Meyer wavelet

Spectrum of the Meyer wavelet (numerically computed).

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

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

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

where

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

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

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

Meyer scale function (numerically computed)

The Meyer scale function is given by

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

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

waveform of the Meyer wavelet (numerically computed)

Close expressions

Valenzuela and de Oliveira give the explicit expressions of Meyer wavelet and scale functions:

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

and

$\psi (t)={\psi }_1(t)+{\psi }_2(t),$

where

${\psi }_1(t)=\frac{\frac{4}{3\pi }(t-\frac{1}{2})\cos [\frac{2\pi }{3}(t-\frac{1}{2})]-\frac{1}{\pi }\sin [\frac{4\pi }{3}(t-\frac{1}{2})]}{(t-\frac{1}{2})-\frac{16}{9}(t-\frac{1}{2}{)}^3},$

${\psi }_2(t)=\frac{\frac{8}{3\pi }(t-\frac{1}{2})\cos [\frac{8\pi }{3}(t-\frac{1}{2})]+\frac{1}{\pi }\sin [\frac{4\pi }{3}(t-\frac{1}{2})]}{(t-\frac{1}{2})-\frac{64}{9}(t-\frac{1}{2}{)}^3}.$

References

• 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. https://archive.org/details/tenlecturesonwav0000daub/page/117. 

External links

• wavelet toolbox
• Matlab implementation