# 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:
•  MaxRecursion 8 number of recursive iterations to use WorkingPrecision MachinePrecision precision to use in internal computations

# Examples

open allclose all

## 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:

Wavelet function for continuous wavelets, including DGaussianWavelet:

Multivariate scaling and wavelet functions are products of univariate ones:

## Options(3)

### MaxRecursion(1)

Plot wavelet function using different levels of recursion:

### WorkingPrecision(2)

By default is used:

Use higher-precision filter computation:

## 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:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2022_waveletpsi, author="Wolfram Research", title="{WaveletPsi}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/WaveletPsi.html}", note=[Accessed: 01-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_waveletpsi, organization={Wolfram Research}, title={WaveletPsi}, year={2010}, url={https://reference.wolfram.com/language/ref/WaveletPsi.html}, note=[Accessed: 01-June-2023 ]}