Meyer wavelet

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]

Spectrum of the Meyer wavelet (numerically computed).

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

where

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

Meyer scale function (numerically computed)

The Meyer scale function is given by

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

and

where

References

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