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

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

where

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

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

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

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

where

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

It is possible to rewrite dyadic discrete wavelet transform as

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

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

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

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