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PaulWavelet[]

represents a Paul wavelet of order 4.

PaulWavelet[n]

represents a Paul wavelet of order n.

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PaulWavelet

PaulWavelet[]

represents a Paul wavelet of order 4.

PaulWavelet[n]

represents a Paul wavelet of order n.

Details

Examples

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Basic Examples  (2)

Wavelet function:

Wolfram Language code: ψ = WaveletPsi[PaulWavelet[4], x];
Wolfram Language code: Plot[{Re[ψ], Im[ψ]}, {x, -2, 2}, PlotRange -> All]
Wolfram Language code: ψ = WaveletPsi[PaulWavelet[6], x];
Wolfram Language code: Plot[{Re[ψ], Im[ψ]}, {x, -2, 2}, PlotRange -> All]

Scope  (3)

Compute a wavelet function:

Wolfram Language code: WaveletPsi[PaulWavelet[3], x]

Wavelet function as a function of order n:

Wolfram Language code: FormulaGrid[list_] := Grid[list, Alignment -> Center, Background -> {None, {{StandardGray, Orange}}}, Dividers -> {None, {Darker[Gray, .6], {False}, Darker[Gray, .6]}}, ItemSize -> {{Scaled[.1], Scaled[.9]}}, ItemStyle -> {{14}, 16}]
Wolfram Language code: FormulaGrid[Table[{k, Simplify@WaveletPsi[PaulWavelet[k], x]}, {k, 1, 5}]]

PaulWavelet is used to perform ContinuousWaveletTransform:

Wolfram Language code: f[t_] := Piecewise[{{Sin[2π 10t], 0 ≤ t < (1/4)}, {Sin[2π 25t], (1/4) ≤ t < (1/2)}, {Sin[2π 50t], (1/2) ≤ t < (3/4)}, {Sin[2π 100t], (3/4) ≤ t ≤ 1}}]
Wolfram Language code: data = Table[f[t], {t, 0, 1, (1/1023)}];
Wolfram Language code: ListLinePlot[data]
Wolfram Language code: cwt = ContinuousWaveletTransform[data, PaulWavelet[4], {8, 12}, Padding -> 0.0, SampleRate -> 1023]

Use WaveletScalogram to get a time scale representation of wavelet coefficients:

Wolfram Language code: WaveletScalogram[cwt]

Use InverseWaveletTransform to reconstruct the signal:

Wolfram Language code: ListLinePlot[{data, InverseContinuousWaveletTransform[cwt]}]

Properties & Relations  (3)

Wavelet function integrates to zero; :

Wolfram Language code: Integrate[WaveletPsi[PaulWavelet[3], x], {x, -∞, ∞}]

Wavelet function and its Fourier transform:

Wolfram Language code: ψ = WaveletPsi[PaulWavelet[4], x]
Wolfram Language code: Plot[{Re[ψ], Im[ψ]}, {x, -5, 5}, PlotRange -> All, PlotStyle -> {Automatic, Dashed}, Frame -> True, GridLines -> Automatic]
Wolfram Language code: Overscript[ψ, ^ ] = FourierTransform[ψ, x, ω, FourierParameters -> {0, -2Pi}]
Wolfram Language code: Plot[Overscript[ψ, ^ ], {ω, -1, 2}, PlotRange -> All]

PaulWavelet does not have a scaling function:

Wolfram Language code: WaveletPhi[PaulWavelet[6], x]
Wolfram Research (2010), PaulWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/PaulWavelet.html.

Text

Wolfram Research (2010), PaulWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/PaulWavelet.html.

CMS

Wolfram Language. 2010. "PaulWavelet." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PaulWavelet.html.

APA

Wolfram Language. (2010). PaulWavelet. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PaulWavelet.html

BibTeX

@misc{reference.wolfram_2026_paulwavelet, author="Wolfram Research", title="{PaulWavelet}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PaulWavelet.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_paulwavelet, organization={Wolfram Research}, title={PaulWavelet}, year={2010}, url={https://reference.wolfram.com/language/ref/PaulWavelet.html}, note=[Accessed: 13-June-2026]}