Empirical wavelet transform (2410.23534v1)

Abstract: Some recent methods, like the Empirical Mode Decomposition (EMD), propose to decompose a signal accordingly to its contained information. Even though its adaptability seems useful for many applications, the main issue with this approach is its lack of theory. This paper presents a new approach to build adaptive wavelets. The main idea is to extract the different modes of a signal by designing an appropriate wavelet filter bank. This construction leads us to a new wavelet transform, called the empirical wavelet transform. Many experiments are presented showing the usefulness of this method compared to the classic EMD.

• The paper's main contribution is the development of an adaptive method that constructs a wavelet filter bank based on the signal's Fourier spectrum.
• It rigorously compares EWT with EMD, showing that EWT provides more reliable and interpretable mode extraction for both synthetic and real-world signals.
• The study highlights future directions such as optimizing Fourier segmentation and extending applications to higher-dimensional data like images.

Empirical Wavelet Transform: A Novel Adaptive Decomposition Method

The paper presents a comprehensive paper of a new adaptive method in signal processing termed the Empirical Wavelet Transform (EWT). This approach is an effort to address the theoretical limitations and practical challenges associated with the Empirical Mode Decomposition (EMD), a widely utilized technique for signal decomposition. Unlike EMD, which lacks a robust theoretical framework, EWT employs a mathematically grounded methodology to extract signal modes by constructing an adaptive wavelet filter bank.

Key Concepts and Methodology

The essence of EWT lies in its ability to build adaptive wavelet functions that are tailored according to the characteristics present within the signal's spectrum. To achieve this, the technique segments the Fourier spectrum of the given signal into distinct bands, each associated with a mode of the signal. These bands are determined by identifying significant features in the signal's frequency domain, specifically by locating local maxima and dynamically defining boundaries between modes. The mathematical foundation of this approach ensures that EWT forms a tight frame, achieving an efficient energy transfer from the signal to its decomposed components.

Comparative Analysis and Experimental Validation

The paper rigorously contrasts EWT against the classical EMD by applying both methods to several test signals, including synthetic and real-world signals such as electrocardiograms and seismic data. The findings demonstrate that EWT consistently provides a more interpretable and reliable characterization of signal modes compared to EMD. EWT's advantage stems from its robust construction, which avoids the over-segmentation issues often experienced with EMD, wherein additional modes are extracted that do not correspond to any meaningful physical phenomena in the signal.

Implications and Future Directions

The introduction of EWT opens new avenues for practical applications within the field of signal processing. Its adaptability makes it suitable for a broad spectrum of applications, including noise reduction, feature extraction, and non-stationary signal analysis. The paper suggests that future work should focus on optimizing the segmentation of the Fourier spectrum to enhance the accuracy of mode detection further. Moreover, exploring the application of EWT in higher-dimensional data, such as images, could extend its utility, potentially inspiring new developments in adaptive transform methods.

In conclusion, the empirical wavelet transform offers a solid theoretical and practical framework for adaptive signal decomposition, overcoming limitations of previous methods like EMD. This advance in signal processing could lead to more precise data analysis in disciplines that require complex signal interpretation, fostering deeper insights and innovations within the computational sciences.