۱- Introduction

۲- Multivariate time-frequency analysis

۳- Multicomponent signals

۴- Inversion and signal decomposition

۵- Decomposition algorithm

۶- Numerical examples

۷- Conclusion

References

Abstract

A solution of the notoriously difficult problem of characterization and decomposition of multicomponent multivariate signals which partially overlap in the joint time-frequency domain is presented. This is achieved based on the eigenvectors of the signal autocorrelation matrix. The analysis shows that the multivariate signal components can be obtained as linear combinations of the eigenvectors that minimize the concentration measure in the time-frequency domain. A gradient-based iterative algorithm is used in the minimization process and for rigor, a particular emphasis is given to dealing with local minima associated with the gradient descent approach. Simulation results over illustrative case studies validate the proposed algorithm in the decomposition of multicomponent multivariate signals which overlap in the time-frequency domain.

Introduction

Signals with time-varying spectral content are not easily characterized by the conventional Fourier analysis. They are commonly studied within the time-frequency (TF) analysis [1]–[۸]. Research in this field has resulted in numerous representations and algorithms which have been almost invariably introduced for the processing of univariate signals, with most frequent characterization through amplitude and frequency-modulated oscillations [6], [9]. Recently, the progress in sensing technology for multidimensional signals has been followed by a growing interest in time-frequency analysis of such multichannel (multivariate and/or multidimensional) data. Namely, developments in sensor technology have made accessible multivariate data. Indeed, the newly introduced concept of modulated bivariate and trivariate data oscillations (3D inertial body sensor, 3D anemometers [9]) and the generalization of this concept to an arbitrary number of channels have opened the way to exploit multichannel signal interdependence in the joint time-frequency analysis [10]– [۱۲]. The concept of multivariate modulated oscillations has been proposed in [10], under the restricting assumption that one common oscillation fits best all individual channel oscillations. In other words, a joint instantaneous frequency (IF) aims to characterize multichannel data by capturing the combined frequency of all individual channels. It is defined as a weighted average of the IFs in all individual channels. The deviation of multivariate oscillations in each channel from the joint IF is characterized by the joint instantaneous bandwidth. With the aim to estimate the joint IF of multichannel signals, the synchrosqueezed transform, a highly concentrated time-frequency representation (TFR) belonging to the class of reassigned TF techniques, has been recently extended to the multivariate model [9]. Following the same aim of extracting the local oscillatory dynamics of a multivariate signal, the wavelet ridge algorithm has also been introduced within the multivariate framework [10]. Another very popular concept, empirical mode decomposition (EMD), has been studied for multivariate data, [18]- [22]. However, successful EMD-based multicomponent signal decomposition is possible only for signals having nonoverlapping components in the TF plane.