Time–frequency analysis of bivariate signals

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.

Symmetry breaking of azimuthal waves: Slow-flow dynamics on the Bloch sphere

“…Then, we inserted a quaternion-based ansatz for the acoustic field in order to find the systems of differential equations describing the dynamics of the slow-flow variables (amplitude, nature angle, preferential direction and temporal phase drift of the azimuthal thermoacoustic modes) for different scenarios: i) thermoacoustic feedback with or without time delay, ii) in presence or not of stochastic forcing by turbulence, iii) with uniform or non-uniform thermoacoustic distribution along the annulus circumference, iv) in presence or not of a mean swirl in the annulus. This quaternion-based ansatz was initially proposed for bi-variate time-series analysis and applied for seismic data processing [42], and subsequently taken up for analysing acoustic pressure data recorded in annular combustors [35]. Classical deterministic and stochastic methods of slow-flow averaging were adapted to this quaternion formalism to obtain nonlinear dynamical systems describing the dynamics of azimuthal modes over time scales that are large compared to the acoustic period.…”

“…with A ∈ R the modulus and (θ, χ, φ) ∈]−π, π]×[−π/4, π/4]×]−π, π] the phase triplet of h (see [42]).…”

A Complete Framework for Linear Filtering of Bivariate Signals

“…We briefly survey the Quaternion Fourier Transform (QFT) first introduced in [26] and further studied in [17]. Recent works [17], [18] have demonstrated the relevance of this QFT to process bivariate signals. In particular the QFT decomposes directly bivariate signals into a sum of polarized monochromatic signals.…”

“…This approach can be extended to wideband signals using a a polarization spectrogram based on a short-time QFT. See [17] for details. For finite energy signals a generalized Parseval-Plancherel theorem gives yields two invariants:…”

“…We have recently introduced a powerful alternative approach to bivariate signal processing [17], [18] using a tailored quaternion Fourier transform (QFT). The proposed framework exhibits an unifying structure by directly connecting usual physical quantities from polarization to well-defined mathematical (quaternion-valued) quantities such as spectral densities, covariances, time-frequency representations, etc.…”