BUILT-IN MATHEMATICA SYMBOL

CoifletWavelet

represents a Coiflet wavelet of order

.

CoifletWavelet[n]
represents a Coiflet wavelet of order n.

MORE INFORMATION

CoifletWavelet defines a family of orthogonal wavelets.

CoifletWavelet[n] is defined for positive integers n between 1 and 5.

The scaling function () and wavelet function () have compact support of length . The scaling function has vanishing moments and wavelet function has vanishing moments.

CoifletWavelet can be used with such functions as DiscreteWaveletTransform, WaveletPhi, WaveletPsi, 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 (12)

Compute primal low-pass filter coefficients:

Primal high-pass filter coefficients:

Lifting filter coefficients:

Generate a function to compute a lifting wavelet transform:

Coiflet scaling function of order 1:

Coiflet scaling function of order 4:

Plot scaling function at different refinement scales:

Coiflet wavelet function of order 1:

Coiflet wavelet of order 4:

Plot wavelet function at different refinement scales:

Compute a DiscreteWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a DiscreteWaveletPacketTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a StationaryWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a 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:

Applications (3)

Approximate a function using Haar wavelet coefficients:

Perform a LiftingWaveletTransform:

Approximate original data by keeping n largest coefficients and thresholding everything else:

Compare the different approximations:

Compute the multiresolution representation of a signal containing an impulse:

Compare the cumulative energy in a signal and its wavelet coefficients:

Compute the ordered cumulative energy in the signal:

The energy in the signal is captured by relatively few wavelet coefficients:

Properties & Relations (11)

Low-pass filter coefficients sum to unity;

:

High-pass filter coefficients sum to zero;

:

Scaling function integrates to unity;

:

In particular,

:

Wavelet function integrates to zero;

:

Wavelet function is orthogonal to the scaling function at the same scale;

:

The low-pass and high-pass filter coefficients are orthogonal;

:

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:

The higher the order n, the flatter the response function at the ends:

Fourier transform of

is given by

:

Frequency response for

is given by

:

The filter is a high-pass filter:

The higher the order n, the flatter the response function at the ends:

Fourier transform of

is given by

:

Possible Issues (1)

CoifletWavelet is restricted to n less than 5:

CoifletWavelet is not defined when n is not a positive machine integer:

Neat Examples (2)

Plot translates and dilations of scaling function:

Plot translates and dilations of wavelet function: