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

# DaubechiesWavelet

 DaubechiesWaveletrepresents a Daubechies wavelet of order . DaubechiesWavelet[n] represents a Daubechies wavelet of order n.
• The scaling function () and wavelet function () have compact support length of 2n. The scaling function has n vanishing moments.
Scaling function:
Wavelet function:
Filter coefficients:
Scaling function:
 Out[1]=
 Out[2]=

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

Filter coefficients:
 Out[1]=
 Scope   (14)
Compute primal low-pass filter coefficients:
Primal high-pass 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:
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:
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:
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   (3)
Approximate a function using Haar wavelet coefficients:
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:
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, :
Wavelet function integrates to zero; :
Wavelet function is orthogonal to the scaling function at the same scale; :
The low-pass and high-pass filter coefficients are orthogonal; :
DaubechiesWavelet[n] has n vanishing moment; :
This means linear signals are fully represented in the scaling functions part ():
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 low-pass 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 high-pass filter:
The higher the order n, the flatter the response function at the ends:
Fourier transform of is given by :
Plot translates and dilations of scaling function:
Plot translates and dilations of wavelet function:
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