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.

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

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

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