represents a Poisson window function of x.


uses the parameter α.


  • PoissonWindow, also known as the exponential window, is a window function typically used in signal processing applications where data needs to be processed in short segments.
  • Window functions have a smoothing effect by gradually tapering data values to zero at the ends of each segment.
  • PoissonWindow[x,α] is equal to  ⅇ^(-2 alpha x) 0<=x<=1/2; ⅇ^(2 alpha x) -1/2<=x<0; 0 TemplateBox[{x}, Abs]>1/2; .
  • PoissonWindow[x] is equivalent to PoissonWindow[x,3].
  • PoissonWindow automatically threads over lists.


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Basic Examples  (3)

Shape of a 1D Poisson window:

Shape of a 2D Poisson window:

Extract the continuous function representing the Poisson window:

Parameterized Poisson window:

Scope  (6)

Evaluate numerically:

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

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

Translated and dilated Poisson window:

2D Poisson window with a circular support:

Discrete Poisson window of length 15:

Discrete 15×11 2D Poisson window:

Applications  (3)

Use the Poisson window to diminish the effect of signal partitioning when computing the spectrogram:

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  (3)

The area under the Poisson window:

Normalize to create a window with unit area:

Fourier transform of the Poisson window:

Power spectrum of the Poisson window:

Fourier transform of the parametrized Poisson window:

Variation of the magnitude spectrum of the Kaiser window as a function of the parameter :

Possible Issues  (1)

2D sampling of Poisson 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), PoissonWindow, Wolfram Language function, https://reference.wolfram.com/language/ref/PoissonWindow.html.


Wolfram Research (2012), PoissonWindow, Wolfram Language function, https://reference.wolfram.com/language/ref/PoissonWindow.html.


Wolfram Language. 2012. "PoissonWindow." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PoissonWindow.html.


Wolfram Language. (2012). PoissonWindow. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PoissonWindow.html


@misc{reference.wolfram_2024_poissonwindow, author="Wolfram Research", title="{PoissonWindow}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/PoissonWindow.html}", note=[Accessed: 21-July-2024 ]}


@online{reference.wolfram_2024_poissonwindow, organization={Wolfram Research}, title={PoissonWindow}, year={2012}, url={https://reference.wolfram.com/language/ref/PoissonWindow.html}, note=[Accessed: 21-July-2024 ]}