outperforms the existing estimator and can reach the Cramer–Rao lower bound under mild Gaussian noise conditions.