Semidefinite relaxation method for unified near-Field and far-Field localization by AOA

Abstract

This paper investigates the unified near-field and far-field localization using angle-of-arrival measurements, where the knowledge that the source is in the near-field or far-field is not required. We propose a semidefinite relaxation method to solve this problem by using the modified polar representation of the source. By approximating the measurement model, we formulate the localization problem as a weighted least squares problem, which is a non-convex constrained optimization problem. This problem is then relaxed to a convex semidefinite program (SDP) by applying semidefinite relaxation. The relaxed SDP problem is tightened by adding a set of second-order cone constraints to reach a tighter mixed semidefinite and second-order cone program. The proposed method provides an accurate position estimate for the near-field source, and yields a precise direction of arrival estimate for the far-field source. To reduce the estimation bias introduced by the approximation of the measurement model, we further derive the analytical expression of the bias and subtract it from the solution to obtain an approximately unbiased estimate. Both simulations and theoretical analysis show that the proposed method reaches the Cramer-Rao lower bound when the noise is not large under Gaussian noise, and has a comparable bias with the maximum likelihood estimator.