Uncertainty quantification framework for wavelet transformation of noise-contaminated signals

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Wavelet Transform (WT) is one of the most important signal processing methods that is being widely used in structural health monitoring (SHM). The primary advantage of WT over other signal processing techniques is the ability to extract and analyze information in both time and frequency domain with adaptive resolutions. However, in most of the applications, measurements are highly contaminated with noise from numerous sources, and the noise contamination will accumulate and propagate through the wavelet transformation. The WT outcome, referred to as wavelet coefficients, will be degraded and the features extracted from the contaminated coefficients will cause misinterpretation and false decisions. In this paper, probabilistic uncertainty quantification (UQ) for wavelet coefficients is investigated to facilitate more meaningful data processing for reliable decision-making. The proposed approach is able to estimate the variance of the wavelet coefficients at different scales and shift values which leads to an estimated probability density function (PDF) that can be used later to improve decision making procedure. A probabilistic analytical framework for quantifying the uncertainty of the wavelet transform estimation is proposed in this context, and the results are validated using the Monte Carlo Simulation (MCS) on simulated signals and systems as well as experimental signals recorded from a vibration structure. The adopted framework is generic to all kinds of WT with different mother wavelets.