WelchWindow

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`WelchWindow`

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WelchWindow[x]

represents a Welch window function of x.

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WelchWindow[x,α]

uses the parameter α.

Details

WelchWindow is a window function 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.

WelchWindow[x,α] is equal to $ (alpha^2-4 x^2)/(alpha^2) -1/2<=x<=1/2; 0 TemplateBox[{x}, Abs]>1/2;$ .
WelchWindow[x] is equivalent to WelchWindow[x,1].
WelchWindow automatically threads over lists.

Examples

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

Shape of a 1D Welch window:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-55nvy5

Out[1]=1

Shape of a 2D Welch window:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-yyuiog

Out[1]=1

Extract the continuous function representing the Welch window:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-i5bqnh

Out[1]=1

Parameterized Welch window:

In[2]:=2

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https://wolfram.com/xid/01zj6vfvu6a-zwjlmu

Out[2]=2

Scope (6)Survey of the scope of standard use cases

Evaluate numerically:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-rfdjbb

Out[1]=1

Shape of a 1D Welch window using a specified parameter:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-y9d2xu

Out[1]=1

Variation of the shape as a function of the parameter α:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-eu37ay

Out[1]=1

Translated and dilated Welch window:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-fzqpy6

Out[1]=1

2D Welch window with a circular support:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-h4a5s3

Out[1]=1

Discrete Welch window of length 15:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-5yhlc9

Out[1]=1

Discrete 15×10 2D Welch window:

In[2]:=2

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https://wolfram.com/xid/01zj6vfvu6a-ivcfcj

Out[2]=2

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

Create a moving-average filter of length 11:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-b4svbb

Out[1]=1

Taper the filter using a Welch window:

In[5]:=5

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https://wolfram.com/xid/01zj6vfvu6a-1k06dy

Log-magnitude plot of the power spectra of the filters:

In[7]:=7

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https://wolfram.com/xid/01zj6vfvu6a-65dzhv

Out[7]=7

Use a window specification to calculate sample PowerSpectralDensity:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-la7hx0

Calculate the spectrum:

In[2]:=2

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https://wolfram.com/xid/01zj6vfvu6a-xvpxo9

Compare to spectral density calculated without a windowing function:

In[3]:=3

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https://wolfram.com/xid/01zj6vfvu6a-012m6s

In[4]:=4

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https://wolfram.com/xid/01zj6vfvu6a-phnp33

Out[4]=4

The plot shows that the window smooths the spectral density:

In[5]:=5

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https://wolfram.com/xid/01zj6vfvu6a-z97d2x

Out[5]=5

Compare to the theoretical spectral density of the process:

In[6]:=6

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https://wolfram.com/xid/01zj6vfvu6a-2bqb4v

Out[6]=6

Use a window specification for time series estimation:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-8tceex

Specify window for spectral estimator:

In[2]:=2

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https://wolfram.com/xid/01zj6vfvu6a-nkbyb7

Out[2]=2

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

The area under the Welch window:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-bvk48s

Out[1]=1

Normalize to create a window with unit area:

In[2]:=2

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https://wolfram.com/xid/01zj6vfvu6a-uvswua

Out[2]=2

Fourier transform of the Welch window:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-hw628m

Out[1]=1

Power spectrum of the Welch window:

In[2]:=2

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https://wolfram.com/xid/01zj6vfvu6a-6mml6t

Out[2]=2

Possible Issues (1)Common pitfalls and unexpected behavior

2D sampling of Welch window will use a different parameter for each row of samples when passed as a symbol to Array:

In[1]:=1

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https://wolfram.com/xid/01zj6vfvu6a-0oh59a

Out[1]=1

Use a pure function instead:

In[2]:=2

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https://wolfram.com/xid/01zj6vfvu6a-iqrxnt

Out[2]=2

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