ExactBlackmanWindow

ExactBlackmanWindow[x]

represents an exact Blackman window function of x.

Details

  • ExactBlackmanWindow 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.
  • ExactBlackmanWindow[x] is equal to  (4620 cos(2 pi x)+715 cos(4 pi x)+3969)/(9304) -1/2<=x<=1/2; 0 TemplateBox[{x}, Abs]>1/2;
  • ExactBlackmanWindow automatically threads over lists.

Examples

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Basic Examples  (3)

Shape of a 1D exact Blackman window:

Shape of a 2D exact Blackman window:

Extract the continuous function representing the exact Blackman window:

Scope  (4)

Evaluate numerically:

Translated and dilated exact Blackman window:

2D exact Blackman window with a circular support:

Discrete exact Blackman window of length 15:

Discrete 15×10 2D exact Blackman window:

Applications  (3)

Create a moving average filter of length 11:

Smooth the filter using a exact Blackman 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  (2)

The area under the exact Blackman window:

Normalize to create a window with unit area:

Fourier transform of the exact Blackman window:

Power spectrum of the exact Blackman window:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_exactblackmanwindow, author="Wolfram Research", title="{ExactBlackmanWindow}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ExactBlackmanWindow.html}", note=[Accessed: 22-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_exactblackmanwindow, organization={Wolfram Research}, title={ExactBlackmanWindow}, year={2012}, url={https://reference.wolfram.com/language/ref/ExactBlackmanWindow.html}, note=[Accessed: 22-February-2025 ]}