TOPICS
Search

Bartlett Function

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
Bartlett

The apodization function

f(x)=1-(|x|)/a
(1)

which is a generalization of the one-argument triangle function. Its full width at half maximum is a.

It has instrument function

I(k)=asinc^2(pika),
(2)

where sinc(x) is the sinc function. The peak of I(k) is a, and the full width at half maximum is given by setting x=pika and numerically solving

sinc^2(x)=1/2
(3)

for x_(1/2), yielding

x_(1/2)=pik_(1/2)a=1.39156.
(4)

Therefore, with L=2a,

FWHM=2k_(1/2)=(0.885895)/a=(1.77179)/L.
(5)

The function I(k) is always positive, so there are no negative sidelobes. The extrema are given by differentiating I(k) with respect to k, defining r=ka, and setting equal to 0,

(cos(2pir)+pixsin(2pir)-1)/(pi^2k^2r)=0.
(6)

Solving this numerically gives minima of 0 at r=1, 2, 3, ..., and sidelobes of 0.047190, 0.01648, 0.00834029, ... at r=1.4303, 2.45892, 3.47089, ....

See also

Apodization Function, Instrument Function, Parzen Apodization Function, Triangle Function

Explore with Wolfram|Alpha

References

Bartlett, M. S. "Periodogram Analysis and Continuous Spectra." Biometrika 37, 1-16, 1950.Blackman, R. B. and Tukey, J. W. The Measurement of Power Spectra, From the Point of View of Communications Engineering. New York: Dover, pp. 98-99, 1959.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 554-556, 1992.

Referenced on Wolfram|Alpha

Bartlett Function

Cite this as:

Weisstein, Eric W. "Bartlett Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BartlettFunction.html

Subject classifications