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