WaveletPsi

WaveletPsi[wave,x]

gives the wavelet function for the symbolic wavelet wave evaluated at x.

WaveletPsi[wave]

gives the wavelet function as a pure function.

Details and Options

  • The wavelet function satisfies the recursion equation , where is the scaling function and are the high-pass filter coefficients.
  • A discrete wavelet transform effectively represents a signal in terms of scaled and translated wavelet functions , where .
  • WaveletPsi[wave,x,"Dual"] gives the dual wavelet function for biorthogonal wavelets such as BiorthogonalSplineWavelet and ReverseBiorthogonalSplineWavelet.
  • The dual wavelet function satisfies the recursion equation , where are the dual high-pass filter coefficients.
  • The following options can be used:
  • MaxRecursion8number of recursive iterations to use
    WorkingPrecisionMachinePrecisionprecision to use in internal computations

Examples

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Basic Examples  (3)

Haar wavelet function:

Daubechies wavelet function:

Mexican hat wavelet function:

Scope  (5)

Compute primal wavelet function:

Dual wavelet function:

Wavelet function for discrete wavelets, including HaarWavelet:

DaubechiesWavelet:

SymletWavelet:

CoifletWavelet:

BiorthogonalSplineWavelet:

ReverseBiorthogonalSplineWavelet:

CDFWavelet:

ShannonWavelet:

BattleLemarieWavelet:

MeyerWavelet:

Wavelet function for continuous wavelets, including DGaussianWavelet:

MexicanHatWavelet:

GaborWavelet:

ShannonWavelet:

MorletWavelet:

PaulWavelet:

Multivariate scaling and wavelet functions are products of univariate ones:

Options  (3)

WorkingPrecision  (2)

By default WorkingPrecision->MachinePrecision is used:

Use higher-precision filter computation:

MaxRecursion  (1)

Plot wavelet function using different levels of recursion:

Properties & Relations  (4)

Wavelet function integrates to zero :

satisfies the recursion equation :

Plot the components and the sum of the recursion:

Frequency response for is given by :

The filter is a high-pass filter:

Fourier transform of is given by :

Neat Examples  (1)

Plot translates and dilations of wavelet function:

Introduced in 2010
 (8.0)