On Random Fields Associated with Analytic Wavelet Transform

TL;DR

This work provides a rigorous statistical framework for analyzing the analytic wavelet transform (AWT) of signals obscured by Gaussian noise, treating the AWT magnitude and phase as interacting random fields on the time-scale domain. It derives marginal and joint distributions, concentration inequalities, and covariance structures for both magnitude and phase, including null and non-null (signal-present) settings, and establishes a robust differentiation and ridge-contour theory for noisy scalograms. A key contribution is a Gaussian approximation result for discretized AWT under non-Gaussian, weakly dependent noise, enabling practical inference from sampled data. The results underpin reliable AWT-based ridge detection, contour regularity, and phase-amplitude considerations, and lay groundwork for algorithmic development in noisy environments. Collectively, the paper offers a comprehensive probabilistic foundation for AWT-based signal analysis with rigorous convergence and approximation guarantees.

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.

Figures (8)

• Figure 1: Flowchart of the structure of this paper. j.p.d.f: joint probability density function.
• Figure 2: Time-scale representations of the clean signal $f$ (left column) and the noisy signal $Y$ (right column). The clean signal $f$, shown in the top-left panel, is a frequency-modulated cosine function. The noisy signal $Y$, shown in the top-right panel, is generated by adding a sample path of Gaussian noise to $f$. The second row displays the magnitudes of the AWT of $f$ and $Y$, i.e., the scalograms $|W_{f}|$ and $|W_{Y}|$. Here, the signals $f$ and $Y$ are sampled at 200 Hz, and the mother wavelet $\psi$ has center frequency $\omega_{\psi}= 80$ Hz, as defined in Assumption \ref{['assump:analytic']}. The third row shows the level curves of the corresponding scalograms. The fourth row presents the fields $\cos(\arg(W_{f}))$ and $\cos(\arg(W_{Y}))$, where $\arg(W_{f})$ and $\arg(W_{Y})$ denote the phases of the AWT of $f$ and $Y$, respectively. The fifth row displays $|W_{f}|\cos(\arg(W_{f}))$ and $|W_{Y}|\cos(\arg(W_{Y}))$, i.e., the multiplication of the amplitude and the cosine of the phase fields.
• Figure 3: (a) Magnitude of $W_{f}(t,s)$ as a function of the scale variable, where $f(t) = \cos(2\pi \phi(t))$ with $\phi(t)=\frac{1}{2}t^{2}$ (i.e., $\phi'(t)=t$). The signal $f$ is sampled at 200 Hz and the AWT of $f$ is evaluated at 5 seconds. Here, we use a mother wavelet $\psi$ with center frequency $\omega_{\psi}=80$ Hz. (b) Relationship between the scale variable and frequency. The peak in (a) occurs approximately at scale $s=40$, corresponding to a frequency of $5$ Hz. (c) Polar plot of the phase of $W_{f}(t,s)$ as a function of the scale variable, i.e., a scatter plot of $(s\cos(\Theta_{Y}(5,s)),s\sin(\Theta_{Y}(5,s)))$ for various $s$. The color gradient indicates the variation in the magnitude $|W_{f}(5,s)|$, normalized to range from zero to one. (d) Polar plot of the peak locations of the histogram of the random variable $\Theta_{Y}(5,s)$. Here, the color is determined by the normalized magnitude $|W_{f}(5,s)|$. The histogram of $\Theta_{Y}(5,s)$ is generated using $10^4$ sample paths of the random process $\Phi$.
• Figure 4: Sample paths of $(s\cos(\Theta_{Y}(t,s)),s\sin(\Theta_{Y}(t,s)))$ for $t=5$, where $Y=f+\Phi$ and $f(t) = \cos(2\pi \phi(t))$ with $\phi(t)=\frac{1}{2}t^{2}$. The color gradient in each subfigure represents the variation in the corresponding sample path of $|W_{Y}(5,s)|$, normalized to range from zero to one.
• Figure 5: Graphs of the hypergeometric-related functions in (\ref{['eq:correlation_mag']})

Theorems & Definitions (22)

• Lemma 1
• Remark 1
• Definition 1
• Proposition 1
• Corollary 1
• Remark 2
• Proposition 2
• Proposition 3
• Proposition 4
• Example 1