This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)

SymletWavelet

SymletWavelet
represents the Symlet wavelet of order 4.
SymletWavelet[n]
represents the Symlet wavelet of order n.
  • SymletWavelet, also known as "least asymmetric" wavelet, defines a family of orthogonal wavelets.
  • 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:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
 
Wavelet function:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
 
Filter coefficients:
In[1]:=
Click for copyable input
Out[1]=
Compute primal low-pass filter coefficients:
Primal high-pass filter coefficients:
Lifting filter coefficients:
Generate a function to compute lifting wavelet transform:
Symlet scaling function of order 4:
SymletWavelet of order 10:
Plot scaling function using different levels of recursion:
Symlet wavelet function of order 4:
SymletWavelet of order 10:
Plot scaling function using 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:
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:
Order 1 SymletWavelet 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; :
Order n of SymletWavelet indicates n vanishing moments; :
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 :
SymletWavelet is restricted to n less than 20:
SymletWavelet is not defined when n is not a positive machine integer:
Plot translates and dilations of scaling function:
Plot translates and dilations of wavelet function:
New in 8