The apodization function

(1)

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

It has instrument function

(2)

where
is the sinc function . The peak of is , and the full width
at half maximum is given by setting and numerically solving

(3)

for ,
yielding

(4)

Therefore, with ,

(5)

The function
is always positive, so there are no negative sidelobes.
The extrema are given by differentiating with respect to , defining , and setting equal to 0,

(6)

Solving this numerically gives minima of 0 at , 2, 3, ..., and sidelobes of 0.047190, 0.01648, 0.00834029,
... at ,
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