represents a Parzen window function of x.


  • ParzenWindow 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.
  • ParzenWindow[x] is equal to  -2 (2 x-1)^3 1/4<x<=1/2; 2 (2 x+1)^3 -1/2<=x<-1/4; -48 x^3-24 x^2+1 -1/4<=x<0; 48 x^3-24 x^2+1 0<=x<=1/4; 0 TemplateBox[{x}, Abs]>1/2; .
  • ParzenWindow automatically threads over lists.


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Basic Examples  (3)

Shape of a 1D Parzen window:

Shape of a 2D Parzen window:

Extract the continuous function representing the Parzen window:

Scope  (4)

Evaluate numerically:

Translated and dilated Parzen window:

2D Parzen window with a circular support:

Discrete Parzen window of length 15:

Discrete 15×11 2D Parzen window:

Applications  (4)

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

Create a moving-average filter of length 21:

Taper the filter using a Parzen 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 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 Parzen window:

Normalize to create a window with unit area:

Fourier transform of the Parzen window:

Power spectrum of the Parzen window:

Wolfram Research (2012), ParzenWindow, Wolfram Language function,


Wolfram Research (2012), ParzenWindow, Wolfram Language function,


Wolfram Language. 2012. "ParzenWindow." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2012). ParzenWindow. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_parzenwindow, author="Wolfram Research", title="{ParzenWindow}", year="2012", howpublished="\url{}", note=[Accessed: 16-July-2024 ]}


@online{reference.wolfram_2024_parzenwindow, organization={Wolfram Research}, title={ParzenWindow}, year={2012}, url={}, note=[Accessed: 16-July-2024 ]}