Meyer wavelet

Quality 0.50 · 2 views · Updated 2 months ago

title: "Meyer wavelet" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["wavelets"] topic_path: "general/wavelets" source: "https://en.wikipedia.org/wiki/Meyer_wavelet" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::figure[src="https://upload.wikimedia.org/wikipedia/commons/3/37/Spectrum_Meyer_wavelet.svg" caption="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

: \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 \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 0 & \text{otherwise}, \end{cases}

where : \nu (x) := \begin{cases} 0 & \text{if } x x & \text{if } 0 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 0 & \text{otherwise}. \end{cases}

::figure[src="https://upload.wikimedia.org/wikipedia/commons/c/c9/Spectrum_Meyer_scalefunction.png" caption="Meyer scale function (numerically computed)"] ::

The Meyer scaling function is given by

: \Phi(\omega) := \begin{cases} \frac{1}{\sqrt{2\pi}} & \text{if } | \omega| \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 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: ::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/0f/Meyer_wavelet.svg" caption="waveform of the Meyer wavelet (numerically computed)"] ::

Closed 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}.

References

References

  1. (1990). "Ondelettes et opérateurs: Ondelettes". Hermann.
  2. (2005). "Wavelet-based cascaded adaptive filter for removing baseline drift in pulse waveforms". IEEE Transactions on Biomedical Engineering.
  3. (1997). "A Fourier-Wavelet Monte Carlo method for fractal random fields". Journal of Computational Physics.
  4. (2007). "Rolling element bearings multi-fault classification based on the wavelet denoising and support vector machine". Mechanical Systems and Signal Processing.
  5. (2015). "Anais de XXXIII Simpósio Brasileiro de Telecomunicações".

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::

wavelets