LanczosWindow

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`LanczosWindow`

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LanczosWindow[x]

represents a Lanczos window function of x.

Details

LanczosWindow, also known as the sinc window, is typically used for antialiasing and resampling.
Window functions are used in applications where data is processed in short segments and have a smoothing effect by gradually tapering data values to zero at the ends of each segment.

LanczosWindow[x] is equal to .
LanczosWindow automatically threads over lists.

Examples

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

Shape of a 1D Lanczos window:

In[1]:=1

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https://wolfram.com/xid/0fq8dxyouu8gi-55nvy5

Out[1]=1

Shape of a 2D Lanczos window:

In[1]:=1

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https://wolfram.com/xid/0fq8dxyouu8gi-yyuiog

Out[1]=1

Extract the continuous function representing the Lanczos window:

In[1]:=1

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https://wolfram.com/xid/0fq8dxyouu8gi-i5bqnh

Out[1]=1

Scope (4)Survey of the scope of standard use cases

Evaluate numerically:

In[1]:=1

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https://wolfram.com/xid/0fq8dxyouu8gi-rfdjbb

Out[1]=1

Translated and dilated Lanczos window:

In[1]:=1

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https://wolfram.com/xid/0fq8dxyouu8gi-fzqpy6

Out[1]=1

2D Lanczos window with a circular support:

In[1]:=1

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https://wolfram.com/xid/0fq8dxyouu8gi-h4a5s3

Out[1]=1

Discrete Lanczos window of length 15:

In[1]:=1

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https://wolfram.com/xid/0fq8dxyouu8gi-5yhlc9

Out[1]=1

Discrete 15×10 2D Lanczos window:

In[2]:=2

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https://wolfram.com/xid/0fq8dxyouu8gi-ivcfcj

Out[2]=2

Applications (4)Sample problems that can be solved with this function

Decrease the size of an image by a factor of 5 using the Lanczos method:

In[2]:=2

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https://wolfram.com/xid/0fq8dxyouu8gi-1dcwws

Out[2]=2

By default, "Lanczos" resampling is used when resampling audio signals:

In[1]:=1

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https://wolfram.com/xid/0fq8dxyouu8gi-gme0eq

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0fq8dxyouu8gi-fl9fwi

Out[2]=2

Use a window specification to calculate sample PowerSpectralDensity:

In[1]:=1

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https://wolfram.com/xid/0fq8dxyouu8gi-la7hx0

Calculate the spectrum:

In[2]:=2

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https://wolfram.com/xid/0fq8dxyouu8gi-xvpxo9

Compare to spectral density calculated without a windowing function:

In[3]:=3

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https://wolfram.com/xid/0fq8dxyouu8gi-012m6s

In[4]:=4

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https://wolfram.com/xid/0fq8dxyouu8gi-phnp33

Out[4]=4

The plot shows that window smooths the spectral density:

In[5]:=5

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https://wolfram.com/xid/0fq8dxyouu8gi-z97d2x

Out[5]=5

Compare to the theoretical spectral density of the process:

In[6]:=6

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https://wolfram.com/xid/0fq8dxyouu8gi-2bqb4v

Out[6]=6

Use a window specification for time series estimation:

In[1]:=1

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https://wolfram.com/xid/0fq8dxyouu8gi-8tceex

Specify window for spectral estimator:

In[2]:=2

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https://wolfram.com/xid/0fq8dxyouu8gi-nkbyb7

Out[2]=2

Properties & Relations (2)Properties of the function, and connections to other functions

The area under the Lanczos window:

In[1]:=1

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https://wolfram.com/xid/0fq8dxyouu8gi-bvk48s

Out[1]=1

Normalize to create a window with unit area:

In[2]:=2

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https://wolfram.com/xid/0fq8dxyouu8gi-uvswua

Out[2]=2

Fourier transform of the Lanczos window:

In[1]:=1

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https://wolfram.com/xid/0fq8dxyouu8gi-hw628m

Out[1]=1

Power spectrum of the Lanczos window:

In[2]:=2

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https://wolfram.com/xid/0fq8dxyouu8gi-6mml6t

Out[2]=2

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