Modulated oscillations in many dimensions

• Published in: Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences physical, and engineering sciences, 2013-02, Vol.371 (1984), p.1-21
• Main Author: Olhede, S. C.
• Format: Article
• Language: English
• Subjects: Amplitude modulation; Dimensionality; Eigenvalues; Ellipses; Fourier transformations; Geometry; Modulated signal processing; Signal bandwidth; Time series; Trajectories
• ISSN: 1364-503X

Modulated oscillations are described via their timevarying amplitude and frequency. For multivariate signals, there is structure in the signal beyond this local amplitude and frequency defined for each signal component, in turn describing the commonality of the components. The multivariate structure encodes how the common oscillation is present in each component signal. This structure will also be evolving. I review the special case of the representation of both bivariate and trivariate oscillations. Additionally, existing results on the general multivariate oscillation are covered. I discuss the difference between a model of a multivariate oscillation compared with other common signal models of phenomena observed in several channels, and how their properties are different. I show how for the multivariate signal the global dimensionality of the signal is built up from local one-dimensional contributions, and introduce the purely unidirectional signal, to quantify how any given signal is different from the closest such signal. I illustrate the properties of the derived representation of the multivariate signal with synthetic examples, and discuss the representation of data from observations in physical oceanography.