Meyer wavelet

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


 * $$ \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
 * $$ \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
 * $$ \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


 * $$ \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:

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


 * $$\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


 * $$\psi(t) = \psi_1(t) + \psi_2(t),$$

where


 * $$\psi_1(t) = \frac{\frac{4}{3\pi}(t - \frac12)\cos[\frac{2\pi}{3}(t - \frac12)] - \frac{1}{\pi}\sin[\frac{4\pi}{3}(t - \frac12)]}{(t - \frac12) - \frac{16}{9}(t - \frac12)^3},$$
 * $$\psi_2(t) = \frac{\frac{8}{3\pi}(t - \frac12)\cos[\frac{8\pi}{3}(t - \frac12)] + \frac{1}{\pi}\sin[\frac{4\pi}{3}(t - \frac12)]}{(t - \frac12) - \frac{64}{9}(t - \frac12)^3}.$$