This research investigates the ability of locating a moving object using the Doppler shifts of a carrier frequency signal sent or re ected by the object and observed by several fixed or moving sensors spatially distributed in the 2-D or 3-D space. The idea was previously studied and several solutions are proposed based on exhaustive grid search or numerical polynomial optimization. We shall formulate the problem as a constrained optimization and propose two efficient solutions. The first is by using linear optimization method to reach a closed-form solution and the second is through semi-definite relaxation technique to achieve a noise resilient estimate. The solutions are derived first for the single-time measurement and then developed to multipletime observations collected during a short time interval in which the object motion is linear. Several scenarios are considered including 2-D and 3-D localization geometry, the sensors are fixed or moving along nonlinear trajectory with random speed, the presence of errors in the carrier frequency and the sensor positions, and the noncooperative object scenario where the frequency of the carrier signal is completely not known. Analysis validates the algebraic closed-form solution in reaching the Cramer- Rao Lower Bound accuracy under Gaussian noise within the small error region. The simulations show good performance for the proposed algorithms and support the theoretical analysis.Includes bibliographical references
