Wavelet transform

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
 ,
where
  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
 ,
where
  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
.

Category:Calculus
Category:Harmonic analysis
Category:Wavelets