BUILT-IN MATHEMATICA SYMBOL

ShannonWavelet

represents the Shannon wavelet evaluated on the equally spaced interval

.

ShannonWavelet[lim]
represents the Shannon wavelet evaluated on the equally spaced interval

.

MORE INFORMATION

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:

Scaling function:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

Wavelet function:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

Filter coefficients:

In[1]:=

Out[1]=

Scope (7)

Compute primal low-pass filter coefficients:

Primal high-pass filter coefficients:

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:

Multivariate scaling and wavelet functions are products of univariate ones:

Properties & Relations (8)

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:

Possible Issues (1)

Due to noncompact support, ShannonWavelet poorly approximates the data:

Use wider interval

to improve wavelet approximation:

Neat Examples (2)

Plot translates and dilations of scaling function:

Plot translates and dilations of wavelet function: