Wavelet

Information

The package Wavelet is a function library developed for Modelica to carry out wavelet transform and related calculations. It is used for the post processing of the simulation results or other data. Online operation, meaning execution of the functions during the simulation process, is not possible.

The numerical calculation of wavelet transform requires that the data be sampled in equidistant time grids. If non-equidistant time grids are used, the data should be firstly converted to equidistant using function interpL in this library. If the data are to be generated by Dymola with model simulation, it is suggested that Equidistant time grid should be checked in the Simulation window through the menu Simulation/Setup.../Output/Output selection.

Most functions in this library do not check the input parameters. It is the task of the user to provide valid parameters. Upon wrong input parameters, unexpected errors might occur.

This library is developed and tested with the demo version of Dymola 2013.

List of the packages:

Detailed information is to be found in each package.

Required libraries:

* Modelica_LinearSystems2: This library is used for displaying the curves.
* Plot3D: This is used for showing the images generated by wavelet continuous transform (CWT). The library Plot3D is delivered with Dymola. In the Dymola demo version, the functionality of Plot3D seems to be limited, such that the 3D surfaces can only be displayed in a 2D manner.

License:

All files in this directory (Wavelet) and in all subdirectories, especially all files that build package "Wavelet" are released under the Modelica License 2.

Developers:

Dr. Michael Gao, michael.gao@tum.de
Mr. Qipeng Hu, qipeng.hu@gmail.com
Mr. Weihua Wang
Mr. Hanko Ipach

Acknowlegement:

Package Content

Name	Description
Examples	Some examples to use this Wavelet Library
MRA	Application: One-dimensional multi-resolution analysis
Denoising	Application: One dimensional signal denoising
Transform	Functions for wavelet transform
Families	Functions about wavelet families
General	General functions
Records	Records, mainly used for graphic user interface
Types	Definitions of enumeration values

Wavelet.Examples

Information

This section provides several examples. They serve as a starting point for the users to start their work with this library. Most examples can be accessed through a graphic interface for setting input parameters and displaying results.

To execute the examples that read simulation result data from MAT files, the two Modelica models (testSigna1 and testSignal2) have to be firstly run in Dymola, so that the MAT files are generated and stored in the hard disk.

Package Content

Name	Description
displayWavelet	Display the wavelet available in this Wavelet Library
cwtChirpCurve	CWT of a chirp signal with outputs shown in curves
cwtChirpImage	CWT of a chirp signal with outputs shown with curves and images
fileData_cwtn	CWT of the simulation result read from a MAT file
fileDataMRA	MRA of the simulation result read from a MAT file
fileDataDenoising	Denoising of the simulation result read from a MAT file
testSignal1	Generates a test signal for analysis
testSignal2	Generates a test signal for analysis

Wavelet.Examples.displayWavelet

Information

This example displays the wavelet and scaling functions of a wavelet family in one diagram. For a wavelet that has no scaling function, only the wavelet function is displayed. For the complex wavelets, the real and imaginary parts are displayed with two curves. The user is able to select a wavelet by its name and set its parameters through a graphic interface.

Please refer to the description of wavelet families for the detailed information about the available wavelets in this library.

Because of the length of discrete Meyer wavelet, the display of it takes a long time.

Inputs

Type	Name	Default	Description
wavletDefinition	wp		Parameters for generating wavelet and scaling functions. See wavFunc() for detailed description of the parameters.

Wavelet.Examples.cwtChirpCurve

Information

This function carries out continuous wavelet transform of a chirp signal with graphic user interface. The data to be analyzed are given as default input values in the function. Bothe the original data and the transform result are shown with curves.

Since simple default values are provided with this function, a direct execution of this function without any extra input parameters will deliver a complete MRA analysis for an example.

Refer to the description of Families for detailed information about the available wavelets.

Inputs

Type	Name	Default	Description
Real	signal[:]	{1.0000,1.0000,0.9999,0.9996...	Vector, the signal to be analyzed
Real	scales[:]	{2,8,32}	Scales
wavletDefinition	wIn		Input parameters for wavelet filters and functions

Wavelet.Examples.cwtChirpImage

Information

This function carries out continuous wavelet transform of a chirp signal with graphic user interface. The data to be analyzed are given as default input values in the function. The original data are shown with a curve. The transform result will be shown in an image with pseudo-color. For showing this image, the 'Plot3D' library has to be available.

Since simple default values are provided with this function, a direct execution of this function without any extra input parameters will deliver a complete MRA analysis for an example.

Refer to the description of Families for detailed information about the available wavelets.

Inputs

Type	Name	Default	Description
Real	signal[:]	{1.0000,1.0000,0.9999,0.9996...	Vector, the signal to be analyzed
Real	scales[:]	{i for i in 1:16}	Scales
wavletDefinition	wIn		Input parameters for wavelet filters and functions

Wavelet.Examples.fileData_cwtn

Information

This function carries out continuous wavelet transform with graphic user interface. The data to be analyzed are to be read from the *.MAT data file that is generated by Dymola. The original data are shown in a curve. The transform result is shown in an image with pseudo-color. For showing this image, the 'Plot3D' library has to be available.

Since simple default values are provided with this function, a direct execution of this function without any extra input parameters will deliver a complete MRA analysis for an example. Due to large computational amount, this example will take some time (usually below 30 seconds) to show the transform results.

Refer to the description of Families for detailed information about the available wavelets.

Inputs

Type	Name	Default	Description
Input data
String	fileName	"testSignal2.mat"	The MAT file containing the signal to be analyzed
String	signalName	"y"	The signal name to be analyzed
Real	t0	0.0	Start time of the signal to be analyzed
Real	t1	1.0	End time of the signal to be analyzed
Real	fs	500.0	Sampling frequency of the signal for analysis. This is used to generate an equidistant time grid with the expected time step. If fs == 0, the simulation time grid will be used, even if it is none-equidistant. Since wavelet analysis has to be done in equidistant time grid, non-equidistant time grids usually cause incorrect results.
Real	scales[:]	{i for i in 1:4:64}	Scales
wavletDefinition	wIn		Input parameters for wavelet filters and functions

Wavelet.Examples.fileDataMRA

Information

This function carries out the complete wavelet MRA with graphic user interface. The data to be analyzed are to be read from the *.MAT data file that is generated by Dymola. Please note that only the wavelets of types 1, 2 and 3 are possible for MRA. Otherwise running errors will occur. The results are displayed with multiple curves. It is possible to show either the wavelet coefficients at different levels or the signals that are reconstructed with the coefficients of different single levels.

Since simple default values are provided with this function, a direct execution of this function without any extra input parameters will deliver a complete MRA analysis for an example.

Refer to the description of Families for detailed information about the available wavelets.

Inputs

Type	Name	Default	Description
Input data
String	fileName	"testSignal2.mat"	The MAT file containing the signal to be analyzed
String	signalName	"y"	The signal name to be analyzed
Real	t0	0.0	Start time of the signal to be analyzed
Real	t1	1.0	End time of the signal to be analyzed
Real	fs	200.0	Sampling frequency of the signal for analysis. This is used to generate an equidistant time grid with the expected time step. If fs == 0, the simulation time grid will be used, even if it is none-equidistant. Since wavelet analysis has to be done in equidistant time grid, non-equidistant time grids usually cause incorrect results.
wavletDefinition	wd		Wavelet definition. Note: Only (bi)orthogonal wavelets can be used for MRA. Valid parameters are wavID (<=7), Nd, Nr and range (dMeyer)
mraParameters	mp		Parameters for MRA

Wavelet.Examples.fileDataDenoising

Information

This function carries out denoising operation to the one-dimensional data read from a specified *.MAT file, which is a simulation database generated by Dymola. The original data, the decomposed data in different levels, the denoised data in all levels and the denoised data are displayed with diagrams. In addition, the noise that is removed from the signal is displayed in a diagram, too.

The default input values provide an example to show the functionality of this function.

Refer to the description of Families for detailed information about the available wavelets.

Inputs

Type	Name	Default	Description
Input data
String	fileName	"testSignal2.mat"	The MAT file containing the signal to be analyzed
String	signalName	"y"	The signal name to be analyzed
Real	t0	0.0	Start time of the signal to be analyzed
Real	t1	1.0	End time of the signal to be analyzed
Real	fs	200.0	Sampling frequency of the signal for analysis. This is used to generate an equidistant time grid with the expected time step. If fs == 0, the simulation time grid will be used, even if it is none-equidistant. Since wavelet analysis has to be done in equidistant time grid, non-equidistant time grids usually cause incorrect results.
wavletDefinition	wd		Wavelet definition. Note: Only (bi)orthogonal wavelets can be used for MRA. Valid parameters are wavID (<=7), Nd, Nr and range (dMeyer)
denoisParameters	dp		Data and parameters for wavelet denoising

Outputs

Type	Name	Description
Real	y[:]	Denoised signal
Real	noise[:]	Noise removed from signal
Real	c[:]	Original wavelet coefficients of all levels, details see wavDec()
Real	cth[:]	Wavelet coefficients of all levels after applying thresholds
Integer	len[:]	Lengths of coefficients for each levels, details see wavDec()

Wavelet.Examples.testSignal1

Information

This simple model generates a sum of three sinusoidal signals with different frequencies and magnitudes.

After running this model in Dymola, all simulation data are saved in testSignal1.MAT file in the hard disk.

Extends from Modelica.Blocks.Interfaces.SignalSource (Base class for continuous signal source).

Parameters

Type	Name	Default	Description
Real	offset	0	Offset of output signal y
Time	startTime	0	Output y = offset for time < startTime [s]

Connectors

Type	Name	Description
output RealOutput	y	Connector of Real output signal

Wavelet.Examples.testSignal2

Information

This simple model generates a sum of sinusoidal signals with different frequencies and magnitudes.

After running this model in Dymola, all simulation data are saved in testSignal2.MAT file in the hard disk.

Extends from Modelica.Blocks.Interfaces.SignalSource (Base class for continuous signal source).

Parameters

Type	Name	Default	Description
Real	offset	0	Offset of output signal y
Time	startTime	0	Output y = offset for time < startTime [s]

Connectors

Type	Name	Description
output RealOutput	y	Connector of Real output signal

Wavelet.MRA

Information

The one dimensional wavelet Multi-Resolution-Analysis (MRA) toolbox is an application function group created on the wavelet transform base library. Wavelet MRA carries out an N-level wavelet decomposition to obtain the wavelet coefficients in N detail levels and one approximation level. The coefficients of every single level are then used to reconstruct the signal separately. The result is a series of signals, each of which represents a certain frequency range of the original signal. In this way, the original signal can be observed in different (N+1) resolutions.

This toolbox provides two main functions for MRA. Function mra carries out numerical calculation and outputs MRA results in a matrix. Function mraGUI provides a graphic user interface to directly display the analysis results. It provides a further possibility to show the wavelet coefficients in every singles levels.

Only orthogonal, biorthogonal and discrete Meyer wavelets are possible for MRA. Please refer to the description of wavelet families for the detailed information about the available wavelets in this library.

Package Content

Name	Description
mraGUI	Graphic user interface for multi-resolution analysis (MRA)
mra	Wavelet multi-resolution analysis (MRA)Graphic user interface for multi-resolution analysis (MRA)
tuneCoef	Tune the wavelet coefficients of all decomposition levels

Wavelet.MRA.mraGUI

Information

This function carries out the complete wavelet MRA with graphic user interface. Please note that only the wavelets of types 1, 2 and 3 are possible for MRA. Otherwise running errors will occur. The results are displayed with multiple curves. It is possible to show either the wavelet coefficients at different levels or the signals that are reconstructed with the coefficients of different single levels.

Since simple default values are provided with this function, a direct execution of this function without any extra input parameters will deliver a complete MRA analysis for an example.

Refer to the description of Families for detailed information about the available wavelets.

Inputs

Type	Name	Default	Description
Input data
Real	u[:]	{0.9594,3.6278,7.6766,8.3415...	Signal to be analyzed
wavletDefinition	wd		Wavelet definition. Note: Only (bi)orthogonal wavelets can be used for MRA. Valid parameters are wavID (<=7), Nd, Nr and range (dMeyer)
mraParameters	mp		Parameters for MRA

Wavelet.MRA.mra

Information

This function carries out the complete wavelet MRA and outputs decomposed signals in all levels after coefficient tuning. Please note that only the wavelets of types 1 (orthogonal), 2 (bi-orthogonal) and 3 (discrete Meyer) are possible for MRA. Otherwise running errors will occur.

Since default values are provided with this function, a direct execution of this function without extra input parameters will deliver a simple example.

Refer to the description of Families for detailed information about the available wavelets.

Inputs

Type	Name	Default	Description
Input data
Real	u[:]	{0.9594,3.6278,7.6766,8.3415...	Signal to be analyzed
wavletDefinition	wd		Wavelet definition. Note: Only (bi)orthogonal wavelets can be used for MRA. Valid parameters are wavID (<=7), Nd, Nr and range (dMeyer)
mraParameters	mp		Parameters for MRA

Outputs

Type	Name	Description
Real	y[:, :]	Reconstructed signals with tuned coefficients in all levels. y[1,:] -- reconstructed signal, y[2,:] -- 1st detail level, y[3,:] -- 2nd detail level, ..., y[N+1,:] -- N-th detail level, y[N+2,:] -- approximation level, where N is the number of the decomposition levels.

Wavelet.MRA.tuneCoef

Information

This function supports wavelet MRA. After wavelet (multi-level) decomposition, the coefficients are usually tuned in order to change the signal magnitude at different levels. This function is able to freely change the magnitude of the coefficients at each level by multiplying them with a given factor, which could be any real value.

The definition of c[:] and len[:] are the same as in the function wavDec().

Inputs

Type	Name	Default	Description
Real	c0[:]	{-0.707,-0.707,-0.707,-0.706...	Original wavelet coefficients of all levels
Integer	len[:]	{13,7,4,2,2}	Lengths of wavelet coefficients for each levels
Real	ratio[:]	{1,0.1,2,-1}	{rD1, rD2, ..., rDN, rAN}, tuning factors of all levels.

Outputs

Type	Name	Description
Real	c1[size(c0, 1)]	Tuned wavelet coefficients of all levels

Wavelet.Denoising

Information

The wavelet denoising toolbox is an application function group created on the wavelet transform base library. Wavelet denoising is based on the consideration that the noise energy in the data is significantly lower than the signal energy, or the part of the data that contains useful information. In this case, the noise is mainly represented with the wavelet coefficients that have small magnitudes. If these coefficients are removed (set to be zeros) and the signal is reconstructed, the noise will be eliminated from the original data. Therefore, wavelet denoising is only suitable for the cases where the signal-to-noise ratio is high.

The thresholds to decide which coefficients are to be removed are usually empirically determined based on the data to be analyzed. However, there are methods (the function thSelect) to automatically select the thresholds by analyzing the input data if the noise has normal distribution property. It has to be noted that the thresholds calculated by thSelect in this toolbox are valid only if the noise has normal (Gaussian) distribution with zero mean and standard deviation of one! Otherwise, the data have to be pre-conditioned or the thresholds have to be manually selected by the user.

The same data processing for denoising could also be used for data compression. The idea is based on such consideration: The wavelet coefficients with small magnitudes contain little information about the original signal. By removing these coefficients, the data amount can be greatly reduced while not much information is lost. If a suitable wavelet is used, most coefficients might have very small magnitudes. Thus, a large compression ratio can be achieved.

Only orthogonal, biorthogonal or discrete Meyer wavelets can be used for wavelet denoising.

Package Content

Name	Description
denoisGUI	Graphic user interface for 1D wavelet denoising
denois	1D wavelet denoising
thSelect	Calculates threshold value for denoising
thApply	Apply soft or hard thresholding

Wavelet.Denoising.denoisGUI

Information

This function is a modified version of the function denois. It provides graphic user interface (GUI) for parameter input and displaying denoising results. Instead of filter banks, the wavelet is specified with its name. The original data, the decomposed data in different levels, the denoised data in all levels and the denoised data are displayed with diagrams. In addition, the noise that is removed from the signal is displayed in a diagram, too.

The default input values provide an example to show the functionality of this function.

Inputs

Type	Name	Default	Description
Input data
Real	u[:]	{-1.3853,1.5788,2.2041,3.982...	Signal to be analyzed
wavletDefinition	wd		Wavelet definition. Note: Only (bi)orthogonal wavelets can be used for MRA. Valid parameters are wavID (<=7), Nd, Nr and range (dMeyer)
denoisParameters	dp		Data and parameters for wavelet denoising

Outputs

Type	Name	Description
Real	y[:]	Denoised signal
Real	noise[:]	Noise removed from signal
Real	c[:]	Original wavelet coefficients of all levels, details see wavDec()
Real	cth[:]	Wavelet coefficients of all levels after applying thresholds
Integer	len[:]	Lengths of coefficients for each levels, details see wavDec()

Wavelet.Denoising.denois

Information

This function carries out the wavelet denoising calculation for a data. The wavelet filter bank must be available for calling this function. This has to be obtained by wavelet filter functions.

It has to be noted that the thresholds calculated by thSelect in this toolbox are valid only if the noise has normal (Gaussian) distribution with zero mean and standard deviation of one! Otherwise, the data have to be pre-conditioned or the thresholds have to be manually selected by the user.

The manual selection of the thresholds has the possibility that the thresholds for the levels can be different. The thresholds are given in the input parameter, th[:], each element of which is used for one level. If a threshold for a level is less than zero, the default threshold automatically calculated by thSelect will be used.

Inputs

Type	Name	Default	Description
Real	u[:]	{-1.3853,1.5788,2.2041,3.982...	Data to be denoised
Integer	decLevel	2	Wavelet decomposition levels
Real	lod[:]	{0.7071067811865476,0.707106...	Wavelet low pass filter for decomposition
Real	hid[:]	{-0.7071067811865476,0.70710...	Wavelet high pass filter for decomposition
Real	lor[:]	{0.7071067811865476,0.707106...	Wavelet low pass filter for reconstruction
Real	hir[:]	{0.7071067811865476,-0.70710...	Wavelet high pass filter for reconstruction
Real	th[:]	{-1,-1}	Thresholds for all decomposition levels with th[1] for the first detail level (highest frequency components)
Boolean	sorh	true	Soft (false) or hard (true) thresholding for all decomposition levels
threshMethod	thMethod	1	Method for automatic threshold calculation

Outputs

Type	Name	Description
Real	y[:]	Denoised data
Real	noise[:]	Noise removed from the data, = u-y
Real	c[:]	Wavelet coefficients before denoising
Real	cth[:]	Wavelet coefficients after denoising
Integer	len[:]	Length of wavelet coefficients of all levels

Wavelet.Denoising.thSelect

Information

This function analyzes the input data and calculates a common threshold for wavelet denoising for all detail levels using the specified methods. Four methods are available:

1. Fixed form: The threshold is calculated as sqrt(2*log(size(u,1))).

2. SURE: The threshold is calculated based on Stein's Unbiased Risk Estimate (SURE) of risk, which is a quadratic loss function.

3. Heuristic SURE: This is a heuristic version of the SURE method. It is actually a combination of the first two methods. For the signal with strong noises, the fixed form method is used. Otherwise, SURE method id used.

4. Minimal maximum: The threshold is estimated to realize the minimum of the maximum mean square error of the observed data. This method is derived from statistic theory.

It has to be noted that the thresholds calculated by thSelect in this toolbox are valid only if the noise has normal (Gaussian) distribution with zero mean and standard deviation of one! Otherwise, it is recommended that the data are to be pre-conditioned or the thresholds be manually selected.

Inputs

Type	Name	Default	Description
Real	u[:]	{-1.3853,1.5788,2.2041,3.982...	Signal vector
threshMethod	thID	1	Threshold calculation method

Outputs

Type	Name	Description
Real	th	Threshold value

Wavelet.Denoising.thApply

Inputs

Type	Name	Default	Description
Real	u[:]	{0,1,2,3,4,5,6,7,8,9}	Input vector
Real	th	5	Threshold value
Boolean	sorh	true	Soft (false) or hard (true) thresholding

Outputs

Type	Name	Description
Real	y[:]	Vector after thresholding

Wavelet.Transform

Information

This library includes the following one-dimensional wavelet transforms and inverse transforms:

1. Continuous wavelet transform: cwt() and cwtn();

2. Discrete wavelet transform: dwt();

3. Inverse discrete wavelet transform: idwt();

4. Multi-level wavelet decomposition: wavDec();

5. Multi-level wavelet reconstruction: wavRec();

6. Single level reconstruction based on multi-level wavelet coefficients: wavRec1();

Package Content

Name	Description
cwt	One-dimensional continuous wavelet transform with a given wavelet function
cwtn	One-dimensional continuous wavelet transform with a given wavelet name
dwt	One-dimensional discrete wavelet transform (one-level decomposition)
idwt	One-dimensional inverse discrete wavelet transform (one-level reconstruction)
wavDec	Wavelet multilevel decomposition
wavRec	Wavelet multilevel reconstruction
wavRec1	Wavelet reconstruction using coefficients from a single level
wavCoef1	Extract the wavelet coefficients of a single level

Wavelet.Transform.cwt

Information

One-dimensional CWT transforms a vector to a two dimensional image. The first dimension (horizontal axis) is the same as the original data. The second dimension (vertical axis) is the scale, similar to frequency. The scales are defined by the input parameter, scales. Larger scale values correspond to lower frequencies.

Refer to the section of wavelet Families for the detailed information about the available wavelets.

The default input values provide a quick example with a simple saw-tooth signal.

Inputs

Type	Name	Default	Description
Real	u[:]	{1.0000,1.0000,0.9999,0.9996...	Signal to be analyzed
Real	scales[:]	{1,4,16}	Scales to be transformed
Real	w[:]	{1,-1}	Wavelet function for the transform
Real	x_w[:]	{0,1}	Regular grid of the wavelet function

Outputs

Type	Name	Description
Real	coefs[size(scales, 1), size(u, 1)]	A two dimensional matrix of wavelet coefficients. Each column corresponds to one element of the input parameter, scales.

Wavelet.Transform.cwtn

Information

This function carries out one-dimensional CWT transform, same as the function, cwt(). The only difference is that cwtn() receives the wavelet name as input.

Refer to the section of wavelet Families for the detailed information about the available wavelets.

The default input values provide a quick example with a simple saw-tooth signal.

Inputs

Type	Name	Default	Description
Input data
Real	u[:]	{1.0000,1.0000,0.9999,0.9996...	Signal to be analyzed
Real	scales[:]	{1,4,16}	Scales for the transformation
wavletDefinition	wIn		Wavelet definition

Outputs

Type	Name	Description
Real	coefs[size(scales, 1), size(u, 1)]	A two dimensional matrix of wavelet coefficients. Each column corresponds to one element of the input parameter, scales.

Wavelet.Transform.dwt

Information

This function calculates discrete wavelet transform (one-level wavelet decomposition) of the input signal, u. Before transformation, the original signal is extended at both ends with zero padding. The length of the two result vectors is floor((size(u,1)+size(lod,1)-1)/2).

Refer to the section of wavelet Families for the detailed information about the available wavelets.

The default input values provide a quick example with a simple saw-tooth signal.

Inputs

Type	Name	Default	Description
Real	u[:]	{1,2,1,2,1,2,3,4,1}	Signal to be analyzed
Real	lod[:]	{0.707,0.707}	Low pass filter for decomposition
Real	hid[size(lod, 1)]	{-0.707,0.707}	High pass filter for decomposition

Outputs

Type	Name	Description
Real	ca[:]	Approximation coefficients, size(ca,1) = ceil(size(u,1)/2)
Real	cd[size(ca, 1)]	Detail coefficients

Wavelet.Transform.idwt

Information

This function calculates inverse discrete wavelet transform (one-level wavelet reconstruction) using the input approximation and detail wavelet coefficients, ca and cd. The coefficient vectors are extended at both ends with zero padding before calculation. The length of the reconstructed data is usually (but not always) equal to the length of the original signal if the length is not specified.

Refer to the section of wavelet Families for the detailed information about the available wavelets.

The default input values provide a quick example to generate a simple saw-tooth signal.

Inputs

Type	Name	Default	Description
Real	ca[:]	{2.121,2.121,2.121,4.95,0.707}	Approximation coefficients
Real	cd[size(ca, 1)]	{-0.707,-0.707,-0.707,-0.707...	Detail coefficients
Real	lor[:]	{0.707,0.707}	Low pass filter for reconstruction
Real	hir[size(lor, 1)]	{0.707,-0.707}	High pass filter for reconstruction
Integer	ny	0	Length of the result, y (the central part will be extracted). If ny == 0, size(y,1) = 2*size(ca,1) - size(lor,1) +2

Outputs

Type	Name	Description
Real	y[:]	Reconstructed signal

Wavelet.Transform.wavDec

Information

This function carries out multi-level wavelet decomposition.

Parameter, len, is a vector with (N+2) elements, where N is the number of levels. It contents the length information of the coefficients in all levels. len[1] is the length of the original signal vector; len[2] is the length of the detail coefficient vector of the first decomposition level; len[3] is the length of the detail coefficient vector of the second decomposition level; and so forth. Finally, len[N+2] contains the length of the approximation coefficient vector. Parameter, c, contains the wavelet coefficients of all levels, firstly the first detail level, then the second detail level and so on, and finally the approximation coefficients. This is described again in detail as follows:

Definition of the outputs

Refer to the section of wavelet Families for the detailed information about the available wavelets.

The default input values provide a quick example with a simple saw-tooth signal.

Inputs

Type	Name	Default	Description
Real	u[:]	{1,2,1,2,1,2,3,4,1,2,3,4,1}	Signal to be analyzed
Integer	N	3	Decomposition level
Real	lod[:]	{0.707,0.707}	Low pass filter for decomposition
Real	hid[size(lod, 1)]	{-0.707,0.707}	High pass filter for decomposition

Outputs

Type	Name	Description
Real	c[:]	Detail and approximation coefficients. For data structure see the information section
Integer	len[N + 2]	Lengths of the coefficient vectors of all levels. For data structure see the information section

Wavelet.Transform.wavRec

Information

This function transforms the wavelet coefficients, which are obtained with multi-level wavelet decomposition, back to the original signal. The definitions of the input parameters, c and len, are same as the function wavDec().

Refer to the section of wavelet Families for the detailed information about the available wavelets.

The default input values provide a quick example to generate a simple saw-tooth signal.

Inputs

Type	Name	Default	Description
Real	c[:]	{-0.707,-0.707,-0.707,-0.706...	Detail and approximation coefficients. For data structure see the information section
Integer	len[:]	{13,7,4,2,2}	Coefficient vector lengths of all levels. For data structure see the information section
Real	lor[:]	{0.707,0.707}	Low pass filter for reconstruction
Real	hir[size(lor, 1)]	{0.707,-0.707}	High pass filter for reconstruction

Outputs

Type	Name	Description
Real	u[:]	Reconstructed signal

Wavelet.Transform.wavRec1

Information

This function carries out a modified wavelet reconstruction: It reconstructs the signal with the wavelet coefficients from only one single level. All other coefficients are set zero. This operation is useful for some wavelet analysis, e.g. multi-resolution analysis and wavelet de-noising.

The definitions of input parameters, c and len, are same as the function wavDec().

Refer to the section of wavelet Families for the detailed information about the available wavelets.

The default input values provide a quick example.

Inputs

Type	Name	Default	Description
Real	c[:]	{-0.707,-0.707,-0.707,-0.706...	Wavelet coefficients of all levels
Integer	len[:]	{13,7,4,2,2}	Lengths of coefficients of all levels
Real	lor[:]	{0.707,0.707}	Low pass filter for reconstruction
Real	hir[size(lor, 1)]	{0.707,-0.707}	High pass filter for reconstruction
Integer	level	2	The level to be reconstructed. 1..N are for the detail levels, where N is the maximum decomposition level. 1 stands for the most detail level (highest frequency). N+1 is for the approximation level.

Outputs

Type	Name	Description
Real	y[:]	Reconstructed signal

Wavelet.Transform.wavCoef1

Information

This function extracts the coefficients of a specified decomposition level from the data structure of the wavelet coefficients. This operation is useful if only the coefficients of a specific level are interested.

The definitions of the input parameters, c and len, are same as the function wavDec().

Refer to the section of wavelet Families for the detailed information about the available wavelets.

The default input values provide a quick example.

Inputs

Type	Name	Default	Description
Real	c[:]	{-0.707,-0.707,-0.707,-0.706...	Wavelet coefficients of all levels
Integer	len[:]	{13,7,4,2,2}	Lengths of coefficient vectors for all levels
Integer	level	2	The level to be reconstructed. 1..N are for the detail levels, where N is the maximum decomposition level. 1 stands for the most detail level (highest frequency). N+1 is for the approximation level.

Outputs

Type	Name	Description
Real	c1[:]	Extracted coefficients

Wavelet.Families

Information

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.

Applicable parameters of the wavelet families

Available orders for biorthogonal and reverse biorthogonal wavelets

For biorthogonal and reverse biorthogonal wavelets, only some combinations of the orders are available. For details, refer to the functions, wavBiorSpline and wavRevBiorSpline.

Wavelet types

All wavelets are categorized into five types. They are defined as follows:

Package Content

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

Wavelet.Families.wavFunc

Information

This function provides a common entry to access all available wavelets in this library. Since different wavelet families require different parameters and return different results, suitable input parameters must be provided. Not applicable parameters for a specific wavelet are omitted by the function.

Refer to Wavelet Families for detailed information about the available wavelets.

Inputs

Type	Name	Default	Description
wavletDefinition	wIn		Input parameters for wavelet filters and functions

Outputs

Type	Name	Description
wavFuncOut	wOut	Output data of the function wavFunc()

Wavelet.Families.scalingWaveFunc

Information

The total length of the functions is (size(lof,1)-1)*2^iteration+1. The function support (non-zero part) is (2^iteration -1)*nf-2^iteration+2, shorter than total length. One 'zero' is then added at the beginning and multiple 'zeros' are added at the end.

Inputs

Type	Name	Default	Description
wavletID	wavID	1	Id-Number of the wavelet. Necessary only for discrete Meyer wavelet
Real	lof[:]	{0.707,0.707}	Wavelet low pass filter
Real	hif[:]	{-0.707,0.707}	Wavelet high pass filter
Integer	iteration	6	Iteration of the calculations for generating the functions

Outputs

Type	Name	Description
Real	x[:]	Equidistant grid of the functions
Real	phi[:]	The scaling function
Real	psi[:]	The wavelet function

Wavelet.Families.wavHaar

Information

This function generate the Haar wavelet filter bank. No input parameters are required.

Outputs

Type	Name	Description
Real	F[:]	Scaling filter
Real	lod[:]	High pass filter for decomposition
Real	hid[:]	Low pass filter for decomposition
Real	lor[:]	High pass filter for reconstruction
Real	hir[:]	Low pass filter for reconstruction

Wavelet.Families.wavDaubechies

Information

This function generate the Daubechies wavelet filter banks for the orders up to 10. If a wrong order number is given, an empty vector is returned.

Inputs

Type	Name	Default	Description
Integer	order	2	Order of the wavelet. order<=10

Outputs

Type	Name	Description
Real	F[:]	Scaling filter
Real	lod[:]	High pass filter for decomposition
Real	hid[:]	Low pass filter for decomposition
Real	lor[:]	High pass filter for reconstruction
Real	hir[:]	Low pass filter for reconstruction

Wavelet.Families.wavSymlets

Information

This function generate the Symlets wavelet filter banks for the orders up to 8. If a wrong order number is given, an empty vector is returned.

Inputs

Type	Name	Default	Description
Integer	order	2	Order of the wavelet. order<=8

Outputs

Type	Name	Description
Real	F[:]	Scaling filter
Real	lod[:]	High pass filter for decomposition
Real	hid[:]	Low pass filter for decomposition
Real	lor[:]	High pass filter for reconstruction
Real	hir[:]	Low pass filter for reconstruction

Wavelet.Families.wavCoiflets

Information

This function generate the Coiflets wavelet filter banks for the orders up to 5. If a wrong order number is given, an empty vector is returned.

Inputs

Type	Name	Default	Description
Integer	order	2	Order of the wavelet. order<=5

Outputs

Type	Name	Description
Real	F[:]	Scaling filter
Real	lod[:]	High pass filter for decomposition
Real	hid[:]	Low pass filter for decomposition
Real	lor[:]	High pass filter for reconstruction
Real	hir[:]	Low pass filter for reconstruction

Wavelet.Families.wavBiorSpline

Information

This function generates the biorthogonal spline wavelet filters, where Nr is the order of the reconstruction filters, and Nd the order of the decomposition filters.

Usually, lodFd and hidFr are used for decomposition, and lorFr and hirFd are used for reconstruction.

Available orders of biorthogonal wavelet filters

Inputs

Type	Name	Default	Description
Integer	Nd	2	Wavelet order for decomposition
Integer	Nr	2	Wavelet order for reconstruction

Outputs

Type	Name	Description
Real	Fd[:]	Scaling filter for decomposition
Real	lodFd[:]	High pass filter for decomposition using filter Fd
Real	hidFd[:]	Low pass filter for decomposition using filter Fd
Real	lorFd[:]	High pass filter for reconstruction using filter Fd
Real	hirFd[:]	Low pass filter for reconstruction using filter Fd
Real	Fr[:]	Scaling filter for reconstruction
Real	lodFr[:]	High pass filter for decomposition using filter Fr
Real	hidFr[:]	Low pass filter for decomposition using filter Fr
Real	lorFr[:]	High pass filter for reconstruction using filter Fr
Real	hirFr[:]	Low pass filter for reconstruction using filter Fr

Wavelet.Families.wavRevBiorSpline

Information

This function generates the biorthogonal spline wavelet filters, where Nr is the order of the reconstruction filters, and Nd the order of the decomposition filters.

Usually, lodFd and hidFr are used for decomposition, and lorFr and hirFd are used for reconstruction.

Available orders of biorthogonal wavelet filters

Inputs

Type	Name	Default	Description
Integer	Nd	2	Wavelet order for decomposition
Integer	Nr	2	Wavelet order for reconstruction

Outputs

Type	Name	Description
Real	Fd[:]	Scaling filter for decomposition
Real	lodFd[:]	High pass filter for decomposition using filter Fd
Real	hidFd[:]	Low pass filter for decomposition using filter Fd
Real	lorFd[:]	High pass filter for reconstruction using filter Fd
Real	hirFd[:]	Low pass filter for reconstruction using filter Fd
Real	Fr[:]	Scaling filter for reconstruction
Real	lodFr[:]	High pass filter for decomposition using filter Fr
Real	hidFr[:]	Low pass filter for decomposition using filter Fr
Real	lorFr[:]	High pass filter for reconstruction using filter Fr
Real	hirFr[:]	Low pass filter for reconstruction using filter Fr

Wavelet.Families.wavDMeyer

Information

This function returns the filter bank of discrete Meyer wavelet, which is an approximation of continuous Meyer wavelet. The filters with the length of 102 points usually provide sufficient accuracy. However, such long filters might cause long calculation time. Therefore, this function provides the possibility to return shorter filters. The trade-off is lower accuracy. Please note that some lengths will cause significantly large errors. The default length is 22.

Inputs

Type	Name	Default	Description
Integer	points	22	Filter length (<= 102). The complete length is used if this variable is <= 0

Outputs

Type	Name	Description
Real	F[:]	Scaling filter
Real	lod[:]	High pass filter for decomposition
Real	hid[:]	Low pass filter for decomposition
Real	lor[:]	High pass filter for reconstruction
Real	hir[:]	Low pass filter for reconstruction

Wavelet.Families.wavMeyer

Information

This function generates the continuous Meyer wavelet and scaling functions. Since Meyer wavelet has theoretically unlimited support, the function range has to be defined with [low, high].

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.

Inputs

Type	Name	Default	Description
Real	low	-8	Lower boundary of the function
Real	high	8	Higher boundary of the function
Integer	points	32	Number of data points of the returned functions

Outputs

Type	Name	Description
Real	x[points]	Regular grid
Real	phi[points]	Scaling function
Real	psi[points]	Wavelet function

Wavelet.Families.wavGaussian

Information

This function generates the continuous Gaussian wavelet function with the order up to 8. It has no scaling function. Since Gaussian wavelet has theoretically unlimited support, the function range has to be defined with [low, high].

Inputs

Type	Name	Default	Description
Real	low	-5	Lower boundary of the function
Real	high	5	Higher boundary of the function
Integer	order	1	Order of the derivative, <=8
Integer	points	32	Number of data points of the returned functions

Outputs

Type	Name	Description
Real	x[points]	Regular grid
Real	psi[points]	Wavelet function

Wavelet.Families.wavMexHat

Information

This function generates the continuous Mexican Hat wavelet function. It has no scaling function. Since Mexican Hat wavelet has theoretically unlimited support, the function range has to be defined with [low, high].

Inputs

Type	Name	Default	Description
Real	low	-5	Lower boundary of the returned function
Real	high	5	Higher boundary of the returned function
Integer	points	32	Number of the data points of the returned function

Outputs

Type	Name	Description
Real	x[points]	Regular grid
Real	psi[points]	The wavelet function

Wavelet.Families.wavMorlet

Information

This function generates the continuous Morlet wavelet function. It has no scaling function. Since Morlet wavelet has theoretically unlimited support, the function range has to be defined with [low, high].

Inputs

Type	Name	Default	Description
Real	low	-4	Lower boundary of the function
Real	high	4	Higher boundary of the function
Integer	points	32	Integer number, length of the returned functions

Outputs

Type	Name	Description
Real	x[points]	Regular grid
Real	psi[points]	Wavelet function

Wavelet.Families.wavXGaussian

Information

This function generates the complex Gaussian wavelet function with the order up to 8. It has no scaling function. Since complex Gaussian wavelet has theoretically unlimited support, the function range has to be defined with [low, high].

Inputs

Type	Name	Default	Description
Real	low	-5	Lower boundary of the function
Real	high	5	Higher boundary of the function
Integer	order	2	Order of the derivative, <=8
Integer	points	32	Number of data points of the returned functions

Outputs

Type	Name	Description
Real	x[points]	Regular grid
Complex	psi[points]	Wavelet function

Wavelet.Families.wavXMorlet

Information

This function generates the complex Morlet wavelet function. It has no scaling function. Since complex Morlet wavelet has theoretically unlimited support, the function range has to be defined with [low, high].

Inputs

Type	Name	Default	Description
Real	low	-4	Lower boundary of the function
Real	high	4	Higher boundary of the function
Integer	points	32	Number of data points of the returned functions
Real	fb	1	Band width
Real	fc	1	Center frequency, e.g. 0.1, 0.5, 1, 1.5, ...

Outputs

Type	Name	Description
Real	x[points]	Regular grid
Complex	psi[points]	Wavelet function

Wavelet.Families.wavXShannon

Information

This function generates the complex Shannon wavelet function. It has no scaling function. Since complex Shannon wavelet has theoretically unlimited support, the function range has to be defined with [low, high].

Inputs

Type	Name	Default	Description
Real	low	-20	Lower boundary of the function
Real	high	20	Higher boundary of the function
Integer	points	32	Number of data points of the returned functions
Real	fb	1	Band width, e.g. 1, 2
Real	fc	1	Center frequency, e.g. 0.1, 0.5, 1, 1.5, ...

Outputs

Type	Name	Description
Real	x[points]	Regular grid
Complex	psi[points]	Wavelet function

Wavelet.Families.wavXFreqBSpline

Information

This function generates the complex Frequency B-Spline wavelet function. It has no scaling function. Since this wavelet has theoretically unlimited support, the function range has to be defined with [low, high].

Inputs

Type	Name	Default	Description
Real	low	-20	Lower boundary of the wavelet function
Real	high	20	Higher boundary of the wavelet function
Integer	order	2	Wavelet order, a natural number, 1, 2, ...
Integer	points	32	Number of data points of the returned functions
Real	fb	1	Band width
Real	fc	1	Center frequency, a real number, e.g. 0.1, 0.5, 1, 1.5, ...

Outputs

Type	Name	Description
Real	x[points]	Regular grid
Complex	psi[points]	Wavelet function

Wavelet.General

Information

This section defines some general functions, which are common utilities for wavelet transformation and other functions.

Package Content

Name	Description
_fft	An external function to carry out FFT
cumSum	Cumulative sum of a vector. Data type is Real.
cumSumInt	Cumulative sum of a vector. Data type is Integer
diff	Difference between every two adjacent elements of a vector
fft	Fast Fourier transform
fftShift	Shift zero-frequency component to center of spectrum
findIndex	Find the location of a value in a monotone vector
filterBank	Get the four wavelet filters based on a given scaling filter
ifft	Inverse fast Fourier transform
innerProduct	Inner product of two same length vectors
interpL	One-dimensional linear interpolation
midVector	Extract the middle part of a vector
nStdIfft	Inverse non-standard 1-D fast Fourier transform
quadReverse	Quadrature mirror of a given vector
sinc	Sinc function
upsample	Up-sampling of a vector (insert a zero after every element except the last one)
wavConv	Fully convolving of a data vector and a filter vector for wavelet transform

Wavelet.General._fft

Inputs

Type	Name	Default	Description
Integer	dir	1	Calculation direction: 1 - forward; -1 - backward
Integer	m	3	Data points = 2^m
Real	x[:]	{i for i in 1:8}	Real part of the data
Real	y[:]	{0,0,0,0,0,0,0,0}	Imaginary part of the data

Outputs

Type	Name	Description
Real	x_out[:]	Real part of the result
Real	y_out[:]	Imaginary part of the result

Wavelet.General.cumSum

Inputs

Type	Name	Default	Description
Real	i_v[:]	{1,2,3,4}	Input vector

Outputs

Type	Name	Description
Real	o_v[size(i_v, 1)]	Output vector

Wavelet.General.cumSumInt

Inputs

Type	Name	Default	Description
Integer	i_v[:]	{1,2,3,4}	Input vector

Outputs

Type	Name	Description
Integer	o_v[size(i_v, 1)]	Output vector

Wavelet.General.diff

Inputs

Type	Name	Default	Description
Real	in_v[:]	{1,2,3,4}	Input vector

Outputs

Type	Name	Description
Real	out_v[size(in_v, 1) - 1]	Output vector

Wavelet.General.fft

Inputs

Type	Name	Default	Description
Complex	x[:]	{Complex(re=1, im=0),Complex...	The vector to be transformed
Integer	n	8	Number of data points to be calculated

Outputs

Type	Name	Description
Complex	y[size(x, 1)]	The result

Wavelet.General.fftShift

Inputs

Type	Name	Default	Description
Complex	x[:]		The vector to be shifted

Outputs

Type	Name	Description
Complex	y[size(x, 1)]	The shifted vector

Wavelet.General.findIndex

Information

This function searches the index of the input vector, u, according to x, so that:

If x is between u[k] and u[k+1], id1 = k and id2 = k+1.

If x is equal to u[k], id1 = id2 = k.

If x is smaller than all elements of u, id1 = id2 = 0.

If x is larger than all elements of u, id1 = id2 = size(u,1)+1.

The searching method is binary search, which has high efficiency for large searching spaces.

Inputs

Type	Name	Default	Description
Real	u[:]	{1,2,3,3,3,4,5,6,7,8}	A monotone vector, in which u[k+1] >= u[k] for all k
Real	x	3	The value to be located in u

Outputs

Type	Name	Description
Integer	id1	The largest index, id1, so that u[id1] <= x
Integer	id2	The smallest index, id2, so that u[id2] >= x

Wavelet.General.filterBank

Inputs

Type	Name	Default	Description
Real	F[:]	{0.5,0.5}	The scaling filter

Outputs

Type	Name	Description
Real	lod[:]	High pass filter for decomposition
Real	hid[:]	Low pass filter for decomposition
Real	lor[:]	High pass filter for reconstruction
Real	hir[:]	Low pass filter for reconstruction

Wavelet.General.ifft

Inputs

Type	Name	Default	Description
Complex	x[:]	{Complex(re=36, im=0),Comple...	The vector to be inverse transformed
Integer	n	8	Number of data points for calculation

Outputs

Type	Name	Description
Complex	y[size(x, 1)]	The result

Wavelet.General.innerProduct

Inputs

Type	Name	Default	Description
Real	x1[:]	{1,2,3,4}	Vector 1
Real	x2[size(x1, 1)]	{1,2,3,4}	Vector 2

Outputs

Type	Name	Description
Real	y	Result

Wavelet.General.interpL

Information

This function finds the values at u1 of the linearly interpolated function of the original data pairs, u and y. Both u and u1 must be monotone, meaning, e.g. u[k]>=u[k-1]. For the elements in u1 whose values are less than u[1], the corresponding elements in y1 are set as y[1]. For the elements in u1 whose values are greater than the last value of u, the corresponding elements in y1 are set as y[end].

Inputs

Type	Name	Default	Description
Real	u[:]	{1,2,3,4}	The variable of the original data
Real	y[size(u, 1)]	{0,1,-1,2}	The function of the original data
Real	u1[:]	{0,0.5,1,1.5,2,2.5,3,3.5,4,4...	The variable of the interpolated function

Outputs

Type	Name	Description
Real	y1[size(u1, 1)]	The interpolated result at the variable values, u1[:]

Wavelet.General.midVector

Inputs

Type	Name	Default	Description
Real	x[:]	{1,2,3,4,5,6}	Original vector
Integer	n	3	Number of the vector elements to be extracted

Outputs

Type	Name	Description
Real	y[:]	The result vector

Wavelet.General.nStdIfft

Inputs

Type	Name	Default	Description
Complex	xhat[:]		The vector to be inverse transformed
Real	low	-8	Lower boundary of the function
Real	high	8	Higher boundary of the function

Outputs

Type	Name	Description
Real	x[size(xhat, 1)]	The recovered signal
Real	t[size(xhat, 1)]	Time grid

Wavelet.General.quadReverse

Information

Changes the signs of the even index entries of the reversed input vector, x, if p is true. If p is false the same holds for odd index entries.

Inputs

Type	Name	Default	Description
Real	x[:]	{1,2,3,4,5,6,7,8}	Vector
Boolean	p	true	If p is true, the signs of the even index entries of the reversed input vector, x, is reversed. Otherwise, odd index entries are reversed.

Outputs

Type	Name	Description
Real	y[size(x, 1)]	Quadrature mirrored elements of the input vector

Wavelet.General.sinc

Inputs

Type	Name	Default	Description
Real	x	pi/2

Outputs

Type	Name	Description
Real	y

Wavelet.General.upsample

Inputs

Type	Name	Default	Description
Real	u[:]	{1,2,3,4}	Input vector

Outputs

Type	Name	Description
Real	y[size(u, 1)*2 - 1]	Output vector with the (double length of u) -1

Wavelet.General.wavConv

Inputs

Type	Name	Default	Description
Real	u[:]	{1,2,3,4}	The data vector
Real	f[:]	{1,2}	The filter vector

Outputs

Type	Name	Description
Real	y[size(u, 1) + size(f, 1) - 1]	The filtered data vector

Wavelet.Records

Package Content

Name	Description
denoisParameters	Parameters for 1D wavelet denoising
mraParameters	Parameters for multi-resolution analysis (MRA)
wavFuncOut	Output data of function, wavFunc()
wavletDefinition	Wavelet definition

Wavelet.Records.denoisParameters

Information

Parameters for carrying out wavelet denoising.

Extends from Modelica.Icons.Record (Icon for records).

Parameters

Type	Name	Default	Description
Integer	decLevel	2	Decomposition levels, <=12
threshMethod	thMethod	1	Select what method is to be used for calculating the threshold
Boolean	sorh	true	Soft (false) or hard (true) thresholding method
Thresholds (>=0) for all levels
Real	thD12	-1	Detail level 12
Real	thD11	-1	Detail level 11
Real	thD10	-1	Detail level 10
Real	thD9	-1	Detail level 9
Real	thD8	-1	Detail level 8
Real	thD7	-1	Detail level 7
Real	thD6	-1	Detail level 6
Real	thD5	-1	Detail level 5
Real	thD4	-1	Detail level 4
Real	thD3	-1	Detail level 3
Real	thD2	-1	Detail level 2
Real	thD1	-1	Detail level 1

Wavelet.Records.mraParameters

Information

Parameters for carrying out wavelet multi-resolution analysis (MRA).

Extends from Modelica.Icons.Record (Icon for records).

Parameters

Type	Name	Default	Description
Wavelet
Integer	decLevel	2	Decomposition levels, <=12
mraDisplay	mraData	2	Select what data type is to be displayed in the decomposed levels
Tuning factor of the coefficients before reconstruction. Levels higher than decLevel are omitted.
Real	rA	1	Approximating coefficients
Real	rD12	1	Detail coefficients, level 12
Real	rD11	1	Detail coefficients, level 11
Real	rD10	1	Detail coefficients, level 10
Real	rD9	1	Detail coefficients, level 9
Real	rD8	1	Detail coefficients, level 8
Real	rD7	1	Detail coefficients, level 7
Real	rD6	1	Detail coefficients, level 6
Real	rD5	1	Detail coefficients, level 5
Real	rD4	1	Detail coefficients, level 4
Real	rD3	1	Detail coefficients, level 3
Real	rD2	1	Detail coefficients, level 2
Real	rD1	1	Detail coefficients, level 1

Wavelet.Records.wavFuncOut

Information

Output parameters of function wavFunc().

Parameters

Type	Name	Default	Description
Output
String	wavName		Wavelet name
Integer	wavType		Wavelet type: 1. orthogonal wavelets; 2. biorthogonal wavelets; 3. wavelet with scaling function; 4. wavelet without scaling function; 5. complex wavelet without scaling function
Real	F1[:]		Orthogonal wavelets: scaling filter; biorthogonal wavelets: scaling filter for decomposition
Real	F2[:]		Only for biorthogonal wavelets: scaling filter for reconstruction
Real	lod[:]		Low pass filter for decomposition
Real	hid[:]		High pass filter for decomposition
Real	lor[:]		Low pass filter for reconstruction
Real	hir[:]		High pass filter for reconstruction
Real	x[:]		Equidistance grid
Real	phi1[:]		Scaling functions
Real	psi1[:]		Wavelet functions
Real	phi2[:]		Scaling functions, only applicable for biorthogonal wavelets
Real	psi2[:]		Wavelet functions, only applicable for biorthogonal wavelets

Wavelet.Records.wavletDefinition

Information

Parameters for wavelet definition.

Refer to the section of wavelet Families for detailed information about the available wavelets.

Extends from Modelica.Icons.Record (Icon for records).

Parameters

Type	Name	Default	Description
Input
wavletID	wavID	2	ID number of the wavelet family to be displayed, 1..15.
Integer	Nd	3	Wavelet order
Integer	Nr	3	Wavelet order for reconstruction, only used for biorthogonal wavelets
Real	fb	1.0	Bandwidth frequency, only for complex wavelets
Real	fc	1.0	Center frequency, only for complex wavelets
Real	range	5.0	Boundary of the wavelet and scaling functions, [-range, range]. Valid for wavelets with theoretically unlimited support. Effective support is used as default value if range==0
Integer	refinement	6	Refinement of the estimated wavelet and scaling functions. For orthogonal and biorthogonal wavelets, it stands for the iteration times to generating functions. The returned function length is (filter_length-1)*2^refinement+1. For other wavelets, the length of the returned functions is 2^refinement.

Wavelet.Types

Information

Enumeration variables are defined here.

waveletID: ID numbers of wavelets

mraDisplay: Data type to be displayed in MRA

threshID: ID numbers of different methods for threshold calculation for denoising

Package Content

Name	Description
mraDisplay	Selection of display data for MRA: 1 - wavelet coefficients; 2 - reconstructed signal
threshMethod	Methods for calculating threshold for denoising: 1 - fixed form; 2 - SURE method; 3 - heuristic SURE; 4 - minimal maximum method
wavletID	Definition of wavelet identifiers: 1 - Haar; 2 - Daubechies; 3 - Symlets; 4 - Coiflets; 5 - Biorthogonal spline; 6 - Reverse biorthogonal spline; 7 - Meyer; 8 - Discrete Meyer; 9 - Gaussian; 10 - Mexican hat; 11 - Morlet; 12 - Complex Gaussian; 13 - Complex Morlet; 14 - Complex Shannon; 15 - Complex frequency B-Spline

Wavelet.Types.mraDisplay

Wavelet.Types.threshMethod

Wavelet.Types.wavletID