BUILT-IN MATHEMATICA SYMBOL

MeyerWavelet

represents the Meyer wavelet of order 3.

MeyerWavelet[n]
represents the Meyer wavelet of order n evaluated on the equally spaced interval

.

MeyerWavelet

represents the Meyer wavelet of order n evaluated on the equally spaced interval

.

MORE INFORMATION

MeyerWavelet defines a family of orthonormal wavelets.

MeyerWavelet[n] is equivalent to MeyerWavelet.

MeyerWavelet 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 . »

The wavelet function () is given by its Fourier transform as .

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:

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 (9)

Compute primal low-pass filter coefficients:

Primal high-pass 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:

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:

Multivariate scaling and wavelet functions are products of univariate ones:

Properties & Relations (10)

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:

Frequency response for

is given by

:

The filter is a high-pass 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: