WelchWindow
✖
WelchWindow
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
.
- WelchWindow[x] is equivalent to WelchWindow[x,1].
- WelchWindow automatically threads over lists.

Examples
open allclose allBasic Examples (3)Summary of the most common use cases

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


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

Extract the continuous function representing the Welch window:

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


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

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

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

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

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

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

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

Translated and dilated Welch window:

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

2D Welch window with a circular support:

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

Discrete Welch window of length 15:

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

Discrete 15×10 2D Welch window:

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

Applications (3)Sample problems that can be solved with this function
Create a moving-average filter of length 11:

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

Taper the filter using a Welch window:

https://wolfram.com/xid/01zj6vfvu6a-1k06dy
Log-magnitude plot of the power spectra of the filters:

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

Use a window specification to calculate sample PowerSpectralDensity:

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

https://wolfram.com/xid/01zj6vfvu6a-xvpxo9
Compare to spectral density calculated without a windowing function:

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

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

The plot shows that the window smooths the spectral density:

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

Compare to the theoretical spectral density of the process:

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

Use a window specification for time series estimation:

https://wolfram.com/xid/01zj6vfvu6a-8tceex
Specify window for spectral estimator:

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

Properties & Relations (2)Properties of the function, and connections to other functions
The area under the Welch window:

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

Normalize to create a window with unit area:

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

Fourier transform of the Welch window:

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

Power spectrum of the Welch window:

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

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:

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


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

Wolfram Research (2012), WelchWindow, Wolfram Language function, https://reference.wolfram.com/language/ref/WelchWindow.html.
Text
Wolfram Research (2012), WelchWindow, Wolfram Language function, https://reference.wolfram.com/language/ref/WelchWindow.html.
Wolfram Research (2012), WelchWindow, Wolfram Language function, https://reference.wolfram.com/language/ref/WelchWindow.html.
CMS
Wolfram Language. 2012. "WelchWindow." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WelchWindow.html.
Wolfram Language. 2012. "WelchWindow." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WelchWindow.html.
APA
Wolfram Language. (2012). WelchWindow. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WelchWindow.html
Wolfram Language. (2012). WelchWindow. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WelchWindow.html
BibTeX
@misc{reference.wolfram_2025_welchwindow, author="Wolfram Research", title="{WelchWindow}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/WelchWindow.html}", note=[Accessed: 05-June-2025
]}
BibLaTeX
@online{reference.wolfram_2025_welchwindow, organization={Wolfram Research}, title={WelchWindow}, year={2012}, url={https://reference.wolfram.com/language/ref/WelchWindow.html}, note=[Accessed: 05-June-2025
]}