Time-frequency analysis of bivariate signals

TL;DR

This paper introduces a quaternion-based time-frequency analysis method for bivariate signals, capturing their geometrical and polarization properties more effectively than traditional separate analyses.

Contribution

It develops a quaternion Fourier transform framework for bivariate signals, enabling joint time-frequency and polarization analysis with proven theorems and practical implementation.

Findings

Provides meaningful time-frequency and polarization representations

Ensures conservation laws and reconstruction formulas

Demonstrates effectiveness with synthetic and real data

Abstract

Many phenomena are described by bivariate signals or bidimensional vectors in applications ranging from radar to EEG, optics and oceanography. The time-frequency analysis of bivariate signals is usually carried out by analyzing two separate quantities, e.g. rotary components. We show that an adequate quaternion Fourier transform permits to build relevant time-frequency representations of bivariate signals that naturally identify geometrical or polarization properties. First, the quaternion embedding of bivariate signals is introduced, similar to the usual analytic signal of real signals. Then two fundamental theorems ensure that a quaternion short term Fourier transform and a quaternion continuous wavelet transform are well defined and obey desirable properties such as conservation laws and reconstruction formulas. The resulting spectrograms and scalograms provide meaningful representations of both the time-frequency and geometrical/polarization content of the signal. Moreover the numerical implementation remains simply based on the use of FFT. A toolbox is available for reproducibility. Synthetic and real-world examples illustrate the relevance and efficiency of the proposed approach.