# MeyerWavelet

represents the Meyer wavelet of order 3.

MeyerWavelet[n]

represents the Meyer wavelet of order n evaluated on the equally spaced interval {-10,10}.

MeyerWavelet[n,lim]

represents the Meyer wavelet of order n evaluated on the equally spaced interval {-lim,lim}.

# Details

• MeyerWavelet defines a family of orthonormal wavelets.
• MeyerWavelet[n] is equivalent to MeyerWavelet[n,8].
• MeyerWavelet[n,lim] is defined for any positive integer n and real limit lim.
• The scaling function () and wavelet function () have infinite support. The functions are symmetric.
• The scaling function () is given by its Fourier transform as . »
• The wavelet function () is given by its Fourier transform as .
• The polynomial is a polynomial of the form , where is the order of the Meyer wavelet.
• MeyerWavelet can be used with such functions as DiscreteWaveletTransform and WaveletPhi, etc.

# Examples

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## Basic Examples(3)

Scaling function:

Wavelet function:

Filter coefficients:

## Scope(9)

### Basic Uses(4)

Compute primal lowpass filter coefficients:

Primal highpass filter coefficients:

Meyer scaling function of order 3:

Meyer scaling function of order 10:

Meyer wavelet function of order 3:

Meyer wavelet function of order 10:

### Wavelet Transforms(4)

Compute a DiscreteWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

MeyerWavelet can be used to perform a DiscreteWaveletPacketTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

MeyerWavelet can be used to perform a StationaryWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

MeyerWavelet can be used to perform a StationaryWaveletPacketTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

### Higher Dimensions(1)

Multivariate scaling and wavelet functions are products of univariate ones:

## Properties & Relations(10)

Lowpass filter coefficients approximately sum to unity; :

Highpass filter coefficients approximately sum to zero; :

Scaling function integrates to unity; :

Wavelet function integrates to zero; :

satisfies the recursion equation :

Plot the components and the sum of the recursion:

satisfies the recursion equation :

Plot the components and the sum of the recursion:

Frequency response for is given by :

The filter is a lowpass filter:

Frequency response for is given by :

The filter is a highpass filter:

Fourier transform of is given by :

Compare the above result with the exact Fourier transform:

Fourier transform of is given by :

Compare the above result with the exact Fourier transform:

## Neat Examples(2)

Plot translates and dilations of scaling function:

Plot translates and dilations of wavelet function:

Wolfram Research (2010), MeyerWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/MeyerWavelet.html.

#### Text

Wolfram Research (2010), MeyerWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/MeyerWavelet.html.

#### CMS

Wolfram Language. 2010. "MeyerWavelet." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MeyerWavelet.html.

#### APA

Wolfram Language. (2010). MeyerWavelet. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MeyerWavelet.html

#### BibTeX

@misc{reference.wolfram_2022_meyerwavelet, author="Wolfram Research", title="{MeyerWavelet}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MeyerWavelet.html}", note=[Accessed: 23-March-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_meyerwavelet, organization={Wolfram Research}, title={MeyerWavelet}, year={2010}, url={https://reference.wolfram.com/language/ref/MeyerWavelet.html}, note=[Accessed: 23-March-2023 ]}