### Hann or Hanning or Raised Cosine

The *Hann window* (or *hanning* or *raised-cosine*
window) is defined based on the settings
and
in
(3.17):

where .

The Hann window and its transform appear in Fig.3.9. The
Hann window can be seen as one period of a cosine ``raised'' so that
its negative peaks just touch zero (hence the alternate name ``raised cosine'').
Since it reaches zero at its endpoints with zero slope, the
discontinuity leaving the window is in the second derivative, or the
*third term* of its Taylor series expansion at an endpoint. As a
result, the side lobes roll off approximately 18 dB per octave. In
addition to the greatly accelerated roll-off rate, the first side lobe
has dropped from
dB (rectangular-window case) down to
dB. The main-lobe width is of course double that of the rectangular
window. For Fig.3.9, the window was computed in Matlab
as `hanning(21)`. Therefore, it is the variant that places the
zero endpoints one-half sample to the left and right of the outermost
window samples (see next section).

**Next Section:**

Matlab for the Hann Window

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Rectangular Window Summary