Wavelet

A Wavelet is a mathematical function used to write down a function or signal in terms of other functions that are simpler to study. Many signal processing tasks can be seen in terms of a wavelet transform. Informally speaking, the signal can be seen under the lens with a magnification given by the scale of the wavelet. In doing so, we can see only the information that is determined by the shape of the wavelet used.

The English term "wavelet" was introduced in the early 1980s by French physicists Jean Morlet and Alex Grossman. They used the French word "ondelette" (which means "small wave"). Later, this word was brought into English by translating "onde" into "wave" giving "wavelet".

Wavelet is (complex) function from the Hilbert space ${\displaystyle \psi \in L^{2}(\mathbb {R} )}$. For practical applications it should satisfy following conditions.

It must have finite energy.

${\displaystyle \int _{-\infty }^{\infty }|\psi (t)|^{2}dt<\infty }$

It must satisfy an admissibility condition.

${\displaystyle \int _{0}^{\infty }{{|{\hat {\psi }}(\omega )|^{2}} \over {\omega }}d\omega <\infty }$, where ${\displaystyle {\hat {\psi }}}$ is a Fourier transform of ${\displaystyle \psi \,}$

Zero mean condition implies from admissibility condition.

${\displaystyle \int _{-\infty }^{\infty }\psi (t)dt=0}$

The function ${\displaystyle \psi \,}$ is called mother wavelet. Its translated (shifted) and dilated (scaled) normalized versions are defined as following.

${\displaystyle \psi _{a,b}(t)={\frac {1}{\sqrt {a}}}\psi \left({{t-b} \over {a}}\right)}$

Original mother wavelet has parameters ${\displaystyle a=1}$ and ${\displaystyle b=0}$. Translation is described by ${\displaystyle b}$ parameter and dilatation by ${\displaystyle a}$ parameter.