# GaussianWindow

represents a Gaussian window function of x.

GaussianWindow[x,σ]

uses standard deviation σ.

# Details

• GaussianWindow 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.
• GaussianWindow[x,α] is equal to .
• is equivalent to GaussianWindow[x,3/10].
• GaussianWindow automatically threads over lists.

# Examples

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## Basic Examples(3)

Shape of a 1D Gaussian window:

Shape of a 2D Gaussian window:

Extract the continuous function representing the Gaussian window:

Parameterized Gaussian window:

## Scope(6)

Evaluate numerically:

Shape of a 1D Gaussian window using a specified standard deviation:

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

Translated and dilated Gaussian window:

2D Gaussian window with a circular support:

Discrete Gaussian window of length 15:

Discrete 15×10 2D Gaussian window:

## Applications(3)

Create a moving average filter of length 11:

Smooth the filter using a Gaussian window:

Log-magnitude plot of the frequency spectrum 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(8)

GaussianWindow[x,] is equivalent to a Dirichlet window:

The area under the Gaussian window:

Normalize to create a window with unit area:

Fourier transform of the Gaussian window:

Power spectrum of the Gaussian window:

Fourier transform of the parametrized Gaussian window:

Variation of the magnitude spectrum as a function of the parameter σ:

Discrete-time Fourier transform of the discrete Gaussian window of length 11:

Magnitude at ω=0:

Magnitude spectrum:

Power spectra for three different window lengths:

Power spectra for three different values of the shape parameter σ:

Power spectra of the Gaussian and rectangular windows:

## Possible Issues(1)

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

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_gaussianwindow, author="Wolfram Research", title="{GaussianWindow}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/GaussianWindow.html}", note=[Accessed: 20-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_gaussianwindow, organization={Wolfram Research}, title={GaussianWindow}, year={2012}, url={https://reference.wolfram.com/language/ref/GaussianWindow.html}, note=[Accessed: 20-July-2024 ]}