Meyer wavelet

File:Spectrum Meyer wavelet.svg

The Meyer wavelet is an orthogonal wavelet proposed by Yves Meyer.{{cite book |last1=Meyer |first1=Yves |title=Ondelettes et opérateurs: Ondelettes |date=1990 |publisher=Hermann |isbn=9782705661250}} As a type of a continuous wavelet, it has been applied in a number of cases, such as in adaptive filters,{{cite journal |last1=Xu |first1=L. |last2=Zhang |first2=D. |last3=Wang |first3=K. |title=Wavelet-based cascaded adaptive filter for removing baseline drift in pulse waveforms |journal=IEEE Transactions on Biomedical Engineering |date=2005 |volume=52 |issue=11 |pages=1973–1975 |doi=10.1109/tbme.2005.856296 |pmid=16285403|hdl=10397/193 |s2cid=6897442 |hdl-access=free }} fractal random fields,{{cite journal |last1=Elliott, Jr. |first1=F. W. |last2=Horntrop |first2=D. J. |last3=Majda |first3=A. J. |title=A Fourier-Wavelet Monte Carlo method for fractal random fields |journal=Journal of Computational Physics |date=1997 |volume=132 |issue=2 |pages=384–408 |doi=10.1006/jcph.1996.5647 |bibcode=1997JCoPh.132..384E|doi-access=free }} and multi-fault classification.{{cite journal |last1=Abbasion |first1=S. |display-authors=etal |title=Rolling element bearings multi-fault classification based on the wavelet denoising and support vector machine |journal=Mechanical Systems and Signal Processing |date=2007 |volume=21 |issue=7 |pages=2933–2945 |doi=10.1016/j.ymssp.2007.02.003 |bibcode=2007MSSP...21.2933A}}

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

: $\backslash Psi(\backslash omega)\; :=\; \backslash begin\{cases\}$

\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

: $\backslash nu\; (x)\; :=\; \backslash begin\{cases\}$

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

: $\backslash nu\; (x)\; :=\; \backslash begin\{cases\}$

x^4 (35 - 84x + 70x^2 - 20x^3) & \text{if } 0 < x < 1, \\

0 & \text{otherwise}.

\end{cases}

File:Spectrum Meyer scalefunction.png

The Meyer scaling function is given by

: $\backslash Phi(\backslash omega)\; :=\; \backslash begin\{cases\}$

\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:

File:Meyer wavelet.svg