Hanning Function


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 a.

It has instrument function


To investigate the instrument function, define the dimensionless parameter u=2pika and rewrite the instrument function as


The half-maximum can then be seen to occur at


so for L=2a, the full width at half maximum is


To find the extrema, take the derivative


and equate to zero. The first two roots are u=7.42023... and 10.7061..., corresponding to the first sidelobe minimum (-0.0267075a) and maximum (0.0084344060a), respectively.

See also

Apodization Function, Hamming Function

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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.

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