This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# BiorthogonalSplineWavelet

 BiorthogonalSplineWavelet represents a biorthogonal spline wavelet of order and dual order . BiorthogonalSplineWaveletrepresents a biorthogonal spline wavelet of order n and dual order m.
• The scaling function () and wavelet function () have compact support. The functions are symmetric.
Primal scaling function:
Primal wavelet function:
Dual scaling function:
Dual wavelet function:
Primal filter coefficients:
Dual filter coefficients:
Primal scaling function:
 Out[1]=

Primal wavelet function:
 Out[1]=

Dual scaling function:
 Out[1]=

Dual wavelet function:
 Out[1]=

Primal filter coefficients:
 Out[1]=

Dual filter coefficients:
 Out[1]=
 Scope   (17)
Compute primal low-pass filter coefficients:
Dual low-pass filter coefficients:
Primal high-pass filter coefficients:
Dual high-pass filter coefficients:
Lifting filter coefficients:
Generate a function to compute lifting wavelet transform:
Primal scaling function:
Dual scaling function:
Plot scaling function using different levels of recursion:
Primal wavelet function:
Dual wavelet function:
Plot scaling function using different levels of recursion:
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:
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:
Multivariate dual scaling and wavelet functions are products of univariate ones:
Low-pass filter coefficients sum to unity; :
High-pass filter coefficients sum to zero; :
Dual low-pass filter coefficients sum to unity; :
Dual high-pass filter coefficients sum to zero; :
Scaling function integrates to unity; :
Dual scaling function integrates to unity; :
Wavelet function integrates to zero; :
Dual wavelet function integrates to zero; :
Scaling function has compact support :
Dual scaling function has compact support :
Corresponding wavelet function has support :
Dual wavelet function has support :
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:
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:
Fourier transform of is given by :
Frequency response for is given by :
The filter is a dual low-pass filter:
Fourier transform of is given by :
Frequency response for is given by :
The filter is a low-pass filter:
Fourier transform of is given by :
Frequency response for is given by :
The filter is a low-pass filter:
Fourier transform of is given by :
Plot translates and dilations of scaling function:
Plot translates and dilations of wavelet function:
New in 8