## Hanning Function

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
 (3)

The first integral is
 (4)

The second integral can be rewritten

 (5)

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 full-width 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)