MeyerWavelet

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 1 TemplateBox[{omega}, Abs]<=(2 pi)/3; cos(1/2 pi nu((3 TemplateBox[{omega}, Abs])/(2 pi)-1)) (2 pi)/3<=TemplateBox[{omega}, Abs]<=(4 pi)/3. »
  • The wavelet function () is given by its Fourier transform as exp((ⅈ omega)/2) sin(pi/2 nu((3 TemplateBox[{omega}, Abs])/(2 pi)-1)) (2 pi)/3<=TemplateBox[{omega}, Abs]<=(4 pi)/3; exp((ⅈ omega)/2) cos(pi/2 nu((3 TemplateBox[{omega}, Abs])/(4 pi)-1)) (4 pi)/3<=TemplateBox[{omega}, Abs]<=(8 pi)/3.
  • 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:

Introduced in 2010
 (8.0)