On Random Fields Associated with Analytic Wavelet Transform
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.
Cite
@article{arxiv.2508.10495,
title = {On Random Fields Associated with Analytic Wavelet Transform},
author = {Gi-Ren Liu and Yuan-Chung Sheu and Hau-Tieng Wu},
journal= {arXiv preprint arXiv:2508.10495},
year = {2025}
}