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DaubechiesWavelet
✖
DaubechiesWavelet

✖

represents a Daubechies wavelet of order 2.

✖

represents a Daubechies wavelet of order n.

Details

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Scaling function:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-gpi6tg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-gb8f9b
Out[2]=2

Wavelet function:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-vc7o0g
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-xdbxfp
Out[2]=2

Filter coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-kjry2h
Out[1]=1

Scope  (14)Survey of the scope of standard use cases

Basic Uses  (8)

Compute primal lowpass filter coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-x5t4az
Out[1]=1

Primal highpass filter coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-tovf4h
Out[1]=1

Lifting filter coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-nxqmse
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-k9eeea
Out[2]=2

Generate a function to compute a lifting wavelet transform:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-nfjqzn
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-ol7ejo
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-7bq5gu
Out[3]=3

Daubechies scaling function of order 2:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-vl9zbf
Out[1]=1

Daubechies scaling function of order 6:

In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-4ybkxk
Out[2]=2

Plot scaling function using different levels of recursion:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-vovrxw
Out[1]=1

Daubechies wavelet function of order 2:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-zncr2c
Out[1]=1

DaubechiesWavelet of order 6:

In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-duatsf
Out[2]=2

Plot wavelet function different levels of recursion:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-hjxsba
Out[1]=1

Wavelet Transforms  (5)

Compute a DiscreteWaveletTransform:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-t8skl0
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-2kxsqf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-xhbmxi
Out[3]=3

View the tree of wavelet coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-gtqy6c
Out[4]=4

Get the dimensions of wavelet coefficients:

In[5]:=5
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-mrjnjd
Out[5]=5

Plot the wavelet coefficients:

In[6]:=6
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-vkcd9f
Out[6]=6

Compute a DiscreteWaveletPacketTransform:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-o34j9k
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-vtrxi2
Out[2]=2

View the tree of wavelet coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-j3n0f1
Out[3]=3

Get the dimensions of wavelet coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-wuwmcl
Out[4]=4

Plot the wavelet coefficients:

In[5]:=5
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-15wf00
Out[5]=5

Compute a StationaryWaveletTransform:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-h4s71h
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-lgbd1m

View the tree of wavelet coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-jbb9sz
Out[3]=3

Get the dimensions of wavelet coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-msff83
Out[4]=4

Plot the wavelet coefficients:

In[5]:=5
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-yakm1p
Out[5]=5

Compute a StationaryWaveletPacketTransform:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-gbl37f
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-4fxpz9

View the tree of wavelet coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-9rgpx
Out[3]=3

Get the dimensions of wavelet coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-ngfrzh
Out[4]=4

Plot the wavelet coefficients:

In[5]:=5
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-q5dqbp
Out[5]=5

Compute a LiftingWaveletTransform:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-4t2boe
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-6vbut6

View the tree of wavelet coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-xwj8z0
Out[3]=3

Get the dimensions of wavelet coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-56ffkm
Out[4]=4

Plot the wavelet coefficients:

In[5]:=5
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-vyacdo
Out[5]=5

Higher Dimensions  (1)

Multivariate scaling and wavelet functions are products of univariate ones:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-rk8e1w
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-tvf11
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-yf2o9
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-s16yjj
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-gmiius
Out[5]=5

Applications  (3)Sample problems that can be solved with this function

Approximate a function using Daubechies wavelet coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-0yl9y

Perform a LiftingWaveletTransform:

In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-6b212t

Approximate original data by keeping n largest coefficients and thresholding everything else:

In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-btzo2q

Compare the different approximations:

In[4]:=4
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-j4qkd4
Out[4]=4

Compute the multiresolution representation of a signal containing an impulse:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-j91b58
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-87cm7i
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-3mqwfq
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-zg56ve
Out[4]=4

Compare the cumulative energy in a signal and its wavelet coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-vo8pov
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-p7milv
Out[2]=2

Compute the ordered cumulative energy in the signal:

In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-d0i4sz
In[4]:=4
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-hqrz4
Out[4]=4

The energy in the signal is captured by relatively few wavelet coefficients:

In[5]:=5
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-7htmqa
In[6]:=6
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-dfvj8p
In[7]:=7
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-ck6x5d
Out[7]=7

Properties & Relations  (13)Properties of the function, and connections to other functions

DaubechiesWavelet[1] is equivalent to HaarWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-4rcqes
Out[1]=1

Lowpass filter coefficients sum to unity; :

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-bs7miu
Out[1]=1

Highpass filter coefficients sum to zero; :

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-jaf8rr
Out[1]=1

Scaling function integrates to unity; :

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-hbb7
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-7n8xl2
Out[2]=2

In particular, :

In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-6zbqqr
Out[3]=3

Wavelet function integrates to zero; :

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-7j8ngf
Out[1]=1

Wavelet function is orthogonal to the scaling function at the same scale; :

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-4ds4zb
Out[1]=1

The lowpass and highpass filter coefficients are orthogonal; :

In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-8vwdxn
Out[2]=2

DaubechiesWavelet[n] has n vanishing moment; :

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-dw8djm
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-gv7ath
Out[2]=2

This means linear signals are fully represented in the scaling functions part ({0}):

In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-i34fvs
In[4]:=4
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-fxiicw
Out[4]=4

Quadratic or higher-order signals are not:

In[5]:=5
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-bspdch
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-bpn9vr
Out[6]=6

satisfies the recursion equation :

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-yjbxzh
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-h6wmys

Plot the components and the sum of the recursion:

In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-c986c4
In[4]:=4
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-jgy4v3
Out[4]=4

satisfies the recursion equation :

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-3enc2f
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-bux40a

Plot the components and the sum of the recursion:

In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-4clwfm
In[4]:=4
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-p99dtx
Out[4]=4

Frequency response for is given by :

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-y5x0mm

The filter is a lowpass filter:

In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-za6vn3
Out[2]=2

The higher the order n, the flatter the response function at the ends:

In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-jynhuu
Out[3]=3

Fourier transform of is given by :

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-idptzz
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-u0b6k1
In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-z65kgr
Out[3]=3

Frequency response for is given by :

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-fujnuf

The filter is a highpass filter:

In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-s5vizp
Out[2]=2

The higher the order n, the flatter the response function at the ends:

In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-q3wbli
Out[3]=3

Fourier transform of is given by :

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-4ndhmw
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-8z0zd4
In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-hyy32u
In[4]:=4
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-f7zvgz
Out[4]=4

Neat Examples  (2)Surprising or curious use cases

Plot translates and dilations of scaling function:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-yz9dxl
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-evsjio
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-p5xgws
Out[3]=3

Plot translates and dilations of wavelet function:

In[1]:=1
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-iu0uje
In[2]:=2
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-b9ooxs
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0e2ykec4pon87s62mh2e-hts69i
Out[3]=3
Wolfram Research (2010), DaubechiesWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/DaubechiesWavelet.html.
✖
Wolfram Research (2010), DaubechiesWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/DaubechiesWavelet.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_daubechieswavelet, author="Wolfram Research", title="{DaubechiesWavelet}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/DaubechiesWavelet.html}", note=[Accessed: 07-March-2025 ]}

✖
@misc{reference.wolfram_2025_daubechieswavelet, author="Wolfram Research", title="{DaubechiesWavelet}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/DaubechiesWavelet.html}", note=[Accessed: 07-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_daubechieswavelet, organization={Wolfram Research}, title={DaubechiesWavelet}, year={2010}, url={https://reference.wolfram.com/language/ref/DaubechiesWavelet.html}, note=[Accessed: 07-March-2025 ]}

✖
@online{reference.wolfram_2025_daubechieswavelet, organization={Wolfram Research}, title={DaubechiesWavelet}, year={2010}, url={https://reference.wolfram.com/language/ref/DaubechiesWavelet.html}, note=[Accessed: 07-March-2025 ]}