Wikipedia:Sandbox

The six-component electromagnetic vector-sensor has been much investigated $[3-11,13-18,20]$ in the recent two decades for diversely polarized direction-of-arrival (DOA) estimation and polarization estimation. This six-component electromagnetic vector-sensor consists of three identical but orthogonally oriented electrically short dipoles, plus three identical but orthogonally oriented magnetically small loops $-$ all spatially collocated in a point-like geometry. This electromagnetic vector-sensor is to distinctly measures all three Cartesian components of the incident electric field and all three Cartesian components of the incident magnetic field - as a $6\times 1$ vector at any one time instant. This would require exceptionally effective isolation of each of the six electromagnetic components from the other five. Mutual coupling could be largely avoided only at considerable hardware cost; and even then, the isolation cannot be perfect. This paper proposes a new direction-finding and polarization estimation algorithm that retains much of the advantages offered by the six-component electromagnetic vector-sensor, but avoiding much of this mutual coupling problem. The advantages of the six-component electromagnetic vector-sensor are numerous: (1) The polarization diversity among the vector sensor's component antennas allows that incident sources to be separated on account of their polarization differences in addition to their azimuth/elevation angular differences. (2) The spatial collocation of all component antennas in the vector sensor means no spatial phase delay in the vector sensor steering vector; hence, near-field sources may be located by an individual vector sensor as well as far-field sources. (3) In a multi-source scenario, each source's three Cartesian direction cosine estimates (and thus each source's azimuth angle estimate and the elevation angle estimate) are automatically paired without further post-processing. Theoretical performance bounds for direction finding using the collocated six-component vector sensors have been defined and derived in [4,5]. A variety of eigenstructure-based direction finding, polarization estimation and tracking schemes [6- 121,14-18] have deployed these collocated six-component vector sensors in diverse array configurations for various signal scenarios using the vector cross-product DOA estimator. This vector cross-product DOA estimator exploits all six Cartesian components of the incident electromagnetic field to estimate the $kth$ source's amplitude-normalized Poynting vector $P_{k}$, which, in turn, gives estimates of the source's elevation angle $\theta _{k}$ (measured from the positive z-axis) and the azimuth angle $\phi _{k}$ (measured from the positive x-axis): Failed to parse (unknown function "\begin{eqnarray}"): {\displaystyle \begin{eqnarray} {\bf p}_k \stackrel{\rm def}{=} \left[\begin{array}{c} p_{x_k} \\ p_{y_k} \\ p_{z_k} \end{array}\right] = \frac{{\bf e}_{k} \times {\bf h}_{k}^{*}}{\left\| {\bf e}_{k} \right\| \hspace{0.2in} \left\| {\bf h}_{k}^{*} \right\|} \stackrel{\rm def}{=} \left[\begin{array}{c} u_k \\ v_k \\ w_k \end{array}\right] = \left[\begin{array}{l} \sin\theta_k \hspace{0.03in} \cos\phi_k \\ \sin\theta_k \hspace{0.03in} \sin\phi_k \\ \cos\theta_k \end{array}\right] \end{eqnarray}} where Failed to parse (unknown function "\hspace"): {\displaystyle \hspace{0.1in}^{*}} denotes complex conjugation, ${\bf {e}}$ and ${\bf {h}}$ respectively denote the source's electric-field vector and magnetic-field vector, $u,v$ and $w$ respectively represent the source's direction-cosines along the $x$-axis, the $y$-axis and the $z$-axis. $\times$ denotes the vector cross product. This vector cross-product DOA estimation approach complements the customary interferometry direction finding approach, which estimates the spatial phase delay among the data sets collected at physically displaced antennas.