TOPICS

# Bartlett 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, ....

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

## Explore with Wolfram|Alpha

More things to try:

## 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