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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:

Valenzuela and de Oliveira ^[5] give the explicit expressions of Meyer wavelet and scale functions:

${\displaystyle \phi ($t)={\begin{cases}{\frac {2}{3}}+{\frac {4}{3\pi }}&t=0,\\{\frac {\sin({\frac {2\pi }{3}}t)+{\frac {4}{3}}t\cos({\frac {4\pi }{3}}t)}{\pi t-{\frac {16\pi }{9}}t^{3}}}&{\text{otherwise}},\end{cases}}}

and

${\displaystyle \psi ($t)=\psi _{1}(t)+\psi _{2}(t),}

where

${\displaystyle \psi _{1}($t)={\frac {{\frac {4}{3\pi }}(t-{\frac {1}{2}})\cos[{\frac {2\pi }{3}}(t-{\frac {1}{2}})]-{\frac {1}{\pi }}\sin[{\frac {4\pi }{3}}(t-{\frac {1}{2}})]}{(t-{\frac {1}{2}})-{\frac {16}{9}}(t-{\frac {1}{2}})^{3}}},}

${\displaystyle \psi _{2}($t)={\frac {{\frac {8}{3\pi }}(t-{\frac {1}{2}})\cos[{\frac {8\pi }{3}}(t-{\frac {1}{2}})]+{\frac {1}{\pi }}\sin[{\frac {4\pi }{3}}(t-{\frac {1}{2}})]}{(t-{\frac {1}{2}})-{\frac {64}{9}}(t-{\frac {1}{2}})^{3}}}.}

• Daubechies, Ingrid (September 1992). Ten Lectures on Wavelets (CBMS-NSF conference series in applied mathematics) (SIAM ed.). Springer-Verlag. pp. 117–119, 137–138, 152–155. ISBN 978-0-89871-274-2.

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