DGaussianWavelet
represents a derivative of Gaussian wavelet of derivative order 2.
represents a derivative of Gaussian wavelet of derivative order n.
Details
- DGaussianWavelet defines a family of non-orthogonal wavelets.
- The wavelet function (
) is given by ![((-1)^(n+1))/(sqrt(TemplateBox[{{n, +, {1, /, 2}}}, Gamma])) (partial^n)/(partialx^n)exp(-(x^2)/2)](Files/DGaussianWavelet.en/1.png "((-1)^(n+1))/(sqrt(TemplateBox[{{n, +, {1, /, 2}}}, Gamma])) (partial^n)/(partialx^n)exp(-(x^2)/2)")
. - DGaussianWavelet can be used with such functions as ContinuousWaveletTransform, WaveletPsi, etc.
Examples
open all close allBasic Examples (1)
Scope (2)
DGaussianWavelet is used to perform ContinuousWaveletTransform:
data = Table[Sin[100 t^2], {t, 0, 1, 1. / 1023}];cwt = ContinuousWaveletTransform[data, DGaussianWavelet[4], {12, 4}, Padding -> 0.0]Use WaveletScalogram to get a time scale representation of wavelet coefficients:
WaveletScalogram[cwt]Use InverseWaveletTransform to reconstruct the signal:
ListLinePlot[{data, Re@InverseContinuousWaveletTransform[cwt]}]Wavelet function as a function of derivative order n:
FormulaGrid[list_] := Grid[list, Alignment -> Center, Background -> {None, {{StandardBlue, StandardGray}}}, Dividers -> {None, {Darker[Gray, .6], {False}, Darker[Gray, .6]}}, ItemSize -> {{Scaled[.1], Scaled[.9]}}, ItemStyle -> {{14}, 16}]FormulaGrid[Table[{k, Simplify@WaveletPsi[DGaussianWavelet[k], x]}, {k, 1, 5}]]Properties & Relations (4)
DGaussianWavelet[2] is the same as MexicanHatWavelet:
Plot[{WaveletPsi[DGaussianWavelet[2], x], WaveletPsi[MexicanHatWavelet[], x]}, {x, -5, 5}, PlotStyle -> {{Red, Thick}, Blue}]Wavelet function integrates to zero; 
:
Integrate[WaveletPsi[DGaussianWavelet[2], x], {x, -∞, ∞}]Integrate[WaveletPsi[DGaussianWavelet[10], x], {x, -∞, ∞}]Wavelet function and its Fourier transform:
ψ = WaveletPsi[DGaussianWavelet[2], x]Plot[ψ, {x, -5, 5}]Overscript[ψ, ^ ] = FourierTransform[ψ, x, ω, FourierParameters -> {0, -2Pi}]Plot[Overscript[ψ, ^ ], {ω, -1, 1}, PlotRange -> All]DGaussianWavelet does not have a scaling function:
WaveletPhi[DGaussianWavelet[4], x]Text
Wolfram Research (2010), DGaussianWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/DGaussianWavelet.html.
CMS
Wolfram Language. 2010. "DGaussianWavelet." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DGaussianWavelet.html.
APA
Wolfram Language. (2010). DGaussianWavelet. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DGaussianWavelet.html