An Iterative Direct Position Determination Approach Based on Doppler Frequency Shifts

Passive position determination of the radiating source is widely applied in wireless sensor networks (WSN). Compared with traditional two-step methods, the direct position determination (DPD) methods estimate the position in a single step and exhibit conspicuous performance superiority at a low signal-to-noise ratio (SNR). Unfortunately, the computational load for the exhaustive maximum likelihood (ML) DPD search is unacceptable for real-time processing. Although the conventional gradient-based iterative DPD method has lower computational complexity, it easily converges to the local maxima. In this article, for a stationary source, we aim to tackle the problem of DPD using Doppler frequency shifts in a computationally attractive way. The waveform of the transmitted signal is a priori knowledge, and the transmitted frequency offset exists as the nuisance parameter. Based on the expectation maximization (EM) concept, we propose an approximately globally-optimal iterative ML DPD algorithm. The proposed method regards the source-receiver vector as the unobserved latent variable, and then by exploiting the Laplace approximation technique and polar coordinate system, the original multi-dimensional grid search is transformed into multiple one-dimensional estimations for polar angle. To guarantee global optimality for the estimation of transmitted frequency offsets, a method different from its counterpart in the iterative cycle is utilized when the local maxima are achieved. Moreover, by correctly simplifying the estimation cost function in the expectation step, both the search of the polar angle and the design of the iteration parameter is avoided, which contributes to prominent computational savings. Simulation results demonstrate that the proposed modified EM-based DPD estimator (MEM-DPD) has high probability of converging to the global maximum. Meanwhile, the estimation performance is very close to the Cramér-Rao lower bound (CRLB) with minor iterations.