Wavelet transform

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Continuous wavelet transform of frequency breakdown signal. Used symlet with 5 vanishing moments.

The wavelet transform is a time-frequency representation of a signal. For example, we use it for noise reduction, feature extraction or signal compression.

Wavelet transform of continuous signal is defined as

\left[W_\psi f\right](a,b) = \frac{1}{\sqrt{a}}\int_{-\infty}^\infty{f(t)\psi^*\left(\frac{t-b}{a}\right)}dt\,,

where

  • \psi is so called mother wavelet,
  • a denotes wavelet dilation,
  • b denotes time shift of wavelet and
  • * symbol denotes complex conjugate.

In case of a = {a_{0}}^{m} and b = {a_{0}}^{m}kT, where a_{0}>1, T>0 and m and k are integer constants, the wavelet transform is called discrete wavelet transform (of continuous signal).

In case of a = 2^m and b = 2^{m}kT, where m>0, the discrete wavelet transform is called dyadic. It is defined as

\left[W_\psi f\right](m,k) = \frac{1}{\sqrt{2^m}}\int_{-\infty}^\infty{f(t)\psi^*\left(2^{-m}t-kT\right)}dt\,,

where

  • m is frequency scale,
  • k is time scale and
  • T is constant which depends on mother wavelet.

It is possible to rewrite dyadic discrete wavelet transform as

\left[W_\psi f\right](m,k) = \int_{-\infty}^\infty{f(t) h_{m}\left(2^{m}kT-t\right)}dt\,,

where h_{m} is impulse characteristic of continuous filter which is identical to {\psi_{m}}^* for given m.

Analogously, dyadic wavelet transform with discrete time (of discrete signal) is defined as

y_{m}[n] = \sum_{k=-\infty}^{\infty} f[k]h_{m}[2^{m}n-k].