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Hann or Hanning or Raised Cosine

The Hann window (or hanning or raised-cosine window) is defined based on the settings $ \alpha=1/2$ and $ \beta=1/4$ in (3.17):

$\displaystyle w_H(n)=w_R(n) \left[ \frac{1}{2} + \frac{1}{2} \cos( \Omega_M n) \right] = w_R(n) \cos^2\left(\frac{\Omega_M}{2} n\right) \protect$ (4.18)

where $ \Omega_M \isdeftext 2\pi/M$ .

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 $ -13$ dB (rectangular-window case) down to $ -31.5$ 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).

Figure 3.9: The Hann window (hanning(21)) and its DTFT.

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Matlab for the Hann Window
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Rectangular Window Summary