Defining Your Own Wavelet
You can define wavelets to plug into the wavelet analysis framework by using the correct template. A wavelet
wave is of the form
, where
wfam is the symbol that indicates the wavelet family and
args provide any necessary specification.
In order to set
wfam as a wavelet family recognized by the system, the property
must be set to
True, where
patt is a pattern that matches acceptable arguments
args.
| WaveletQ | set to True if the symbol is a user wavelet |
Wavelet initialization property.
Both orthogonal and biorthogonal user wavelets are supported. Orthogonal wavelets are indicated by setting the property
and, correspondingly, biorthogonal wavelets are set using the property
.
| "OrthogonalQ" | set to True if the wavelet is orthogonal |
| "BiorthogonalQ" | set to True if the wavelet is biorthogonal |
Wavelet properties.
To compute primal low-pass filter coefficients, the property
must be set; here
prec indicates the precision of filter coefficients. Similarly, to compute dual low-pass filter coefficients, the property
must be set. Both properties
and
are expected to return a list of the form
, where
is the index and
is the corresponding filter coefficient. If a list of the form
is returned, it is assumed that index
starts from 0. An error message is generated if the filter coefficients
are not numeric and indices
are not integers.
Examples
Franklin Wavelet
Define a family of Franklin wavelets.
Franklin wavelet coefficients are given by the integral.
The above definition of a user wavelet can now be used to perform various wavelet operations.
Compute the filter coefficients.
| Out[10]= |  |
The scaling function is computed using the recursive equation
, where
represents the low-pass filter coefficients.
Compute the scaling function.
| Out[11]= |  |
Perform a wavelet transform.
| Out[12]= |  |
| Out[13]= |  |
Legendre Wavelet
Following is an example of the Legendre wavelet.
Although Legendre wavelets are not orthogonal, to be able to perform a wavelet transform, you need to set it to
True.
Specify the function to compute low-pass filter coefficients.
Compute the filter coefficients.
| Out[12]= |  |
| Out[13]= |  |
Compute the scaling function.
| Out[15]= |  |
The wavelet function is computed using the recursive equation
, where
represents the high-pass filter coefficients.
Compute the wavelet function.
| Out[17]= |  |
Perform a wavelet transform.
| Out[153]= |  |
| Out[154]= |  |
Le Gall Wavelet
Generate a Le Gall wavelet.
Define
and
properties.
Use the Le Gall wavelet for thresholding.
| Out[7]= |  |
| Out[8]= |  |
| Out[9]= |  |
