BUILT-IN MATHEMATICA SYMBOL

BattleLemarieWavelet

represents the Battle-Lemarié wavelet of order

.

BattleLemarieWavelet[n]
represents the Battle-Lemarié wavelet of order n evaluated on equally spaced interval

.

BattleLemarieWavelet

represents the Battle-Lemarié wavelet of order n evaluated on equally spaced interval

.

MORE INFORMATION

BattleLemarieWavelet defines a family of orthogonal wavelets based on orthonormalization of B-splines of degree n.

BattleLemarieWavelet[n] is equivalent to BattleLemarieWavelet.

The scaling function () and wavelet function () have infinite support with an exponential decay outside the interval -lim to lim. The functions are continuously differentiable.

BattleLemarieWavelet can be used with such functions as DiscreteWaveletTransform, 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:

Battle-Lemarié scaling function of order 2:

Battle-Lemarié scaling function of order 5:

Battle-Lemarié wavelet function of order 2:

Battle-Lemarié wavelet function of order 5:

Compute a DiscreteWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

BattleLemarieWavelet can be used to perform a DiscreteWaveletPacketTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

BattleLemarieWavelet can be used to perform a StationaryWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

BattleLemarieWavelet 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 (11)

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;

:

For even order n, scaling function is symmetrical about 1/2:

For even order n, wavelet function is antisymmetrical about 1/2:

For odd order n, scaling function is symmetrical about 0:

For odd order n, wavelet function is symmetrical about 1/2:

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

:

Fourier transform of

is given by

:

Possible Issues (1)

BattleLemarieWavelet is restricted to n less than 15:

BattleLemarieWavelet 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: