Time-frequency-polarization Analysis of Bivariate Signals

Bivariate signals arise in many fields such as oceanography, seismology or radar. A bivariate signal (e.g. the electromagnetic wave field) can be either described by a time-evolving real vector [u(t), v(t)] or equivalently by the complex signal x(t) = u(t)+ iv(t), where u(t), v(t) are its components. In many applications, extracting physically interpretable parameters (e.g. polarization properties) from bivariate signals is of interest. This Ph.D. work aims at providing novel time-frequency methods to process bivariate signals in the most general case, that is possibly multicomponent and nonstationary. Many authors have used augmented representations to extract geometric or polarization properties for nonstationary bivariate signals, see [2–5] and references therein. A well-known method is the rotary spectrum analysis, which decomposes a bivariate signal at each frequency into clockwise and counter-clockwise rotating components. Existing methods have in common that they rely on the use of the classical complex Fourier Transform (FT). However the classical FT of complex signals has no Hermitian symmetry, which prevents from using directly standard timefrequency tools such as the analytic signal. We show that using the Quaternion Fourier Transform (QFT), an alternate definition of the FT, makes it possible to extend standard time-frequency representations to process bivariate signals. Fundamental theorems are derived, illustrating the relevance of the approach. It is algebraic and takes geometrical properties into account by using quaternionic algebra.