Higher-Order Properties of Analytic Wavelets

TL;DR

This paper investigates how higher-order properties of analytic wavelets affect their performance, identifying wavelet functions like Airy wavelets that offer superior time localization and signal analysis capabilities.

Contribution

It provides a detailed analysis of generalized Morse wavelets, highlighting the impact of asymmetry and proposing wavelets with optimized properties for signal analysis.

Findings

Airy wavelets outperform Morlet wavelets in high time localization

Wavelet asymmetry influences localization and bias

Generalized Morse wavelets can be tailored to signal characteristics

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 "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.