⚡ electrical engineering

A Classification-Based Adaptive Bayesian Wavelet Packet Denoising Method for Additive Gaussian-Impulse Mixed Noise in Industrial Sensor and Measurement Signals

This paper proposes a classification-based adaptive Bayesian wavelet packet transform (CABWPT) method that utilizes noise classification, SNR-adaptive hybrid thresholding, and Bayesian optimization to effectively denoise industrial sensor and measurement signals corrupted by non-stationary additive Gaussian-impulse mixed noise.

Original authors: hao ying, zhou zhou, yangsen wen, pengpeng he

Published 2026-09-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: hao ying, zhou zhou, yangsen wen, pengpeng he

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the noisy world of industrial machinery and human speech, signals are rarely pure. A vibration sensor on a factory floor or a microphone recording a conversation often captures a messy mixture of the desired sound and unwanted interference. This interference usually comes in two forms: a steady, humming background static and sudden, sharp spikes of static that pop up unpredictably. When engineers try to clean these signals to find hidden faults in machines or to make speech clearer, they face a difficult problem. Traditional tools for cleaning signals were built on the assumption that noise is always that steady, humming static. When the noise includes those sharp, unpredictable spikes, these old tools often fail, either leaving too much noise behind or accidentally cutting away parts of the real signal they were meant to save.

Researchers at Beibu Gulf University have developed a new method to solve this specific problem of mixed noise. Their approach, described in a recent study, treats the noise not as a single, uniform problem but as a situation that changes depending on what kind of noise is present. Instead of using a single, rigid rule to clean the signal, their system first takes a moment to listen and identify the character of the noise. It checks whether the noise is the steady kind, the spiky kind, or a combination of both. Once it knows what it is dealing with, it switches to the specific cleaning tool best suited for that type. This allows the system to be much more precise, removing the unwanted static while keeping the important details of the original sound intact, whether that sound is a machine vibration or a human voice.

The core of this new method is a process of classification and adaptation. Imagine a signal as a complex tapestry of threads. The researchers' system first looks at the highest-frequency threads, which are usually where the noise hides the most. By measuring how "spiky" and how "lopsided" the noise is, the system can sort the noise into one of four categories: steady, spiky, lopsided, or a mix. For each category, the system has a different strategy. If the noise is steady, it uses one type of mathematical filter. If the noise is spiky, it uses a different one that is better at ignoring those sudden pops. If the noise is a mix, it blends these strategies together. This prevents the system from making a common mistake where it gets confused by the spikes and ends up cutting away the real signal, or conversely, leaving the spikes untouched because it thinks they are part of the signal.

To ensure the cleaning process is as effective as possible, the researchers combined two different ways of deciding how much noise to remove. One way is very careful and looks at each small piece of the signal individually, while the other is a broad, general rule that works well for the whole signal. The new method mixes these two approaches, adjusting the balance depending on how loud the noise is compared to the signal. When the noise is very loud, the system leans more on the careful, individual approach. When the noise is quieter, it relies more on the broad rule. This balance is managed by a set of strict rules that prevent the system from being too aggressive and damaging the signal. For instance, the system constantly checks to make sure it hasn't removed too much of the original sound, and if it has, it automatically backs off.

After the initial cleaning, the researchers apply a final polishing step. Sometimes, even after the main noise is gone, there are faint echoes or distortions left behind. The system uses a technique that looks at the overall structure of the remaining signal to smooth out these imperfections, ensuring the final result is as close to the original as possible. To find the perfect settings for all these steps, the researchers used a computerized trial-and-error process that tested thousands of combinations to find the best settings for different types of signals. They tested their method on two very different types of data: vibration signals from ball bearings in a factory and recordings of human speech.

The results showed that this new method outperformed ten other existing techniques across a wide range of conditions. In tests with factory vibrations, where the noise was particularly harsh, the new method was the only one that successfully improved the signal quality at all noise levels, including the very quietest ones where other methods actually made the signal worse. In the speech tests, it also performed better than the others, preserving the clarity of the voice while removing the background noise. The study found that the most significant improvements came from the system's ability to correctly identify the noise type and its smart way of balancing different cleaning strategies. By adapting to the specific nature of the noise rather than forcing a single solution on every situation, the researchers created a robust tool that can handle the messy, unpredictable reality of industrial and acoustic environments.

Technical Summary: A Classification-Based Adaptive Bayesian Wavelet Packet Denoising Method for Additive Gaussian-Impulse Mixed Noise

Problem Statement
Industrial sensor and measurement signals, such as mechanical vibration and speech, are frequently corrupted by additive Gaussian-impulse mixed noise. While Wavelet Packet Transform (WPT) is a standard tool for signal denoising due to its multi-resolution capabilities, existing thresholding methods often fail under these non-Gaussian conditions. Classical approaches like VisuShrink and BayesShrink rely on Gaussian assumptions or universal thresholds that do not account for the heavy-tailed characteristics of impulse noise. Consequently, these methods suffer from noise estimation mismatches and threshold degradation, leading to significant performance loss, particularly when impulse components are present or at higher Signal-to-Noise Ratios (SNR) where signal leakage inflates noise estimates. Furthermore, deep learning alternatives often lack interpretability and require labeled data that is scarce in industrial settings.

Methodology
The authors propose a Classification-Based Adaptive Bayesian Wavelet Packet Transform (CABWPT) framework designed to handle additive Gaussian-impulse mixed noise through a five-step process:

Signal Type Classification and Basis Selection: The input signal is first classified as either "mechanical vibration" or "speech" using a weighted scoring classifier based on crest factor, skewness, and kurtosis. This classification dictates the selection of the optimal wavelet basis (e.g., Db20 for vibration, Sym8 for speech).
Noise Classification and Robust Estimation: The highest-frequency subband is analyzed to classify noise into four types: Gaussian, Laplacian, Impulsive, and Mixed, using joint kurtosis-skewness statistics. Each type is matched with a dedicated noise variance estimator (e.g., MAD for Gaussian, IQR for Laplacian, trimmed mean for Impulsive). Additionally, a percentile-hybrid correction mechanism is applied at high SNR levels to prevent signal leakage from inflating noise estimates.
SNR-Adaptive Hybrid Thresholding: A hybrid threshold is constructed by fusing BayesShrink and VisuShrink strategies. The fusion weights are SNR-dependent, favoring BayesShrink at low SNRs and gradually incorporating VisuShrink's robustness as SNR increases. This process is governed by a Multi-Constraint Hierarchical Adjustment (MCHA) mechanism, which enforces frequency-layer adaptation, impulse protection, and a hard constraint on the signal retention rate (set at 0.85) to prevent over-processing and signal energy loss.
Post-Processing: For mechanical signals with specific periodicity and low SNR, SVD-Hankel post-processing is applied to suppress residual spectral leakage via low-rank approximation. Strict energy normalization is then applied to ensure reconstruction fidelity, preventing energy inflation or distortion.
Hyperparameter Optimization: A Bayesian optimization framework (offline optimization, online inference) is employed to globally optimize five core hyperparameters (fusion weights, scaling factors, etc.) to maximize denoising performance across diverse scenarios.

Key Contributions

Noise Classification Framework: The introduction of a joint kurtosis-skewness classifier that enables the selection of optimal noise estimators tailored to specific noise distributions (Gaussian, Laplacian, Impulsive, Mixed), addressing the statistical mismatch in traditional methods.
SNR-Adaptive Hybrid Thresholding: A novel fusion strategy that dynamically balances BayesShrink and VisuShrink based on SNR levels, governed by the MCHA mechanism to prevent both under-thresholding (noise retention) and over-thresholding (signal loss).
High-SNR Correction: A percentile-hybrid correction mechanism specifically designed to mitigate noise estimation errors caused by signal leakage in high-SNR environments.
Comprehensive Optimization: The integration of Bayesian optimization for global hyperparameter tuning and the combination of SVD-Hankel post-processing with strict energy normalization to ensure high-fidelity reconstruction.

Experimental Results
The method was evaluated on two benchmark datasets: the CWRU bearing vibration dataset and the TIMIT speech corpus, under additive Gaussian-impulse mixed noise at SNRs ranging from 0 dB to 15 dB.

Performance on CWRU (Vibration): The proposed CABWPT method achieved the highest SNR improvement ( $\Delta$ SNR) across all tested levels. Notably, at 15 dB, where all ten competing methods (including classical wavelet and low-rank decomposition methods) produced negative $\Delta$ SNR (degrading the signal), CABWPT maintained a positive improvement of +1.23 dB. At 0 dB, it achieved +5.13 dB, outperforming the second-best method (ASVD) by 0.73 dB.
Performance on TIMIT (Speech): The method demonstrated superior performance at 0 dB (+6.65 dB) and 5 dB (+5.16 dB). At 15 dB, where noise is nearly Gaussian, the performance gap narrowed, with CABWPT achieving +1.95 dB compared to BayesShrink's +2.17 dB, a difference the authors note is within statistical variation.
Ablation Studies: Experiments confirmed that Bayesian optimization provided the most consistent performance gains. The High-SNR correction module was critical for maintaining performance at high SNR levels, while Hybrid Thresholding was essential for mid-range SNRs. SVD-Hankel post-processing showed the most significant contribution at low SNRs.
Computational Efficiency: The full method with Bayesian optimization is computationally intensive, requiring approximately 91 seconds per CWRU signal segment and 45 seconds per TIMIT segment. However, a lightweight mode without online optimization reduces runtime to ~1.2 seconds for CWRU and ~9.0 seconds for TIMIT segments, with only a moderate performance cost (0.1–1.7 dB), making the lightweight variant viable for real-time applications.

Significance and Claims
The paper claims that the CABWPT method provides a robust framework for high-fidelity signal reconstruction under realistic, non-stationary industrial noise conditions. By moving beyond the Gaussian assumption through classification-based adaptation, the method addresses the fundamental limitations of existing wavelet thresholding techniques when faced with mixed noise. The authors assert that this approach offers substantial practical value for improving the diagnostic reliability of mechanical fault detection systems and the perceptual quality of speech enhancement applications, particularly in scenarios where labeled data is scarce. The study emphasizes that the multi-module architecture allows the system to handle distinct failure modes across the entire SNR range, ensuring that signal features are preserved without artificial distortion, while the lightweight mode offers a computationally feasible alternative for time-sensitive applications.

More like this