# BattleLemarieWavelet

represents the BattleLemarié wavelet of order 3.

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

BattleLemarieWavelet[n,lim]

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

# 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:

BattleLemarié scaling function of order 2:

BattleLemarié scaling function of order 5:

BattleLemarié wavelet function of order 2:

BattleLemarié 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:

Wolfram Research (2010), BattleLemarieWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/BattleLemarieWavelet.html.

#### Text

Wolfram Research (2010), BattleLemarieWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/BattleLemarieWavelet.html.

#### CMS

Wolfram Language. 2010. "BattleLemarieWavelet." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BattleLemarieWavelet.html.

#### APA

Wolfram Language. (2010). BattleLemarieWavelet. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BattleLemarieWavelet.html

#### BibTeX

@misc{reference.wolfram_2024_battlelemariewavelet, author="Wolfram Research", title="{BattleLemarieWavelet}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/BattleLemarieWavelet.html}", note=[Accessed: 10-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_battlelemariewavelet, organization={Wolfram Research}, title={BattleLemarieWavelet}, year={2010}, url={https://reference.wolfram.com/language/ref/BattleLemarieWavelet.html}, note=[Accessed: 10-August-2024 ]}