An Apodization Function, also called the Hann Function, frequently used to reduce Aliasing in
Fourier Transforms. The illustrations above show the Hanning function, its Instrument
Function, and a blowup of the Instrument Function sidelobes. The Hanning function is given by

(1) 
The Instrument Function for Hanning apodization can also be written

(2) 
Its Full Width at Half Maximum is . It has Apparatus Function
The first integral is

(4) 
The second integral can be rewritten
Combining (4) and (5) gives

(6) 
To find the extrema, define
and rewrite (6) as

(7) 
Then solve

(8) 
to find the extrema. The roots are and 10.7061, giving a peak Negative sidelobe of and a peak
Positive sidelobe (in units of ) of 0.00843441. The peak in units of is 1, and the fullwidth at half maximum is
given by setting (7) equal to 1/2 and solving for , yielding

(9) 
Therefore, with , the Full Width at Half Maximum is

(10) 
See also Apodization Function, Hamming Function
© 19969 Eric W. Weisstein
19990525