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):
![$\displaystyle w_H(n)=w_R(n) \left[ \frac{1}{2} + \frac{1}{2} \cos( \Omega_M n) \right] = w_R(n) \cos^2\left(\frac{\Omega_M}{2} n\right) \protect$](http://www.dsprelated.com/josimages_new/sasp2/img362.png)
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).
![\includegraphics[width=4in]{eps/hanningWindow}](http://www.dsprelated.com/josimages_new/sasp2/img365.png)
Next Section:
Matlab for the Hann Window
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Rectangular Window Summary
