Wavelet transform

mathematical technique used in data compression and analysis

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

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

Wavelet transform of continuous signal is defined as



  • is so called mother wavelet,
  • denotes wavelet dilation,
  • denotes time shift of wavelet and
  • symbol denotes complex conjugate.

In case of and , where , and and are integer constants, the wavelet transform is called discrete wavelet transform (of continuous signal).

In case of and , where , the discrete wavelet transform is called dyadic. It is defined as



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

It is possible to rewrite dyadic discrete wavelet transform as


where is impulse characteristic of continuous filter which is identical to for given .

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