Meyer wavelet explained

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

\nu as\Psi(\omega) : = \begin{cases} (1)/(\sqrt{ 2 \pi) } \sin\left(\frac \nu \left(\frac -1\right)\right) e^ & \text 2 \pi /3<|\omega|< 4 \pi /3, \\ \frac \cos\left(\frac \nu \left(\frac-1\right)\right) e^ & \text 4 \pi /3<| \omega|< 8 \pi /3, \\ 0 & \text,\end

where

\nu (x) : = \begin{cases} 0 & if x < 0 , \\ x & if 0 < x < 1 , \\ 1 & 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 - 84 x + 70 x ^{2} - 20 x ^{3)} & if 0 < x < 1 , \\ 0 & otherwise . \end{cases}

The Meyer scale function is given by

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

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

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

and

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

where

\psi _{1(t)} = ((4)/((t - (12 )\cos[ (2 \pi)/(3))/((t) - (12 )])/(-) (1)/(\pi) \sin[ (4 \pi)/(3) (t - (12 )])/() 3 \pi))/((t - (12 ))/(-) (16)/(9) (t - (12 ))/(^{3})) , \psi _{2(t)} = ((8)/((t - (12 )\cos[ (8 \pi)/(3))/((t) - (12 )])/(+) (1)/(\pi) \sin[ (4 \pi)/(3) (t - (12 )])/() 3 \pi))/((t - (12 ))/(-) (64)/(9) (t - (12 ))/(^{3})) .

References

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

External links

• wavelet toolbox
• Matlab implementation

Notes and References

1. Book: Meyer . Yves . Ondelettes et opérateurs: Ondelettes . 1990 . Hermann . 9782705661250.
2. Xu . L. . Zhang . D. . Wang . K. . Wavelet-based cascaded adaptive filter for removing baseline drift in pulse waveforms . IEEE Transactions on Biomedical Engineering . 2005 . 52 . 11 . 1973–1975 . 10.1109/tbme.2005.856296 . 16285403. 10397/193 . 6897442 . free .
3. Elliott, Jr. . F. W. . Horntrop . D. J. . Majda . A. J. . A Fourier-Wavelet Monte Carlo method for fractal random fields . Journal of Computational Physics . 1997 . 132 . 2 . 384–408 . 10.1006/jcph.1996.5647 . 1997JCoPh.132..384E. free .
4. Abbasion . S. . etal . Rolling element bearings multi-fault classification based on the wavelet denoising and support vector machine . Mechanical Systems and Signal Processing . 2007 . 21 . 7 . 2933–2945 . 10.1016/j.ymssp.2007.02.003 . 2007MSSP...21.2933A.
5. Book: Valenzuela . Victor Vermehren . Anais de XXXIII Simpósio Brasileiro de Telecomunicações . de Oliveira . H. M. . Close expressions for Meyer Wavelet and Scale Function . 1502.00161 . 4 . 2015. 10.14209/SBRT.2015.2 . 88513986 .