BlackmanWindow

BlackmanWindow[x]

represents a Blackman window function of x.

Details

  • BlackmanWindow 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.
  • The Blackman window is a close approximation of the ExactBlackmanWindow. The truncated coefficients result in an improved high-frequency roll-off but at the cost of larger spectral side lobes.
  • BlackmanWindow[x] is equal to  1/(50) (25 cos(2 pi x)+4 cos(4 pi x)+21) -1/2<=x<=1/2; 0 TemplateBox[{x}, Abs]>1/2; .
  • BlackmanWindow automatically threads over lists.

Examples

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Basic Examples  (3)

Shape of a 1D Blackman window:

Shape of a 2D Blackman window:

Extract the continuous function representing the Blackman window:

Scope  (4)

Evaluate numerically:

Translated and dilated Blackman window:

2D Blackman window with a circular support:

Discrete Blackman window of length 15:

Discrete 15×10 2D Blackman window:

Applications  (4)

Create a lowpass FIR filter with cutoff frequency of and length 21:

Taper the filter using a Blackman window to improve stopband attenuation:

Normalize:

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

Filter a white noise signal using the Blackman window method:

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  (5)

The area under the Blackman window:

Normalize to create a window with unit area:

Fourier transform of the Blackman window:

Power spectrum of the Blackman window:

Discrete Blackman window of length 15:

Normalize so the coefficients add up to 1:

Discrete-time Fourier transform of a normalized discrete Blackman window of length 15:

Magnitude spectrum:

Power spectra of the Blackman and rectangular windows:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_blackmanwindow, author="Wolfram Research", title="{BlackmanWindow}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/BlackmanWindow.html}", note=[Accessed: 22-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_blackmanwindow, organization={Wolfram Research}, title={BlackmanWindow}, year={2012}, url={https://reference.wolfram.com/language/ref/BlackmanWindow.html}, note=[Accessed: 22-November-2024 ]}