On the Analytic Wavelet Transform

TL;DR

This paper derives an exact expression for the analytic wavelet transform of real signals, accounting for amplitude and frequency modulation, and provides a framework to optimize wavelet properties for accurate signal analysis.

Contribution

It introduces a general, exact formulation of the analytic wavelet transform that incorporates local modulation effects and guides wavelet selection for improved signal estimation.

Findings

Derived closed-form expressions for modulation functions using Bell polynomials.

Identified conditions for matching wavelet properties to signal variability.

Provided a method to quantify and minimize bias in wavelet ridge analysis.

Abstract

An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed, resolving the time-dependent effects of non-negligible amplitude and frequency modulation. The analytic signal is first locally represented as a modulated oscillation, demodulated by its own instantaneous frequency, and then Taylor-expanded at each point in time. The terms in this expansion, called the instantaneous modulation functions, are time-varying functions which quantify, at increasingly higher orders, the local departures of the signal from a uniform sinusoidal oscillation. Closed-form expressions for these functions are found in terms of Bell polynomials and derivatives of the signal's instantaneous frequency and bandwidth. The analytic wavelet transform is shown to depend upon the interaction between the signal's instantaneous modulation functions and frequency-domain derivatives of the wavelet, inducing a hierarchy of departures of the transform away from a perfect representation of the signal. The form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal, in which case our expressions simplify considerably. One may then quantify the time-varying bias associated with signal estimation via wavelet ridge analysis, and choose wavelets to minimize this bias.