On Random Fields Associated with Analytic Wavelet Transform

TL;DR

This paper provides a statistical analysis of the analytic wavelet transform's magnitude and phase as random fields in noisy environments, deriving distributions, inequalities, and bounds relevant for signal processing applications.

Contribution

It offers the first systematic statistical characterization of AWT magnitude and phase as random fields, including distributional results and bounds under noise.

Findings

Derived marginal and joint distributions of AWT magnitude and phase

Established concentration inequalities based on SNR

Provided bounds on ridge detection probability

Abstract

Despite the broad application of the analytic wavelet transform (AWT), a systematic statistical characterization of its magnitude and phase as inhomogeneous random fields on the time-frequency domain when the input is a random process remains underexplored. In this work, we study the magnitude and phase of the AWT as random fields on the time-frequency domain when the observed signal is a deterministic function plus additive stationary Gaussian noise. We derive their marginal and joint distributions, establish concentration inequalities that depend on the signal-to-noise ratio (SNR), and analyze their covariance structures. Based on these results, we derive an upper bound on the probability of incorrectly identifying the time-scale ridge of the clean signal, explore the regularity of scalogram contours, and study the relationship between AWT magnitude and phase. Our findings lay the groundwork for developing rigorous AWT-based algorithms in noisy environments.