This section defines all available wavelets in this library. Since different wavelets have different properties, the functions for different wavelets require different input parameters and generate different output data. All available wavelets in this library with the parameters are listed in the following table.
Parameter, Nd, specifies the wavelet order. Nr is only used for biorthogonal wavelets, where Nd is for decomposition and Nr for reconstruction.
For the wavelets with theoretically unlimited support, e.g. Meyer, Gaussian, etc., the function range is set as effective support, [-es, es], if 'range' is not specified (==0).
Parameters, fb and fc, are used for some complex wavelets. These are the so-called frequency parameters defining the oscillation properties of the wavelets.
'phi' stands for scaling function, and 'psi' for wavelet function. A biorthogonal wavelet has two scaling and wavelet functions. The first function set is for forward transformation (decomposition); the second set is for inverse transformation (reconstruction).
Symbol, O, means the corresponding parameter is applicable, - not applicable, for a specific wavelet.
For generating Meyer wavelets, an external C-function saved in fft_c.c file has to be called. This C-file has to be available in the current working directory or the searching path of Modelica must include the directory of fft_c.c file.
| Wavelet | wavID | wavType | Nd | Nr | es | range | fb | fc | phi1 | psi1 | phi2 | psi2 |
| Haar | 1 | 1 | - | - | - | - | - | - | O | O | - | - |
| Daubechies | 2 | 1 | 1..20 | - | - | - | - | - | O | O | - | - |
| Symlets | 3 | 1 | 1..20 | - | - | - | - | - | O | O | - | - |
| Coiflets | 4 | 1 | 1..5 | - | - | - | - | - | O | O | - | - |
| Biorthogonal spline | 5 | 2 | 1..6 | 1..9 | - | - | - | - | O | O | O | O |
| Reverse biorthogonal spline | 6 | 2 | 1..9 | 1..6 | - | - | - | - | O | O | O | O |
| Discrete Meyer | 7 | 3 | - | - | 8 | >=0 | - | - | O | O | - | - |
| Meyer | 8 | 3 | - | - | 8 | >=0 | - | - | O | O | - | - |
| Gaussian | 9 | 4 | 1..8 | - | 5 | >=0 | - | - | - | O | - | - |
| Mexican hat | 10 | 4 | - | - | 5 | >=0 | - | - | - | O | - | - |
| Morlet | 11 | 4 | - | - | 4 | >=0 | - | - | - | O | - | - |
| Complex Gaussian | 12 | 5 | 1..8 | - | 5 | >=0 | - | - | - | O | - | - |
| Complex Morlet | 13 | 5 | - | - | 4 | >=0 | >0 | >0 | - | O | - | - |
| Complex Shannon | 14 | 5 | - | - | 20 | >=0 | >0 | >0 | - | O | - | - |
| Complex frequency B-spline | 15 | 5 | 1,2,... | - | 20 | >=0 | >0 | >0 | - | O | - | - |
For biorthogonal and reverse biorthogonal wavelets, only some combinations of the orders are available. For details, refer to the functions, wavBiorSpline and wavRevBiorSpline.
All wavelets are categorized into five types. They are defined as follows:
| Type | Description |
| 1 | Orthogonal wavelets |
| 2 | Biorthogonal wavelets |
| 3 | Non-(bi)orthogonal wavelets with scaling function |
| 4 | Non-(bi)orthogonal wavelets without scaling function |
| 5 | Complex wavelets (without scaling function) |
| Name | Description |
|---|---|
| wavFunc | Returns the filter bank and functions of a specific wavelet family |
| scalingWaveFunc | Generate the scaling and wavelet functions using wavelet filters |
| wavHaar | Haar wavelet filters |
| wavDaubechies | Daubechies wavelet filters |
| wavSymlets | Symlets wavelet filters |
| wavCoiflets | Coiflets wavelet filters |
| wavBiorSpline | Biorthogonal spline wavelet filters |
| wavRevBiorSpline | Reverse biorthogonal spline wavelet filters |
| wavDMeyer | Discrete Meyer wavelet filters |
| wavMeyer | Meyer wavelet function. The fft_c.c file must be accessible. |
| wavGaussian | Gaussian wavelet function |
| wavMexHat | Mexican hat wavelet function |
| wavMorlet | Morlet wavelet function |
| wavXGaussian | Complex Gaussian wavelet function |
| wavXMorlet | Complex Morlet wavelet function |
| wavXShannon | Complex Shannon wavelet function |
| wavXFreqBSpline | Complex Frequency B-Spline wavelet |