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

Abstract

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.

Resonant absorption of ULF waves near the ion cyclotron frequency: A simulation study

“…From the real‐time histories of E x and E y , the two circularly‐polarized components at each point can be extracted, which was discussed by Kodera et al [1977]. The right‐( E R ) and the left‐handed( E L ) circularly‐polarized electric fields are defined as where the asterisk denotes the complex conjugate.…”

Location of Pc 1–2 waves relative to the magnetopause

“…Since the full vector is measured, the magnetic field can be broken into right-hand, left-hand, and compressional components by transforming to mean field coordinates (with z along the mean field), Fouriertransforming, and recombining the complex Fourier transforms of the x and y components into B L,R = B x ± iB y before computing the power spectrum. This procedure is outlined by Kodera et al (1977) and used in many subsequent papers (e.g. LaBelle and Treumann, 1992, who apply the technique to AMPTE/IRM data).…”

Two‐dimensional hybrid code simulation of electromagnetic ion cyclotron waves of multi‐ion plasmas in a dipole magnetic field

“…As in H&D, we use the method by Kodera et al [1977] to apply a Discrete Fourier Transform (DFT) on the complex signal C ( q , r , t n ) = δB r ( q , r , t n ) + iδB s ( q , r , t n ), where t n is the sampling time, and δB r ( δB s ) is the r ( s ) component of the fluctuating magnetic field. The power is defined as where ω k is the discrete Fourier frequency, ( q , r , ω k ) is the DFT of C ( q , r , t n ), *( q , r , ω k ) is the complex conjugate, and we use + (‐) to indicate power for which ω k > 0 ( ω k < 0).…”