HaarWavelet

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)

DaubechiesWavelet[1] 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:

Introduced in 2010
 (8.0)