Higher-Order Properties of Analytic Wavelets

The influence of higher-order wavelet properties on the analytic wavelet transform behavior is investigated, and wavelet functions offering advantageous performance are identified. This is accomplished through detailed investigation of the generalized Morse wavelets, a two-parameter family of exactl...

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Bibliographic Details
Published in:IEEE transactions on signal processing 2009-01, Vol.57 (1), p.146-160
Main Authors: Lilly, J.M., Olhede, S.C.
Format: Article
Language:English
Subjects:
Amplitude estimation
Amplitude modulation
Applied sciences
Asymmetry
Behavior
Continuous wavelet transforms
Earth
Exact sciences and technology
Frequency estimation
Hilbert transform
Information, signal and communications theory
instantaneous frequency
Localization
Mathematical analysis
Miscellaneous
Morlet wavelet
Position (location)
ridge analysis
Signal analysis
Signal processing
Studies
Telecommunications and information theory
time-frequency analysis
Wavelet
Wavelet analysis
Wavelet domain
wavelet transform
Wavelet transforms
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Summary:The influence of higher-order wavelet properties on the analytic wavelet transform behavior is investigated, and wavelet functions offering advantageous performance are identified. This is accomplished through detailed investigation of the generalized Morse wavelets, a two-parameter family of exactly analytic continuous wavelets. The degree of time/frequency localization, the existence of a mapping between scale and frequency, and the bias involved in estimating properties of modulated oscillatory signals, are proposed as important considerations. Wavelet behavior is found to be strongly impacted by the degree of asymmetry of the wavelet in both the frequency and the time domain, as quantified by the third central moments. A particular subset of the generalized Morse wavelets, recognized as deriving from an inhomogeneous Airy function, emerge as having particularly desirable properties. These ldquoAiry waveletsrdquo substantially outperform the only approximately analytic Morlet wavelets for high time localization. Special cases of the generalized Morse wavelets are examined, revealing a broad range of behaviors which can be matched to the characteristics of a signal.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2008.2007607