# TukeyWindow

TukeyWindow[x]

represents a Tukey window function of x.

TukeyWindow[x,α]

uses the parameter α.

# Details • TukeyWindow, also known as the cosine-tapered 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.
• • TukeyWindow[x,α] is equal to .
• TukeyWindow[x] is equivalent to TukeyWindow[x,2/3].
• TukeyWindow automatically threads over lists.

# Examples

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## Basic Examples(3)

Shape of a 1D Tukey window:

Shape of a 2D Tukey window:

Extract the continuous function representing the Tukey window:

Parameterized Tukey window:

## Scope(6)

Evaluate numerically:

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

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

Translated and dilated Tukey window:

2D Tukey window with a circular support:

Discrete Tukey window of length 15:

Discrete 15×10 2D Tukey window:

## Applications(3)

Use the Tukey 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 the 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(10)

TukeyWindow[x,1] is equivalent to a Hann window:

TukeyWindow[x,0] is equivalent to a box window:

TukeyWindow[x,1] is equivalent to a Hann window:

Tukey window is a convolution of a unit pulse with a raised cosine:

The area under the Tukey window:

Normalize to create a window with unit area:

Fourier transform of the Tukey window:

Power spectrum of the Tukey window:

Fourier transform of the parametrized Tukey window:

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

Discrete-time Fourier transform of the discrete Tukey 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 :

## Possible Issues(1)

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