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On harmonic analysis of vector-valued signals

Sangwine, Stephen J (2017) 'On harmonic analysis of vector-valued signals.' Mathematical Methods in the Applied Sciences, 40 (1). 22 - 30. ISSN 0170-4214

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Abstract

A vector‐valued signal in N dimensions is a signal whose value at any time instant is an N‐dimensional vector, that is, an element of urn:x-wiley:mma:media:mma3938:mma3938-math-0001. The sum of an arbitrary number of such signals of the same frequency is shown to trace an ellipse in N‐dimensional space, that is, to be confined to a plane. The parameters of the ellipse (major and minor axes, represented by N‐dimensional vectors; and phase) are obtained algebraically in terms of the directions of oscillation of the constituent signals, and their phases. It is shown that the major axis of the ellipse can always be determined algebraically. That is, a vector, whose value can be computed algebraically (without decisions or comparisons of magnitude) from parameters of the constituent signals, always represents the major axis of the ellipse. The ramifications of this result for the processing and Fourier analysis of signals with vector values or samples are discussed, with reference to the definition of Fourier transforms, particularly discrete Fourier transforms, such as have been defined in several hypercomplex algebras, including Clifford algebras. The treatment in the paper, however, is entirely based on signals with values in urn:x-wiley:mma:media:mma3938:mma3938-math-0002. Although the paper is written in terms of vector signals (which are taken to include images and volumetric images), the analysis clearly also applies to a superposition of simple harmonic motions in N dimensions.

Item Type: Article
Uncontrolled Keywords: vector signals, Fourier analysis, harmonic analysis
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science and Health > Computer Science and Electronic Engineering, School of
Depositing User: Jim Jamieson
Date Deposited: 23 Jun 2016 13:24
Last Modified: 21 Sep 2018 16:15
URI: http://repository.essex.ac.uk/id/eprint/16954

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