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BattleLemarieWavelet
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BattleLemarieWavelet

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represents the Battle–Lemarié wavelet of order 3.

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represents the Battle–Lemarié wavelet of order n evaluated on equally spaced interval {-10,10}.

✖

represents the Battle–Lemarié wavelet of order n evaluated on equally spaced interval {-lim,lim}.

Details

Examples

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

Scaling function:

In[1]:=1
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https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-4lu6z4
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-kcilqx
Out[2]=2

Wavelet function:

In[1]:=1
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https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-b0xj5n
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-exatgm
Out[2]=2

Filter coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-bwbc06
Out[1]=1

Scope  (9)Survey of the scope of standard use cases

Basic Uses  (4)

Compute primal lowpass filter coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-knpxcc
Out[1]=1

Primal highpass filter coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-fx28gt
Out[1]=1

Battle–Lemarié scaling function of order 2:

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-vl9zbf
Out[1]=1

Battle–Lemarié scaling function of order 5:

In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-4ybkxk
Out[2]=2

Battle–Lemarié wavelet function of order 2:

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-5oc3h8
Out[1]=1

Battle–Lemarié wavelet function of order 5:

In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-hr24zd
Out[2]=2

Wavelet Transforms  (4)

Compute a DiscreteWaveletTransform:

In[1]:=1
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https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-t8skl0
In[2]:=2
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https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-8smgf7
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-59wiq3
Out[3]=3

View the tree of wavelet coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-td6tmt
Out[4]=4

Get the dimensions of wavelet coefficients:

In[5]:=5
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-mrjnjd
Out[5]=5

Plot the wavelet coefficients:

In[6]:=6
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-w7te5
Out[6]=6

BattleLemarieWavelet can be used to perform a DiscreteWaveletPacketTransform:

In[1]:=1
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https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-3wocue
In[2]:=2
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https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-22i9c
Out[2]=2

View the tree of wavelet coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-nwhfbi
Out[3]=3

Get the dimensions of wavelet coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-wuwmcl
Out[4]=4

Plot the wavelet coefficients:

In[5]:=5
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-15wf00
Out[5]=5

BattleLemarieWavelet can be used to perform a StationaryWaveletTransform:

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-h4s71h
In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-lgbd1m

View the tree of wavelet coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-jbb9sz
Out[3]=3

Get the dimensions of wavelet coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-msff83
Out[4]=4

Plot the wavelet coefficients:

In[5]:=5
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-yakm1p
Out[5]=5

BattleLemarieWavelet can be used to perform a StationaryWaveletPacketTransform:

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-gbl37f
In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-4fxpz9

View the tree of wavelet coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-9rgpx
Out[3]=3

Get the dimensions of wavelet coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-ngfrzh
Out[4]=4

Plot the wavelet coefficients:

In[5]:=5
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-q5dqbp
Out[5]=5

Higher Dimensions  (1)

Multivariate scaling and wavelet functions are products of univariate ones:

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-rk8e1w
In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-tvf11
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-yf2o9
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-s16yjj
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-gmiius
Out[5]=5

Properties & Relations  (11)Properties of the function, and connections to other functions

Lowpass filter coefficients approximately sum to unity; :

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-bs7miu
Out[1]=1

Highpass filter coefficients approximately sum to zero; :

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-jaf8rr
Out[1]=1

Scaling function integrates to unity; :

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-hbb7
In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-7n8xl2
Out[2]=2

Wavelet function integrates to zero; :

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-29ja4i
In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-7j8ngf
Out[2]=2

For even order n, scaling function is symmetrical about 1/2:

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-hul8b9
Out[1]=1

For even order n, wavelet function is antisymmetrical about 1/2:

In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-5sdm7w
Out[2]=2

For odd order n, scaling function is symmetrical about 0:

In[3]:=3
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-ukjz2w
Out[3]=3

For odd order n, wavelet function is symmetrical about 1/2:

In[4]:=4
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-pvtxlt
Out[4]=4

satisfies the recursion equation :

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-yjbxzh
In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-h6wmys

Plot the components and the sum of the recursion:

In[3]:=3
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-c986c4
In[4]:=4
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-jgy4v3
Out[4]=4

satisfies the recursion equation :

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-3enc2f
In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-bux40a

Plot the components and the sum of the recursion:

In[3]:=3
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-4clwfm
In[4]:=4
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-p99dtx
Out[4]=4

Frequency response for is given by :

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-y5x0mm

The filter is a lowpass filter:

In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-za6vn3
Out[2]=2

Frequency response for is given by :

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-fujnuf

The filter is a highpass filter:

In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-s5vizp
Out[2]=2

Fourier transform of is given by :

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-idptzz
In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-u0b6k1
In[3]:=3
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-z65kgr
Out[3]=3

Fourier transform of is given by :

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-4ndhmw
In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-8z0zd4
In[3]:=3
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-hyy32u
In[4]:=4
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-f7zvgz
Out[4]=4

Possible Issues  (1)Common pitfalls and unexpected behavior

BattleLemarieWavelet is restricted to n less than 15:

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-5wyn6y
Out[1]=1

BattleLemarieWavelet is not defined when n is not a positive machine integer:

In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-2551ne
Out[2]=2

Neat Examples  (2)Surprising or curious use cases

Plot translates and dilations of scaling function:

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-yz9dxl
In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-evsjio
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-p5xgws
Out[3]=3

Plot translates and dilations of wavelet function:

In[1]:=1
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-iu0uje
In[2]:=2
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-b9ooxs
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ejtb1f0liytxkecyq7xy-hts69i
Out[3]=3
Wolfram Research (2010), BattleLemarieWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/BattleLemarieWavelet.html.
✖
Wolfram Research (2010), BattleLemarieWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/BattleLemarieWavelet.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_battlelemariewavelet, author="Wolfram Research", title="{BattleLemarieWavelet}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/BattleLemarieWavelet.html}", note=[Accessed: 21-April-2025 ]}

✖
@misc{reference.wolfram_2025_battlelemariewavelet, author="Wolfram Research", title="{BattleLemarieWavelet}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/BattleLemarieWavelet.html}", note=[Accessed: 21-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_battlelemariewavelet, organization={Wolfram Research}, title={BattleLemarieWavelet}, year={2010}, url={https://reference.wolfram.com/language/ref/BattleLemarieWavelet.html}, note=[Accessed: 21-April-2025 ]}

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@online{reference.wolfram_2025_battlelemariewavelet, organization={Wolfram Research}, title={BattleLemarieWavelet}, year={2010}, url={https://reference.wolfram.com/language/ref/BattleLemarieWavelet.html}, note=[Accessed: 21-April-2025 ]}