ShannonWavelet

ShannonWavelet[]

represents the Shannon wavelet evaluated on the equally spaced interval {-10,10}.

ShannonWavelet[lim]

represents the Shannon wavelet evaluated on the equally spaced interval {-lim,lim}.

Details

  • ShannonWavelet defines a family of orthonormal wavelets.
  • ShannonWavelet[lim] is defined for any positive real lim.
  • The scaling function () and wavelet function () have infinite support. The functions are symmetric.
  • The scaling function () is given by .
  • The wavelet function () is given by .
  • ShannonWavelet 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  (7)

Basic Uses  (2)

Compute primal lowpass filter coefficients:

Primal highpass filter coefficients:

Wavelet Transforms  (4)

Compute a DiscreteWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

ShannonWavelet can be used to perform DiscreteWaveletPacketTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

ShannonWavelet can be used to perform StationaryWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

ShannonWavelet can be used to perform 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  (8)

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:

With wider interval {-lim,lim}, the frequency response function approaches ideal frequency response:

Frequency response for is given by :

The filter is a highpass filter:

With wider interval {-lim,lim}, the frequency response function approaches ideal frequency response:

Possible Issues  (1)

Due to noncompact support, ShannonWavelet poorly approximates the data:

Use wider interval {-lim,lim} to improve wavelet approximation:

Neat Examples  (2)

Plot translates and dilations of scaling function:

Plot translates and dilations of wavelet function:

Introduced in 2010
 (8.0)