Source Localization by Frequency Measurements in Unknown Signal Propagation Speed Environments

Abstract

Signal propagation speed may not be available in practical localization scenarios. This paper addresses both the moving and stationary source localization problems by using frequency measurements when the signal propagation speed is unavailable. The task is fulfilled by exploiting the Doppler frequency shift coming from the relative motion between the source and a network of mobile sensors. Owing to the complicated nonlinear measurement model and the absence of the knowledge about signal propagation speed, it is difficult to conduct localization. To address this problem, we first transform the measurement model to a tractable form, from which to formulate a constrained weighted least squares (CWLS) problem. Afterwards, we propose two methods to solve the non-convex CWLS problem. The first method relaxes the CWLS problem into a semidefinite programming (SDP) problem by applying semidefinite relaxation (SDR). The second method adopts the alternating estimation procedure by utilizing the prior information about the nominal value of the signal propagation speed. Two subproblems are formulated and solved in an alternate manner, one in estimating the source position and velocity by applying SDR and the other the signal propagation speed by an explicit expression. Furthermore, we conduct the mean square error (MSE) analysis to show that the CWLS solution is able to reach the Cramér-Rao lower bound (CRLB) performance. Simulation results validate the expected performance of the proposed methods.