represents a Welch window function of x.


uses the parameter α.


  • 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.


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Basic Examples  (3)

Shape of a 1D Welch window:

Shape of a 2D Welch window:

Extract the continuous function representing the Welch window:

Parameterized Welch window:

Scope  (6)

Evaluate numerically:

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

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

Translated and dilated Welch window:

2D Welch window with a circular support:

Discrete Welch window of length 15:

Discrete 15×10 2D Welch window:

Applications  (3)

Create a moving-average filter of length 11:

Taper the filter using a Welch window:

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

Use a window specification to calculate sample PowerSpectralDensity:

Calculate the spectrum:

Compare to spectral density calculated without a windowing function:

The plot shows that the window smooths the spectral density:

Compare to the theoretical spectral density of the process:

Use a window specification for time series estimation:

Specify window for spectral estimator:

Properties & Relations  (2)

The area under the Welch window:

Normalize to create a window with unit area:

Fourier transform of the Welch window:

Power spectrum of the Welch window:

Possible Issues  (1)

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

Use a pure function instead:

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.


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. Retrieved from https://reference.wolfram.com/language/ref/WelchWindow.html


@misc{reference.wolfram_2024_welchwindow, author="Wolfram Research", title="{WelchWindow}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/WelchWindow.html}", note=[Accessed: 22-May-2024 ]}


@online{reference.wolfram_2024_welchwindow, organization={Wolfram Research}, title={WelchWindow}, year={2012}, url={https://reference.wolfram.com/language/ref/WelchWindow.html}, note=[Accessed: 22-May-2024 ]}