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WaveletFilterCoefficients
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WaveletFilterCoefficients

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gives the filter coefficients for the symbolic wavelet wave of type filt.

Details and Options

  • WaveletFilterCoefficients[wave,filt] gives a list of the form {{n,cn},{n+1,cn+1},…}, where n is the index and cn the corresponding filter coefficient.
  • For orthogonal wavelets possible filters filt include: "PrimalLowpass" and "PrimalHighpass".
  • The primal highpass filter coefficients satisfy , where are primal lowpass filter coefficients.
  • The scaling function and wavelet function satisfy the relations:
  • (primal) scaling refinement equation
    (primal) wavelet refinement equation
  • For biorthogonal wavelets, possible filters filt include: "PrimalLowpass", "PrimalHighpass", "DualLowpass", and "DualHighpass".
  • The primal highpass filter coefficients satisfy , where are dual lowpass filter coefficients. Dual highpass filter coefficients satisfy , where are primal lowpass filter coefficients.
  • The primal scaling function and wavelet function satisfy the relations:
  • (primal) scaling refinement equation
    (primal) wavelet refinement equation
  • The dual scaling function and dual wavelet function satisfy:
  • (dual) scaling refinement equation
    (dual) wavelet refinement equation
  • For discrete wavelets with compact support, you can also produce LiftingFilterData objects used for LiftingWaveletTransform as well as generating compiled standalone wavelet transform code. The following filt values can be used:
  • "LiftingFilter"default lifting filter
    "AllLiftingFilter"all possible lifting filters
    "BestLiftingFilter"most stable lifting filter
  • With the option setting WorkingPrecision->prec, filter coefficients are computed using precision prec. By default, WorkingPrecision->MachinePrecision is used.

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Primal lowpass filter coefficients:

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-lmmump
Out[1]=1

Primal highpass filter coefficients:

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-0e2bs6
Out[1]=1

Dual lowpass filter coefficients:

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-raub53
Out[1]=1

Dual highpass filter coefficients:

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-m71igw
Out[1]=1

Scope  (5)Survey of the scope of standard use cases

Compute primal lowpass filter coefficients:

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-x5t4az
Out[1]=1

Primal highpass filter coefficients:

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-tovf4h
Out[1]=1

Dual lowpass filter coefficients:

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-ozjyj6
Out[1]=1

Dual highpass filter coefficients:

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-bzlj15
Out[1]=1

"Properties" gives a list of available properties:

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-jmfswx
Out[1]=1

Properties for BiorthogonalSplineWavelet:

In[2]:=2
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-8se554
Out[2]=2

Options  (2)Common values & functionality for each option

WorkingPrecision  (2)

By default WorkingPrecision->MachinePrecision is used:

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-jj9fjg
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-6vi03j
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-fste7p
Out[3]=3

Use higher-precision filter computation:

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-oxpk71
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-f6wruc
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-cwufym
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-yp62kx
Out[4]=4

Properties & Relations  (3)Properties of the function, and connections to other functions

Lowpass filter coefficients sum to unity; :

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-bs7miu
Out[1]=1

Highpass filter coefficients sum to zero; :

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-jaf8rr
Out[1]=1

The lowpass and highpass filter coefficients are orthogonal; :

In[1]:=1
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https://wolfram.com/xid/0bmvb95zs3yjyoig9jq-8vwdxn
Out[1]=1
Wolfram Research (2010), WaveletFilterCoefficients, Wolfram Language function, https://reference.wolfram.com/language/ref/WaveletFilterCoefficients.html.
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Wolfram Research (2010), WaveletFilterCoefficients, Wolfram Language function, https://reference.wolfram.com/language/ref/WaveletFilterCoefficients.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_waveletfiltercoefficients, author="Wolfram Research", title="{WaveletFilterCoefficients}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/WaveletFilterCoefficients.html}", note=[Accessed: 27-March-2025 ]}

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@misc{reference.wolfram_2025_waveletfiltercoefficients, author="Wolfram Research", title="{WaveletFilterCoefficients}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/WaveletFilterCoefficients.html}", note=[Accessed: 27-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_waveletfiltercoefficients, organization={Wolfram Research}, title={WaveletFilterCoefficients}, year={2010}, url={https://reference.wolfram.com/language/ref/WaveletFilterCoefficients.html}, note=[Accessed: 27-March-2025 ]}

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@online{reference.wolfram_2025_waveletfiltercoefficients, organization={Wolfram Research}, title={WaveletFilterCoefficients}, year={2010}, url={https://reference.wolfram.com/language/ref/WaveletFilterCoefficients.html}, note=[Accessed: 27-March-2025 ]}