# 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: