CauchyWindow

CauchyWindow[x]

represents a Cauchy window function of x.

CauchyWindow[x,α]

uses the parameter α.

Details

  • CauchyWindow, also known as the Abel 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.
  • CauchyWindow[x,α] is equal to  1/(4 alpha^2 x^2+1) -1/2<=x<=1/2; 0 TemplateBox[{x}, Abs]>1/2; .
  • CauchyWindow[x] is equivalent to CauchyWindow[x,3].
  • CauchyWindow automatically threads over lists.

Examples

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Basic Examples  (3)

Shape of a 1D Cauchy window:

Shape of a 2D Cauchy window:

Extract the continuous function representing the Cauchy window:

Parameterized Cauchy window:

Scope  (6)

Evaluate numerically:

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

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

Translated and dilated Cauchy window:

2D Cauchy window with a circular support:

Discrete Cauchy window of length 15:

Discrete 15×10 2D Cauchy window:

Applications  (2)

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)

CauchyWindow[x,0] is equivalent to a Dirichlet window:

The area under the Cauchy window:

Normalize to create a window with unit area:

Fourier transform of the Cauchy window:

Power spectrum of the Cauchy window:

Possible Issues  (1)

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_cauchywindow, author="Wolfram Research", title="{CauchyWindow}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/CauchyWindow.html}", note=[Accessed: 22-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_cauchywindow, organization={Wolfram Research}, title={CauchyWindow}, year={2012}, url={https://reference.wolfram.com/language/ref/CauchyWindow.html}, note=[Accessed: 22-February-2025 ]}