An apodization function , also called the Hann function, frequently used to reduce leakage in discrete
Fourier transforms . The illustrations above show the Hanning function, its instrument function , and a blowup of the instrument
function sidelobes. It is named after the Austrian meteorologist Julius von Hann
(Blackman and Tukey 1959, pp. 98-99). The Hanning function is given by

Its full width at half maximum is .

It has instrument function

To investigate the instrument function, define the dimensionless parameter and rewrite the instrument function as

(5)

The half-maximum can then be seen to occur at

(6)

so for ,
the full width at half maximum is

(7)

To find the extrema, take the derivative

(8)

and equate to zero. The first two roots are and 10.7061..., corresponding to the first sidelobe
minimum ( )
and maximum ( ),
respectively.

See also Apodization Function ,

Hamming Function
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References Blackman, R. B. and Tukey, J. W. "Particular Pairs of Windows." §B.5 in The
Measurement of Power Spectra, From the Point of View of Communications Engineering.
New York: Dover, pp. 14-15 and 95-100, 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 Hanning Function
Cite this as:
Weisstein, Eric W. "Hanning Function."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/HanningFunction.html

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