Wavelet Toolbox

Choose the Wavelet Family Full Name

The full name of the wavelet family, fn, must be a string. Predefined wavelet family names are Haar, Daubechies, Symlets, Coiflets, BiorSplines, ReverseBior, Meyer, DMeyer, Gaussian, Mexican_hat, Morlet, Complex Gaussian, Shannon, Frequency B-Spline, and Complex Morlet.

Choose the Wavelet Family Short Name

The short name of the wavelet family, fsn, must be a string of four characters or less. Predefined wavelet family short names are haar, db, sym, coif, bior, rbio, meyr, dmey, gaus, mexh, morl, cgau, fbsp, and cmor.

Determine the Wavelet Type

We distinguish five types of wavelets:

1. Orthogonal wavelets with FIR filters

1. These wavelets can be defined through the scaling filter w. Predefined families of such wavelets include Haar, Daubechies, Coiflets, and Symlets.

2. Biorthogonal wavelets with FIR filters

1. These wavelets can be defined through the two scaling filters wr and wd, for reconstruction and decomposition respectively. The BiorSplines wavelet family is a predefined family of this type.

3. Orthogonal wavelets without FIR filter, but with scale function

1. These wavelets can be defined through the definition of the wavelet function and the scaling function. The Meyer wavelet family is a predefined family of this type.

4. Wavelets without FIR filter and without scale function

1. These wavelets can be defined through the definition of the wavelet function. Predefined families of such wavelets include Morlet and Mexican_hat.

5. Complex wavelets without FIR filter and without scale function

1. These wavelets can be defined through the definition of the wavelet function. Predefined families of such wavelets include Complex Gaussian and Shannon.

Define the Orders of Wavelets Within the Given Family

If a family contains many wavelets, the short name and the order are appended to form the wavelet name. Argument nums is a string containing the orders separated with blanks. This argument is not used for wavelet families that only have a single wavelet (Haar, Meyer, and Morlet for example).

For example, for the first Daubechies wavelets,

• fsn = 'db'
  nums = '1 2 3'
  

yield the three wavelets db1, db2, and db3.

For the first BiorSplines wavelets,

• fsn = 'bior'
  nums = '1.1 1.3 1.5 2.2'
  

yield the four wavelets bior1.1, bior1.3, bior1.5, and bior2.2.

Build a MAT-File or M-File

The wavemngr command requires a file argument, which is a string containing a MAT-file or M-file name.

If a family contains many wavelets, an M-file must be defined and must be of a specific form that depends on the wavelet type. The specific M-file formats are described in the remainder of this section.

If a family contains a single wavelet, then a MAT-file can be defined for wavelets of type 1. It must have the wavelet family short name (fsn) argument as its name and must contain a single variable whose name is fsn and whose value is the scaling filter. An M-file can also be defined as discussed below.

Type 1 (Orthogonal with FIR Filter)

The syntax of the first line in the M-file must be

• function w = file(wname)
  

where the input argument wname is a string containing the wavelet name, and the output argument w is the corresponding scaling filter.

The filter w must be of even length; otherwise, it is zero-padded by the toolbox.

For predefined wavelets, the scaling filter is of sum 1. For a new wavelet, the normalization is free (except 0 of course) since the toolbox uses a suitably normalized version of this filter.

Examples of such M-files for predefined wavelets are dbwavf.m for Daubechies, coifwavf.m for coiflets, and symwavf.m for symlets.

Type 2 (Biorthogonal with FIR Filter)

The syntax of the first line in the M-file must be

• function [wr,wd] = file(wname)
  

where the input argument wname is a string containing the wavelet name and the output arguments wr and wd are the corresponding reconstruction and decomposition scaling filters, respectively.

The filters wr and wd must be of the same even length. In general, initial biorthogonal filters do not meet these requirements, so they are zero-padded by the toolbox.

For predefined wavelets, the scaling filters are of sum 1. For a new wavelet, the normalization is free (except 0 of course) since the toolbox uses a suitably normalized version of these filters.

The M-file biorwavf.m (for BiorSplines) is an example of an M-file for a type 2 predefined wavelet family.

Type 3 (Orthogonal with Scale Function)

The syntax of the first line in the M-file must be

• function [phi,psi,t] = file(lb,ub,n,wname)
  

which returns values of the scaling function phi and of the wavelet function psi on t, a regular n-point grid of the interval [lb ub].

The argument wname is optional (see Note below).

The M-file meyer.m is an example of an M-file for a type 3 predefined wavelet family.

Type 4 or Type 5 (No FIR Filter; No Scale Function)

The syntax of the first line in the M-file must be

• function [psi,t] = file(lb,ub,n,wname)
  

or

• function [psi,t] = file(lb,ub,n,wname, "additional arguments")
  

which returns values of the wavelet function psi on t, a regular n-point grid of the interval [lb ub].

The argument wname is optional (see Note below).

Examples of type 4 M-files for predefined wavelet families are mexihat.m (for Mexican_hat) and morlet.m (for Morlet).

Examples of type 5 M-files for predefined wavelet families are shanwavf.m (for Shannon) and cmorwavf.m (for Complex Morlet).

Note    For the types 3, 4, and 5, the wname argument can be optional. It is only required if the new wavelet family contains more than one wavelet and if you plan to use this new family in the GUI mode. For the types 4 and 5, a complete example of using the "additional arguments" can be found looking at the reference page for the fbspwavf function.

Define the Effective Support

This definition is required only for wavelets of types 3, 4, and 5, since they are not compactly supported.

Defining the effective support means specifying an upper and lower bound. For example, for some predefined wavelet families, we have the following.

Family	Lower Bound (lb)	Upper Bound (ub)
Meyer	-8	8
Mexican_hat	-5	5
Morlet	-4	4

Preparing to Add a New Wavelet Family

Adding a New Wavelet Family

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