# Wavelet transform

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

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

$\left[W_\psi f\right](a,b) = \frac{1}{\sqrt{a}}\int_{-\infty}^\infty{f(t)\psi^*\left(\frac{t-b}{a}\right)}dt\,$,

where

• $\psi$ is so called mother wavelet,
• $a$ denotes wavelet dilation,
• $b$ denotes time shift of wavelet and
• $*$ symbol denotes complex conjugate.

In case of $a = {a_{0}}^{m}$ and $b = {a_{0}}^{m}kT$, where $a_{0}>1$, $T>0$ and $m$ and $k$ are integer constants, the wavelet transform is called discrete wavelet transform (of continuous signal).

In case of $a = 2^m$ and $b = 2^{m}kT$, where $m>0$, the discrete wavelet transform is called dyadic. It is defined as

$\left[W_\psi f\right](m,k) = \frac{1}{\sqrt{2^m}}\int_{-\infty}^\infty{f(t)\psi^*\left(2^{-m}t-kT\right)}dt\,$,

where

• $m$ is frequency scale,
• $k$ is time scale and
• $T$ is constant which depends on mother wavelet.

It is possible to rewrite dyadic discrete wavelet transform as

$\left[W_\psi f\right](m,k) = \int_{-\infty}^\infty{f(t) h_{m}\left(2^{m}kT-t\right)}dt\,$,

where $h_{m}$ is impulse characteristic of continuous filter which is identical to ${\psi_{m}}^*$ for given $m$.

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

$y_{m}[n] = \sum_{k=-\infty}^{\infty} f[k]h_{m}[2^{m}n-k]$.