# HaarWavelet

represents a Haar wavelet.

# Details • HaarWavelet defines a family of orthonormal wavelets.
• The scaling function ( ) and wavelet function ( ) have compact support lengths of 1. They have 1 vanishing moment and are symmetric.
• The scaling function ( ) is given by . »
• The wavelet function ( ) is given by . »
• HaarWavelet 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(10)

### Basic Uses(4)

Compute primal lowpass filter coefficients:

Primal highpass filter coefficients:

Lifting filter coefficients:

Generate function to compute lifting wavelet transform:

### Wavelet Transforms(5)

Compute a DiscreteWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

HaarWavelet can be used to perform a DiscreteWaveletPacketTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

HaarWavelet can be used to perform a StationaryWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

HaarWavelet can be used to perform a StationaryWaveletPacketTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

HaarWavelet can be used to perform 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(4)

Approximate a function using Haar wavelet coefficients:

Perform a LiftingWaveletTransform:

Approximate original data by keeping 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:

Compare range and distribution of wavelet coefficients:

Plot distribution of wavelet coefficients:

Compare with wavelet coefficients plotted along a common axis:

## Properties & Relations(15)

is equivalent to HaarWavelet:

Lowpass filter coefficients sum to unity; :

Highpass filter coefficients sum to zero; :

Scaling function integrates to unity; :

In particular, :

Haar scaling function is orthogonal to its shift; :

Wavelet function integrates to zero; :

Haar wavelet function is orthogonal to its shift; :

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

The lowpass and highpass filter coefficients are orthogonal; :

HaarWavelet has one vanishing moment; :

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

Linear or higher-order signals are not: satisfies the recursion equation :

Symbolically verify recursion:

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 highpass filter:

Fourier transform of is given by :

## Neat Examples(2)

Plot translates and dilations of scaling function:

Plot translates and dilations of wavelet function: