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# BattleLemarieWavelet

 BattleLemarieWavelet represents the Battle-Lemarié wavelet of order . BattleLemarieWavelet[n]represents the Battle-Lemarié wavelet of order n evaluated on equally spaced interval . BattleLemarieWaveletrepresents the Battle-Lemarié wavelet of order n evaluated on equally spaced interval .
• BattleLemarieWavelet defines a family of orthogonal wavelets based on orthonormalization of B-splines of degree n.
• The scaling function () and wavelet function () have infinite support with an exponential decay outside the interval -lim to lim. The functions are continuously differentiable.
Scaling function:
Wavelet function:
Filter coefficients:
Scaling function:
 Out[1]=
 Out[2]=

Wavelet function:
 Out[1]=
 Out[2]=

Filter coefficients:
 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:
View the tree of wavelet coefficients:
Get the dimensions of wavelet coefficients:
Plot the wavelet coefficients:
View the tree of wavelet coefficients:
Get the dimensions of wavelet coefficients:
Plot the wavelet coefficients:
View the tree of wavelet coefficients:
Get the dimensions of wavelet coefficients:
Plot the wavelet coefficients:
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:
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 :
BattleLemarieWavelet is restricted to n less than 15:
BattleLemarieWavelet is not defined when n is not a positive machine integer:
Plot translates and dilations of scaling function:
Plot translates and dilations of wavelet function:
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