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Empirical wavelet transform

Jerome Gilles

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.

Empirical wavelet transform

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.

Paper Structure

This paper contains 21 sections, 1 theorem, 24 equations, 27 figures, 1 table.

Table of Contents

  1. Introduction
  2. Existing approaches
  3. Empirical Mode Decomposition
  4. Wavelets approaches
  5. Empirical Wavelets
  6. Definition
  7. Segmentation of the Fourier spectrum
  8. Frame
  9. Empirical wavelet transform
  10. Experiments
  11. Test signals
  12. Simulated $f_{Sig1}$
  13. Simulated $f_{Sig2}$
  14. Simulated $f_{Sig3}$
  15. Real signals $f_{Sig4}$ and $f_{Sig5}$
  16. ...and 6 more sections

Key Result

Proposition 1

If $\gamma<\min_n\left(\frac{\omega_{n+1}-\omega_n}{\omega_{n+1}+\omega_n}\right)$, then the set $\{\phi_1(t),\{\psi_n(t)\}_{n=1}^N\}$ is a tight frame of $L^2({\mathbb{R}})$.

Figures (27)

  • Figure 1: EMD: basic IMF detection. Envelopes detection on top (thin continuous: $f$, dashed: $\bar{f}$ and $\underline{f}$ and thick continuous: $m$). On bottom: the first IMF candidate $r_1$.
  • Figure 2: On top: dyadic wavelet tiling of the frequency line. On bottom: a wavelet packet like tiling.
  • Figure 3: Partitioning of the Fourier axis
  • Figure 4: On left: Fourier transform of the scaling function for $\nu_n=1,\gamma=0.5$. On right: Fourier transform of the wavelet function for $\nu_n=1,\nu_{n+1}=2.5,\gamma=0.2$.
  • Figure 5: Periodicity of the filter bank
  • ...and 22 more figures

Theorems & Definitions (1)

  • Proposition 1