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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.10495 |
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| _version_ | 1866912778349969408 |
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| author | Liu, Gi-Ren Sheu, Yuan-Chung Wu, Hau-Tieng |
| author_facet | Liu, Gi-Ren Sheu, Yuan-Chung Wu, Hau-Tieng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10495 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Random Fields Associated with Analytic Wavelet Transform Liu, Gi-Ren Sheu, Yuan-Chung Wu, Hau-Tieng Statistics Theory Probability 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. |
| title | On Random Fields Associated with Analytic Wavelet Transform |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2508.10495 |