ConnesWindow

ConnesWindow[x]

represents a Connes window function of x.

ConnesWindow[x,α]

uses the parameter α.

Details

  • ConnesWindow 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.
  • ConnesWindow[x,α] is equal to  ((alpha^2-4 x^2)^2)/(alpha^4) -1/2<=x<=1/2; 0 TemplateBox[{x}, Abs]>1/2; .
  • ConnesWindow[x] is equivalent to ConnesWindow[x,1].
  • ConnesWindow automatically threads over lists.

Examples

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Basic Examples  (3)

Shape of a 1D Connes window:

Shape of a 2D Connes window:

Extract the continuous function representing the Connes window:

Parameterized Connes window:

Scope  (6)

Evaluate numerically:

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

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

Translated and dilated Connes window:

2D Connes window with a circular support:

Discrete Connes window of length 15:

Discrete 15×10 2D Connes window:

Applications  (3)

Create a moving-average filter of length 11:

Taper the filter using a Connes window:

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

Use a window specification to calculate a 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 the window for the spectral estimator:

Properties & Relations  (2)

The area under the Connes window:

Normalize to create a window with unit area:

Fourier transform of the Connes window:

Power spectrum of the Connes window:

Possible Issues  (1)

2D sampling of a Connes 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), ConnesWindow, Wolfram Language function, https://reference.wolfram.com/language/ref/ConnesWindow.html.

Text

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

BibTeX

@misc{reference.wolfram_2021_conneswindow, author="Wolfram Research", title="{ConnesWindow}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ConnesWindow.html}", note=[Accessed: 04-December-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_conneswindow, organization={Wolfram Research}, title={ConnesWindow}, year={2012}, url={https://reference.wolfram.com/language/ref/ConnesWindow.html}, note=[Accessed: 04-December-2021 ]}

CMS

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

APA

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