Complex representation of a polarized signal and its application to the analysis of ULF waves

We define the complex signal s = x + jy, associated with a real signal having two orthogonal components x and y. This signal follows simple transformation rules when the measuring coordinate system is changed. Then we divide it into two other complex signals s+ and s− of opposite polarities, and we study the correspondence which exists between s+ and s− and the concept of an analytical signal. We demonstrate that the polarization parameters of the original signal are given by very simple expressions in terms of s+ or s− or in terms of their Fourier transforms S+ and S−. When the Fourier analysis is made on S+ and S−, the ellipticity, the sense of rotation, and the orientation of the major axis of the polarization ellipse are easily displayed as a function of frequency. The results are valid whatever the coordinate system in which the signal has been measured, even when this system is rotating with respect to a fixed frame of reference (as is often the case for space measurements). In the case of a three-dimensional signal an instantaneous polarization vector is defined, whose components are Pk = |SιJ+|² - |SιJ−|². When it is integrated over time or frequency, this definition conserves the power of the signal and does present the correct relationships with the imaginary part of the coherency matrix. Applications of this formalism to the polarization analysis of artificial and natural signals are presented. Both numerical and analog procedures for obtaining the two circular components of the polarized power are described. When it is applied to the study of ULF waves, this method demonstrates the complexity of the polarization pattern of Pc 1 events.