KaiserWindow

KaiserWindow[x]

represents a Kaiser window function of x.

KaiserWindow[x,α]

uses the parameter α.

Details

  • KaiserWindow, also know as KaiserBessel window, is a window function typically used for finite impulse response (FIR) filter design and spectral analysis.
  • 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.
  • KaiserWindow[x,α] is equivalent to for -x and 0 otherwise.
  • KaiserWindow[x] is equivalent to KaiserWindow[x,3].
  • KaiserWindow automatically threads over lists.

Examples

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Basic Examples  (3)

Shape of a 1D Kaiser window:

Shape of a 2D Kaiser window:

Extract the continuous function representing the Kaiser window:

Parameterized Kaiser window:

Scope  (8)

Evaluate numerically:

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

Translated and dilated Kaiser window:

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

Shape of a 2D Kaiser window:

Two-dimensional Kaiser window with a circular support:

Discrete Kaiser window of length 15:

Discrete 15×10 2D Kaiser window:

Applications  (5)

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

Create a lowpass FIR filter with cutoff frequency of and length 15:

Apply a Kaiser window to the filter to improve stopband attenuation:

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

Filter a white noise signal using the Kaiser window method:

Use a window specification to calculate sample PowerSpectralDensity of an ARMA process:

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  (7)

The area under the Kaiser window:

Normalize to create a window with unit area:

Fourier transform of the Kaiser window:

Power spectrum of the Kaiser window:

Fourier transform of the parametrized Kaiser window:

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

Discrete-time Fourier transform of the discrete Kaiser 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 α:

Power spectra of the Kaiser and rectangular windows:

Possible Issues  (1)

Two-dimensional sampling of Kaiser 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), KaiserWindow, Wolfram Language function, https://reference.wolfram.com/language/ref/KaiserWindow.html.

Text

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

BibTeX

@misc{reference.wolfram_2020_kaiserwindow, author="Wolfram Research", title="{KaiserWindow}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/KaiserWindow.html}", note=[Accessed: 14-April-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_kaiserwindow, organization={Wolfram Research}, title={KaiserWindow}, year={2012}, url={https://reference.wolfram.com/language/ref/KaiserWindow.html}, note=[Accessed: 14-April-2021 ]}

CMS

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

APA

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