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
In case of formula_6 and formula_7, where formula_8, formula_9 and formula_10 and formula_11 are integer constants, the wavelet transform is called discrete wavelet transform (of continuous signal).
In case of formula_12 and formula_13, where formula_14, the discrete wavelet transform is called dyadic. It is defined as
where
It is possible to rewrite dyadic discrete wavelet transform as
where formula_20 is impulse characteristic of continuous filter which is identical to formula_21 for given formula_10.
Analogously, dyadic wavelet transform with discrete time (of discrete signal) is defined as