Analyzing Scalogram Ridges in the Presence of Noise

TL;DR

This work develops a rigorous probabilistic framework for ridges in scalograms under Gaussian noise by modeling the observed signal as $Y=f+\Phi$, with $f$ following an adaptive harmonic model. It defines ridge curves as set-valued random processes based on the scalogram, proves singleton and upper-hemicontinuity properties, and derives exponential tail bounds on ridge deviations from the clean case using new maximal inequalities for complex-valued AWT, extending Borell-TIS and Dudley-type results to the complex domain. The theory is complemented by numerical simulations illustrating ridge behavior and the impact of SNR, and it discusses implications for ridge extraction algorithms and potential extensions to nonstationary noise and change points. Overall, the paper provides a solid theoretical foundation for the stability and continuity of scalogram ridges in noisy nonstationary signals, with practical guidance for robust ridge-based IF estimation. $s_Y(t)=\arg\max_{s>0} S_Y(t,s)$ and $S_Y(t,s)=|W_Y(t,s)|^2$ are central objects, and the results connect spectral energy, noise structure, and ridge geometry in a unified framework.$

Abstract

While ridges in the scalogram, determined by the squared modulus of analytic wavelet transform (AWT), is a widely accepted concept and utilized in nonstationary time series analysis, their behavior in noisy environments remains underexplored. Our object is to provide a theoretical foundation for scalogram ridges by defining ridges as a potentially set-valued random process connecting local maxima of the scalogram along the scale axis and analyzing their properties when the signal fulfills the adaptive harmonic model and is contaminated by stationary Gaussian noise. In addition to establishing several key properties of the AWT for random processes, we investigate the probabilistic characteristics of the resulting random ridge points in the scalogram. Specifically, we establish the uniqueness property of the ridge point at individual time instances and prove the upper hemicontinuity of the ridge random process. Furthermore, we derive bounds on the probability that the deviation between the ridges of noisy and clean signals exceeds a specified threshold, and these bounds depend on the signal-to-noise ratio. To achieve these ridge deviation results, we derive maximal inequalities for the complex modulus of nonstationary Gaussian processes, leveraging classical tools such as the Borell-TIS inequality and Dudley's theorem, which might be of independent interest.

Figures (10)

• Figure 1: The top section illustrates a signal $f$, composed of two frequency-modulated sinusoids, along with its 3-dimensional and image representations in the time-scale domain. Here, the mother wavelet used has center frequency $\omega_{\psi}= 80$ Hz, as defined in (\ref{['def:centralfrequency_psi']}). Hence, the signal $f$'s IF lies approximately within the interval $[\frac{\omega_{\psi}}{58},\frac{\omega_{\psi}}{10}]$ Hz. The bottom section presents the same for its noise-affected counterpart $Y$, with a signal-to-noise ratio of -8.98 dB. Here, $Y$ is the sum of $f$ and a sample path of the Gaussian process $\Phi$. For the single-component case, see Figure \ref{['fig:wavelet_potential_ridge_Surface plot:one']} in the appendix.
• Figure 2: Example of the complex modulus of the AWT for a 30-second noise-contaminated signal composed of two frequency-modulated sinusoids. The first row displays the frequency-modulated signal $f$, while the second row shows the noise, a sample from the stationary Gaussian process $\Phi$, identical to that in Figure \ref{['fig:wavelet_potential_ridge_Surface plot']}. The third row illustrates the complex modulus of the AWT, i.e., the spectral amplitude, of the noise-contaminated signal $Y$, where $Y=f+\Phi$. In the last three columns, the black curves represent the spectral amplitude of $Y$ at specific times, corresponding to cross-sections of the image in the third row. For comparison, the spectral amplitudes of $f$ are shown in gray. For the single-component case, see Figure \ref{['fig:wavelet_ridge:one']} in the appendix.
• Figure 3: Histogram of the random variable $\frac{\partial^{2}S_{Y}}{\partial s^{2}}\left(t, s_{Y}(t)\right)$, where $Y(t) = f(t)+\Phi(t)$ and $\Phi$ is a stationary Gaussian process. The time variable is fixed throughout the experiment, and the histogram is based on $10^5$ realizations of $\Phi$.
• Figure 4: Example of ridge points (local maxima) and a ridge curve $s_{Y}$, where $Y = f+\Phi$, and $s_{f}$. The signal $f$ and the realization of $\Phi$ are identical to those in Figure \ref{['fig:wavelet_potential_ridge_Surface plot']}. The blue boxes in the second row highlight the discontinuities in the ridge curve $s_{Y}$ caused by the presence of noise. For the single-component case, refer to Figure \ref{['fig:wavelet_potential_ridge:one']} in the appendix.
• Figure 5: The relationship between the scale region $B_{m}(t)$ and the interval $[\underline{\mathfrak{i}}_{m}(t),\overline{\mathfrak{i}}_{m}(t)]$ considered in Theorem \ref{['mainresult:deviation']}.

Theorems & Definitions (22)

• Theorem 1
• Example 1
• Proposition 1
• Example 2
• Corollary 1
• Definition 1
• Remark 1
• Definition 2
• Theorem 2
• Theorem 3