# CoifletWavelet

represents a Coiflet wavelet of order 2.

represents a Coiflet wavelet of order n.

# Details • CoifletWavelet defines a family of orthogonal wavelets.
• is defined for positive integers n between 1 and 5.
• The scaling function ( ) and wavelet function ( ) have compact support of length . The scaling function has vanishing moments and wavelet function has vanishing moments.
• CoifletWavelet can be used with such functions as DiscreteWaveletTransform, WaveletPhi, WaveletPsi, etc.

# Examples

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## Basic Examples(3)

Scaling function:

Wavelet function:

Filter coefficients:

## Scope(12)

### Basic Uses(7)

Compute primal lowpass filter coefficients:

Primal highpass filter coefficients:

Lifting filter coefficients:

Generate a function to compute a lifting wavelet transform:

Coiflet scaling function of order 1:

Coiflet scaling function of order 4:

Plot scaling function at different refinement scales:

Coiflet wavelet function of order 1:

Coiflet wavelet of order 4:

Plot wavelet function at different refinement scales:

### Wavelet Transforms(4)

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:

### Higher Dimensions(1)

Multivariate scaling and wavelet functions are products of univariate ones:

## Applications(3)

Approximate a function using Haar 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(11)

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; : 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 :

## Possible Issues(1)

CoifletWavelet is restricted to n less than 5: CoifletWavelet is not defined when n is not a positive machine integer: ## Neat Examples(2)

Plot translates and dilations of scaling function:

Plot translates and dilations of wavelet function: