Higher-Order Properties of Analytic Wavelets

• 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
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2008.2007607

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.