Effect of Sensor Motion on Time Delay and Doppler Shift Localization: Analysis and Solution

This paper investigates active localization of a stationary object using time delay alone or together with Doppler shift measurements observed by a number of dynamic monostatic sensors having collocated transmitters and receivers, where the sensor motion during the interrogation period is not negligible. Non-negligible motion effect appears when the sensor speed relative to the signal propagation speed is significant, such as in an acoustic or an underwater environment. The motion effect causes the signal sent and received locations different, with their separation proportional to the two-way propagation time delay that is dependent on the object location. We first provide new measurement models for time delay and Doppler shift in the presence of sensor motion. They come as recursive equations and their non-recursive forms are derived. Analysis of the motion effect based on the Cramér-Rao lower bound (CRLB) is conducted, and the performance loss in terms of the location bias and variance when ignoring the sensor motion effect is evaluated. Two kinds of estimators for the object location are next proposed by exploiting the non-recursive forms of the model equations. One is a closed-form solution by algebraic evaluation and the other is a semi-definite relaxation solution by convex optimization. The first is computationally efficient and is shown to achieve the CRLB performance over the small error region under Gaussian noise, while the second can handle the positioning problem better when the noise level is high. Simulations validate the theoretical analysis and the performance of the proposed estimators.