# DaubechiesWavelet

represents a Daubechies wavelet of order 2.

represents a Daubechies wavelet of order n.

# Details • DaubechiesWavelet defines a family of orthogonal wavelets.
• is defined for any positive integer n.
• The scaling function ( ) and wavelet function ( ) have compact support length of 2n. The scaling function has n vanishing moments.
• DaubechiesWavelet 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(14)

### Basic Uses(8)

Compute primal lowpass filter coefficients:

Primal highpass filter coefficients:

Lifting filter coefficients:

Generate a function to compute a lifting wavelet transform:

Daubechies scaling function of order 2:

Daubechies scaling function of order 6:

Plot scaling function using different levels of recursion:

Daubechies wavelet function of order 2:

DaubechiesWavelet of order 6:

Plot wavelet function 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:

## Applications(3)

Approximate a function using Daubechies wavelet coefficients:

Perform a LiftingWaveletTransform:

Approximate original data by keeping n largest coefficients and thresholding everything else:

Compare the different approximations:

Compute the multiresolution representation of a signal containing an impulse:

Compare the cumulative energy in a signal and its wavelet coefficients:

Compute the ordered cumulative energy in the signal:

The energy in the signal is captured by relatively few wavelet coefficients:

## Properties & Relations(13)

is equivalent to HaarWavelet:

Lowpass filter coefficients sum to unity; :

Highpass filter coefficients sum to zero; :

Scaling function integrates to unity; :

In particular, :

Wavelet function integrates to zero; :

Wavelet function is orthogonal to the scaling function at the same scale; :

The lowpass and highpass filter coefficients are orthogonal; :

has n vanishing moment; :

This means linear signals are fully represented in the scaling functions part ({0}):

Quadratic or higher-order signals are not: 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:

The higher the order n, the flatter the response function at the ends:

Fourier transform of is given by :

Frequency response for is given by :

The filter is a highpass filter:

The higher the order n, the flatter the response function at the ends:

Fourier transform of is given by :

## Neat Examples(2)

Plot translates and dilations of scaling function:

Plot translates and dilations of wavelet function: