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Setting <math>a_0 = 0.5</math> produces a '''Hann window''':
:<math>w[n] = 0.5\; \left[1 - \cos \left ( \frac{2 \pi n}{N} \right) \right] = \sin^2 \left ( \frac{\pi n}{N} \right),</math><ref name=MWhann/>
named after [[Julius von Hann]], and sometimes referred to as ''Hanning'', which derived from the verb "to Hann".{{Citation needed|date=June 2025}} It is also known as the '''raised cosine''', because of its similarity to a [[raised-cosine distribution]].
This function is a member of both the [[#Cosine-sum windows|cosine-sum]] and [[#Power-of-sine/cosine_windows|power-of-sine]] families. Unlike the [[#Hann and Hamming windows|Hamming window]], the end points of the Hann window just touch zero. The resulting [[Spectral leakage|side-lobes]] roll off at about 18 dB per octave.<ref name=JOShann/>
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