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BattleLemarieWavelet

BattleLemarieWavelet[]

represents the Battle–Lemarié wavelet of order 3.

BattleLemarieWavelet[n]

represents the Battle–Lemarié wavelet of order n evaluated on equally spaced interval {-10,10}.

BattleLemarieWavelet[n,lim]

represents the Battle–Lemarié wavelet of order n evaluated on equally spaced interval {-lim,lim}.

Details

BattleLemarieWavelet defines a family of orthogonal wavelets based on orthonormalization of B-splines of degree n.
BattleLemarieWavelet[n] is equivalent to BattleLemarieWavelet[n,10].
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:

Scope (9)

Basic Uses (4)

Compute primal lowpass filter coefficients:

Primal highpass 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:

Wavelet Transforms (4)

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:

Higher Dimensions (1)

Multivariate scaling and wavelet functions are products of univariate ones:

Properties & Relations (11)

Lowpass filter coefficients approximately sum to unity; :

Highpass 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 lowpass filter:

Frequency response for is given by :

The filter is a highpass 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:

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