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

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

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

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

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

$\left[W_{\psi }f\right](m,k)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)h_m(2^{m}kT-t)dt$,

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

$y_m[n]=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}f[k]h_m[2^{m}n-k]$.