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
- <math>\left[W_\psi f\right](a,b) = \frac{1}{\sqrt{a}}\int_{-\infty}^\infty{f(t)\psi^*\left(\frac{t-b}{a}\right)}dt\,</math>,
where
- <math>\psi</math> is so called mother wavelet,
- <math>a</math> denotes wavelet dilation,
- <math>b</math> denotes time shift of wavelet and
- <math>*</math> symbol denotes complex conjugate.
In case of <math>a = {a_{0}}^{m}</math> and <math>b = {a_{0}}^{m}kT</math>, where <math>a_{0}>1</math>, <math>T>0</math> and <math>m</math> and <math>k</math> are integer constants, the wavelet transform is called discrete wavelet transform (of continuous signal).
In case of <math>a = 2^m</math> and <math>b = 2^{m}kT</math>, where <math>m>0</math>, the discrete wavelet transform is called dyadic. It is defined as
- <math>\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\,</math>,
where
- <math>m</math> is frequency scale,
- <math>k</math> is time scale and
- <math>T</math> is constant which depends on mother wavelet.
It is possible to rewrite dyadic discrete wavelet transform as
- <math>\left[W_\psi f\right](m,k) = \int_{-\infty}^\infty{f(t) h_{m}\left(2^{m}kT-t\right)}dt\,</math>,
where <math>h_{m}</math> is impulse characteristic of continuous filter which is identical to <math>{\psi_{m}}^*</math> for given <math>m</math>.
Analogously, dyadic wavelet transform with discrete time (of discrete signal) is defined as
- <math>y_{m}[n] = \sum_{k=-\infty}^{\infty} f[k]h_{m}[2^{m}n-k]</math>.