This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# ShannonWavelet

 ShannonWavelet represents the Shannon wavelet evaluated on the equally spaced interval . ShannonWavelet[lim]represents the Shannon wavelet evaluated on the equally spaced interval .
• 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 .
Scaling function:
Wavelet function:
Filter coefficients:
Scaling function:
 Out[1]=
 Out[2]=

Wavelet function:
 Out[1]=
 Out[2]=

Filter coefficients:
 Out[1]=
 Scope   (7)
Compute primal low-pass filter coefficients:
Primal high-pass filter coefficients:
View the tree of wavelet coefficients:
Get the dimensions of wavelet coefficients:
Plot the wavelet coefficients:
View the tree of wavelet coefficients:
Get the dimensions of wavelet coefficients:
Plot the wavelet coefficients:
View the tree of wavelet coefficients:
Get the dimensions of wavelet coefficients:
Plot the wavelet coefficients:
View the tree of wavelet coefficients:
Get the dimensions of wavelet coefficients:
Plot the wavelet coefficients:
Multivariate scaling and wavelet functions are products of univariate ones:
Low-pass filter coefficients approximately sum to unity; :
High-pass 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 low-pass filter:
With wider interval , the frequency response function approaches ideal frequency response:
Frequency response for is given by :
The filter is a high-pass filter:
With wider interval , the frequency response function approaches ideal frequency response:
Due to noncompact support, ShannonWavelet poorly approximates the data:
Use wider interval to improve wavelet approximation:
Plot translates and dilations of scaling function:
Plot translates and dilations of wavelet function:
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