Hann or Hanning or Raised Cosine

The Hann window (or hanning or raised-cosine window) is defined based on the settings and in (3.1):

The Hann window and its transform appear in Fig.3.3. The Hann window can be seen as one period of a cosine ``raised'' so that its negative peaks just touch zero (hence the 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.3.3, 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.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.