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

MeyerWavelet

MeyerWavelet
represents the Meyer wavelet of order 3.
MeyerWavelet[n]
represents the Meyer wavelet of order n evaluated on the equally spaced interval .
MeyerWavelet
represents the Meyer wavelet of order n evaluated on the equally spaced interval .
  • MeyerWavelet is defined for any positive integer n and real limit lim.
  • The scaling function () and wavelet function () have infinite support. The functions are symmetric.
  • The scaling function () is given by its Fourier transform as .  »
  • The wavelet function () is given by its Fourier transform as .
  • The polynomial is a polynomial of the form , where is the order of the Meyer wavelet.
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:
Meyer scaling function of order 3:
Meyer scaling function of order 10:
Meyer wavelet function of order 3:
Meyer wavelet function of order 10:
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:
MeyerWavelet 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:
Multivariate scaling and wavelet functions are products of univariate ones:
Low-pass filter coefficients approximately sum to unity; :
High-pass filter coefficients approximately sum to zero; :
Scaling function integrates to unity; :
Wavelet function integrates to zero; :
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:
Frequency response for is given by :
The filter is a high-pass filter:
Fourier transform of is given by :
Compare the above result with the exact Fourier transform:
Fourier transform of is given by :
Compare the above result with the exact Fourier transform:
Plot translates and dilations of scaling function:
Plot translates and dilations of wavelet function:
New in 8