This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 MATHEMATICA COMPATIBILITY INFORMATION
WaveletExplorer
As of Mathematica 8, the functionality of the Wavelet Explorer add-on has been integrated into the Mathematica kernel.
Wavelet Filters
The following is a list of filters available in Wavelet Explorer, along with the equivalent form in Mathematica 8.
 HaarFilter[] WaveletFilterCoefficients[HaarWavelet[]] DaubechiesFilter[n] WaveletFilterCoefficients[DaubechiesWavelet[n]] LeastAsymmetricFilter[n] WaveletFilterCoefficients[SymletWavelet[n]] CoifletFilter[n] WaveletFilterCoefficients[CoifletWavelet[n]] ShannonFilter[lim] WaveletFilterCoefficients[ShannonWavelet[lim]] MeyerFilter[n,lim] WaveletFilterCoefficients[MeyerWavelet[n,lim]] SplineFilter[n,lim] WaveletFilterCoefficients[BattleLemarieWavelet[n,lim]] BiorthogonalSplineFilter[n,m] WaveletFilterCoefficients[BiorthogonalSplineWavelet[n,m]] HighpassFilter[h] WaveletFilterCoefficients[wave,"PrimalHighpass"]
Built-in function equivalents.
To compute wavelet coefficients, use the built in function WaveletFilterCoefficients:
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Note that all wavelet coefficients are scaled by relative to the results from Wavelet Explorer, so to get the equivalent values, you must multiply the result by :
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To compute highpass filter coefficients, use the argument to WaveletFilterCoefficients:
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Scaling and Wavelet Functions
The following is a list of functions available in Wavelet Explorer, along with the equivalent form in Mathematica 8.
 ScalingFunction[filt,j] WaveletPhi[wave] Wavelet[wave,j] WaveletPsi[wave] ShannonPhi[t] WaveletPhi[ShannonWavelet[lim],t] ShannonPsi[t] WaveletPsi[ShannonWavelet[lim],t] MeyerPhi[n,t,lim] WaveletPhi[MeyerWavelet[n,lim],t] MeyerPsi[n,t,lim] WaveletPsi[MeyerWavelet[n,lim],t] SplinePhi[n,t,lim] WaveletPhi[BattleLemarieWavelet[n,lim],t] SplinePsi[n,t,lim] WaveletPsi[BattleLemarieWavelet[n,lim],t] BSpline[n,t] BSplineBasis[{n,{u1,u2,...}},0,t] DScalingFunction[filt,jmax,m] Dt[WaveletPhi[wave,t],{t,m}] DWavelet[filt,jmax,m] Dt[WaveletPsi[wave,t],{t,m}]
Built-in function equivalents.
The functionality of is now available by using WaveletPhi:
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To find the derivative of the scaling function, use Dt and WaveletPhi:
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The functionality of is now available by using Dt and WaveletPsi:
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Compute higher order derivatives of scaling and wavelet function:
The InterpolatingFunction outputted from WaveletPhi & WaveletPsi has InterpolationOrder set to . Hence the second derivative comes out to be 0.
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Resampling and interpolating with a higher InterpolationOrder resolves the issue:
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The functionality of is now available by using built in function BSplineBasis:
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Wavelet Transforms
The following is a list of wavelet transforms available in Wavelet Explorer, along with the equivalent form in Mathematica 8.
 WaveletTransform[data,filt,j] DiscreteWaveletTransform[data,wave,j] InverseWaveletTransform[wd,filt] InverseWaveletTransform[dwd] WaveletPacketCoefficients[data,filt,j] DiscreteWaveletPacketTransform[data,filt,j] WaveletPacketTransform[data,filt,l] WaveletBestBasis[DiscreteWaveletPacketTransform[...]] InverseWaveletPacketTransform[wpdata,filt] InverseWaveletTransform[dwd]
Built-in function equivalents. The function is not directly supported with built in functionality.
The functionality of is now available by using DiscreteWaveletTransform:
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To compute packet transform use DiscreteWaveletPacketTransform:
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Use InverseWaveletTransform to compute the inverse:
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The functionality of can be replicated as follows:
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Sine & Cosine Transforms
The following is a list of functions available in Wavelet Explorer, along with the equivalent form in Mathematica 8.
 CosTransform[data,n, BasisType→m] FourierDCT[data,m] SinTransform[data,n,BasisType→m] FourierDST[data,m] InverseCosTransform[cdata] FourierDCT[cdata,m] InverseSinTransform[sdata] FourierDST[sdata,m]
Built-in function equivalents. The functions , , , , , , and are not directly supported with built in functionality.
To compute , use the built in function FourierDST:
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with specifed second argument:
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In dimension functionality of can be replicated as follows:
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Similarly we can wite using FourierDCT:
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Other Utilities
The following is a list of functions available in Wavelet Explorer, along with the equivalent form in Mathematica 8.
 PlotCoefficients[wd] WaveletListPlot[dwd] PhaseSpacePlot[wd] WaveletScalogram[dwd] ShowBasisPosition[wd] DiscreteWaveletData[...]["BestBasisBlockView"] PlotCoefficients2D[wd] WaveletMatrixPlot[dwd] ShowBasisPosition2D[wd] DiscreteWaveletData[...]["BestBasisBlockView"] WaveletCompress[wd,...] WaveletThreshold[dwd,tspec]
Built-in function equivalents. The functions and are not directly supported with built in functionality.
To plot wavelet coefficients use WaveletScalogram:
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Use WaveletThreshold for data compression:
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The function can be written as follows:
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