Modulated Oscillations in Three Dimensions

The analysis of the fully three-dimensional and time-varying polarization characteristics of a modulated trivariate, or three-component, oscillation is addressed. The use of the analytic operator enables the instantaneous three-dimensional polarization state of any square-integrable trivariate signa...

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Bibliographic Details
Published in:IEEE transactions on signal processing 2011-12, Vol.59 (12), p.5930-5943
Main Author: Lilly, J. M.
Format: Article
Language:English
Subjects:
Applied sciences
Bandwidth
Detection, estimation, filtering, equalization, prediction
Ellipses
Exact sciences and technology
Frequency modulation
Information, signal and communications theory
Instantaneous bandwidth
instantaneous frequency
Mathematical analysis
Modulation, demodulation
nonstationary signal analysis
Oscillations
Oscillators
Planes
Polarization
Polarization characteristics
Seismic phenomena
Signal and communications theory
Signal representation. Spectral analysis
Signal, noise
Studies
Telecommunications and information theory
Three dimensional
three-component signal
Time frequency analysis
trivariate signal
Wavelet transforms
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Summary:The analysis of the fully three-dimensional and time-varying polarization characteristics of a modulated trivariate, or three-component, oscillation is addressed. The use of the analytic operator enables the instantaneous three-dimensional polarization state of any square-integrable trivariate signal to be uniquely defined. Straightforward expressions are given which permit the ellipse parameters to be recovered from data. The notions of instantaneous frequency and instantaneous bandwidth, generalized to the trivariate case, are related to variations in the ellipse properties. Rates of change of the ellipse parameters are found to be intimately linked to the first few moments of the signal's spectrum, averaged over the three signal components. In particular, the trivariate instantaneous bandwidth-a measure of the instantaneous departure of the signal from a single pure sinusoidal oscillation-is found to contain five contributions: three essentially two-dimensional effects due to the motion of the ellipse within a fixed plane, and two effects due to the motion of the plane containing the ellipse. The resulting analysis method is an informative means of describing nonstationary trivariate signals, as is illustrated with an application to a seismic record.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2011.2164914