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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Veröffentlicht in:	IEEE transactions on signal processing Jg. 57; H. 1; S. 146 - 160
Hauptverfasser:	Lilly, J.M., Olhede, S.C.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York, NY IEEE 01.01.2009 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:	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 Performance evaluation Time-frequency analysis Modulated signal wavelet transform Mapping Hilbert transformation ridge analysis Wavelets Hilbert transform Time domain method Asymmetry Wavelet transformation Signal processing Instantaneous frequency Localization
ISSN:	1053-587X, 1941-0476
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Abstract	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.
AbstractList	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 "Airy wavelets" 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. 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. 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.
Author	Olhede, S.C. Lilly, J.M.
Author_xml	– sequence: 1 givenname: J.M. surname: Lilly fullname: Lilly, J.M. organization: Earth & Space Res., Seattle, WA – sequence: 2 givenname: S.C. surname: Olhede fullname: Olhede, S.C. organization: Dept. of Stat. Sci., Univ. Coll. London, London
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Cites_doi	10.1098/rspa.2002.1049 10.1007/s002020100104 10.1088/0266-5611/4/3/009 10.1109/5.135376 10.1109/78.611186 10.1109/18.119728 10.1109/78.558469 10.1109/TBME.2002.807322 10.1016/j.pocean.2003.08.013 10.1109/MSP.2005.1550194 10.1007/978-1-4612-4142-3 10.5194/npg-13-467-2006 10.1137/1022086 10.1109/TBME.2006.889185 10.1109/TSP.2002.804066 10.1098/rspa.2002.1050 10.1007/978-1-4471-0341-7 10.1006/acha.1998.0255 10.1109/TIT.2004.842706 10.1109/78.502323
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Keywords	Performance evaluation Time-frequency analysis Modulated signal wavelet transform Mapping Hilbert transformation ridge analysis Wavelets Hilbert transform Time domain method Asymmetry Wavelet transformation Signal processing Instantaneous frequency Localization
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Snippet	The influence of higher-order wavelet properties on the analytic wavelet transform behavior is investigated, and wavelet functions offering advantageous... 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...
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SubjectTerms	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
Title	Higher-Order Properties of Analytic Wavelets
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