# CDFWavelet

represents a CohenDaubechiesFeauveau wavelet of type "9/7".

CDFWavelet["type"]

represents a CohenDaubechiesFeauveau wavelet of type "type".

# Details • CDFWavelet defines a set of biorthogonal wavelets.
• The following "type" forms can be used:
•  "5/3" used in lossless JPEG2000 compression "9/7" used in lossy JPEG2000 compression
• The scaling function ( ) and wavelet function ( ) have compact support. The functions are symmetric.
• CDFWavelet 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(16)

### Basic Uses(10)

Compute primal lowpass filter coefficients:

Dual lowpass filter coefficients:

Primal highpass filter coefficients:

Dual highpass filter coefficients:

Lifting filter coefficients:

Generate 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(1)

Multivariate scaling and wavelet functions are products of univariate ones:

## Properties & Relations(16)

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; : 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: