Meyer wavelet

The Meyer wavelet is an orthogonal wavelet proposed by Yves Meyer.^[1] As a type of a continuous wavelet, it has been applied in a number of cases, such as in adaptive filters,^[2] fractal random fields,^[3] and multi-fault classification.^[4]

The Meyer wavelet is infinitely differentiable with infinite support and defined in frequency domain in terms of function ${\displaystyle \nu }$ as

${\displaystyle \Psi (\omega ):={\begin{cases}{\frac {1}{\sqrt {2\pi }}}\sin \left({\frac {\pi }{2}}\nu \left({\frac {3|\omega |}{2\pi }}-1\right)\right)e^{j\omega /2}&{\text{if }}2\pi /3<|\omega |<4\pi /3,\\{\frac {1}{\sqrt {2\pi }}}\cos \left({\frac {\pi }{2}}\nu \left({\frac {3|\omega |}{4\pi }}-1\right)\right)e^{j\omega /2}&{\text{if }}4\pi /3<|\omega |<8\pi /3,\\0&{\text{otherwise}},\end{cases}}}$

where

${\displaystyle \nu (x):={\begin{cases}0&{\text{if }}x<0,\\x&{\text{if }}0<x<1,\\1&{\text{if }}x>1.\end{cases}}}$

There are many different ways for defining this auxiliary function, which yields variants of the Meyer wavelet. For instance, another standard implementation adopts

${\displaystyle \nu (x):={\begin{cases}x^{4}(35-84x+70x^{2}-20x^{3})&{\text{if }}0<x<1,\\0&{\text{otherwise}}.\end{cases}}}$

The Meyer scale function is given by

${\displaystyle \Phi (\omega ):={\begin{cases}{\frac {1}{\sqrt {2\pi }}}&{\text{if }}|\omega |<2\pi /3,\\{\frac {1}{\sqrt {2\pi }}}\cos \left({\frac {\pi }{2}}\nu \left({\frac {3|\omega |}{2\pi }}-1\right)\right)&{\text{if }}2\pi /3<|\omega |<4\pi /3,\\0&{\text{otherwise}}.\end{cases}}}$

In the time domain, the waveform of the Meyer mother-wavelet has the shape as shown in the following figure: