On Random Fields Associated with Analytic Wavelet Transform

Gi-Ren Liu; Yuan-Chung Sheu; Hau-Tieng Wu

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arXiv · 2508.10495

On Random Fields Associated with Analytic Wavelet Transform

Gi-Ren Liu·Yuan-Chung Sheu·Hau-Tieng Wu

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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.

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Gi-Ren Liu, Yuan-Chung Sheu, Hau-Tieng Wu. 2025-08-14. On Random Fields Associated with Analytic Wavelet Transform. https://arxiv.org/abs/2508.10495

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