# BiorthogonalSplineWavelet

represents a biorthogonal spline wavelet of order 4 and dual order 2.

represents a biorthogonal spline wavelet of order n and dual order m.

# Examples

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## Basic Examples(6)

Primal scaling function:

Primal wavelet function:

Dual scaling function:

Dual wavelet function:

Primal filter coefficients:

Dual filter coefficients:

## Scope(17)

### Basic Uses(10)

Compute primal lowpass filter coefficients:

Dual lowpass filter coefficients:

Primal highpass filter coefficients:

Dual highpass 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:

### Wavelet Transforms(5)

Compute a DiscreteWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a DiscreteWaveletPacketTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a StationaryWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a StationaryWaveletPacketTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a LiftingWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

### Higher Dimensions(2)

Multivariate scaling and wavelet functions are products of univariate ones:

Multivariate dual scaling and wavelet functions are products of univariate ones:

## Properties & Relations(18)

is equivalent to HaarWavelet:

Lowpass filter coefficients sum to unity; :

Highpass filter coefficients sum to zero; :

Dual lowpass filter coefficients sum to unity; :

Dual highpass 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 {n1,n2}:

Dual scaling function has compact support {nd1,nd2}:

Corresponding wavelet function has support ({n1- nd2+1)/2,(n2- nd1+1)/2}:

Dual wavelet function has support ({nd1- n2+1)/2,(nd2- n1+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:

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:

Fourier transform of is given by :

Frequency response for is given by :

The filter is a dual lowpass filter:

Fourier transform of is given by :

Frequency response for is given by :

The filter is a lowpass filter:

Fourier transform of is given by :

Frequency response for is given by :

The filter is a lowpass filter:

Fourier transform of is given by :

## Neat Examples(2)

Plot translates and dilations of scaling function:

Plot translates and dilations of wavelet function:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2022_biorthogonalsplinewavelet, author="Wolfram Research", title="{BiorthogonalSplineWavelet}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/BiorthogonalSplineWavelet.html}", note=[Accessed: 22-March-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_biorthogonalsplinewavelet, organization={Wolfram Research}, title={BiorthogonalSplineWavelet}, year={2010}, url={https://reference.wolfram.com/language/ref/BiorthogonalSplineWavelet.html}, note=[Accessed: 22-March-2023 ]}