Methods And Applications Of Geometric Algebra Model For Vector Antenna Array Signal

In this paper, the techniques of electromagnetic (EM) vector-sensor and/or its array modeling and EM source localization are discussed in the framework of the geometric algebra of Euclidean 3-space (G3).To facilitate subsequent discussion, we first define a new kind of matrix, nominated as G3-matrix, whose entries are multivectors in the G3. Two fundamental analysis tools, i.e., the complex representation and quaternion adjoint matrices of the G3-matrx, are then given. By means of such two tools, some primary topics of the G3-matrix analysis, such as the Hermitian transposes, inverses, eigenvalues and eigenvectors, ranks, and singular value decompositions, are investigated for the later array signal modeling and processing convenience.A new array model (called as G-MODEL), aiming at fully embodying the orthogonality among the EM signals, is first proposed based on the G3 formulation of Maxwell equations. Using the G-MODEL, a novel array data covariance matrix is defined by the geometric products in the G3 and then analyzed. The analytical results show that the six-component measurement noise of a vector-sensor can naturally be whitened if the noise cross-correlations between the different axial electric and magnetic components are equal to one another. Compared with the known best quad-quaternion model, the new covariance matrix model results in a reduction of half memory requirements while the amount of divisions is reduced to 1/2, multiplications and additions reduced to almost 1/7.With the reported G-MODEL and G3-matrix manipulation techniques, the algorithms of direction of arrival (DOA) estimation by using the measurements of an EM vector-sensor and/or its array are also presented. For the single-source single-vector-sensor case, a weighted inner product (WIP) estimator, focusing on estimating the propagation direction vector of the incident EM waves, is proposed based on the minimum mean square error (MMSE) fusion rule. Statistical analyses show that the performances of the WIP estimator are always better than its traditional cross product counterpart. For the vector-sensor array case, a geometric algebra version of MUSIC estimator (G-MUSIC) is derived. Analytical results reveal that our G-MUSIC estimator allows estimating the source's arrival angles without considering their polarizations. To'decorrelate'the incident signals, a complex representation matrix averaging (CRMA) algorithm is also developed to extend the subspace based techniques to the fully correlated sources. Unlike most existing methods, such as spatial smoothing and forward-backward averaging, the CMRA algorithm does not decrease the array aperture and can be applied to arbitrary array geometry.In general, the work in this paper demonstrates that the G3 is a very powerful tool for the EM vector-sensor array signal description and processing. The modeling technique presented herein perfectly matches the mathematical object's physical nature (i.e., the orthogonal relationships among the EM signals) and the mathematical tool (i.e., the geometric algebra of Euclidean 3-space). The G-MODEL and its excellent DOA estimation performances make us believe that the G3 can also be served as an effective tool in other areas of array signal processing, for example, beamforming.