Application of the quaternion algebra in the transmission of polarization-manipulated radio signals

The potential possibilities of using the elliptically polarized electromagnetic wave as an information carrier are constantly in the focus of attention of engineers developing digital data transmission systems. The logic of the operation of individual blocks of the data transmission system is determined by the use of a particular mathematical apparatus. In this paper, it was proposed to consider the transmission of modulation symbols as the rotation of the system of spheres, the radii of which are determined by the amplitudes of the elliptically polarized electromagnetic wave. The rotation of the spheres is considered from the point of view of the application of quaternion transformations. A quadrature amplitude-polarization modulator as part of a digital data transmission system with specific values of amplitude A, initial phase φ, ellipticity angle α and orientation angle of the polarization ellipse β of the elliptically polarized electromagnetic wave of the transmitted modulation symbol provides a transition from the previous modulation symbol to the next one with simultaneous calculation of the values of the in-phase and quadrature components. The paper presents an analytical solution to the quaternion equation for finding the angle of rotation and coordinates of the point through which the axis of rotation of the spheres passes during the transmission of digital data. The analytical calculations were supported by the construction of a mathematical model of binary sequence transmission. The results of mathematical modeling show that the initial sequence of binary numbers intended for transmission can be written to the modulator's memory in the form of an initial number and a transition matrix composed of normalized quaternions q. The practical significance of this work lies in the fact that the obtained mathematical formulas of quaternion transformations can be used as the basis for the program code in the development of the quadrature amplitude-polarization modulator.