A Weighted Inner Product Estimator in the Geometric Algebra of Euclidean 3-Space for Source Localization Using an EM Vector-sensor

In this paper, the source localization by utilizing the measurements of a single electromagnetic (EM) vector-sensor is investigated in the framework of the geometric algebra of Euclidean 3-space. In order to describe the orthogonality among the electric and magnetic measurements, two multivectors of the geometric algebra of Euclidean 3-space (G3) are used to model the outputs of a spatially collocated EM vector-sensor. Two estimators for the wave propagation vector estimation are then formulated by the inner product between a vector and a bivector in the G3. Since the information used by the two estimators is different, a weighted inner product estimator is then proposed to fuse the two estimators together in the sense of the minimum mean square error (MMSE). Analytical results show that the statistical performances of the weighted inner product estimator are always better than its traditional cross product counterpart. The efficacy of the weighted inner product estimator and the correctness of the analytical predictions are demonstrated by simulation results.