This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# HaarWavelet

 HaarWavelet represents a Haar wavelet.
• 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 . »
Scaling function:
Wavelet function:
Filter coefficients:
Scaling function:
 Out[1]=
 Out[2]=

Wavelet function:
 Out[1]=
 Out[2]=

Filter coefficients:
 Out[1]=
 Scope   (10)
Compute primal low-pass filter coefficients:
Primal high-pass filter coefficients:
Lifting filter coefficients:
Generate function to compute lifting wavelet transform:
View the tree of wavelet coefficients:
Get the dimensions of wavelet coefficients:
Plot the wavelet coefficients:
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:
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:
Multivariate scaling and wavelet functions are products of univariate ones:
 Applications   (4)
Approximate a function using Haar wavelet coefficients:
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:
DaubechiesWavelet is equivalent to HaarWavelet:
Low-pass filter coefficients sum to unity; :
High-pass 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 low-pass and high-pass filter coefficients are orthogonal; :
HaarWavelet has one vanishing moment; :
This means constant signals are fully represented in the scaling functions part ():
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 low-pass filter:
Fourier transform of is given by :
Frequency response for is given by :
The filter is a high-pass filter:
Fourier transform of is given by :
Plot translates and dilations of scaling function:
Plot translates and dilations of wavelet function:
New in 8