Accurate Semidefinite Relaxation Method for Elliptic Localization With Unknown Transmitter Position

Abstract

Elliptic localization where a transmitter actively sending out a signal to locate an object from its echo appears in many practical systems including multiple-input-multiple-output (MIMO) radars and wireless communications. In this article, the elliptic localization problem when the transmitter position is not known is addressed. We propose an effective method for joint estimation of the object and transmitter positions. Using the range measurements from the direct paths between the transmitter and the receivers together with the indirect paths through the object, we formulate a non-convex weighted least squares (WLS) minimization problem. The non-convex nature of this problem makes it difficult for an iterative search algorithm to reach the optimal solution, implying that good estimate is not guaranteed. To handle this difficulty, we propose to solve the formulated WLS localization problem by applying the semidefinite relaxation (SDR) technique, resulting in a convex semidefinite program (SDP). The proposed SDR method is then extended to the more general case when multiple transmitters at unknown positions are used and the receiver positions are subject to errors. Simulation results show that the proposed method can attain the Cramer-Rao lower bound (CRLB) accuracy under Gaussian noise and has superior performance over the existing method.