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\begin{align}
w_0(n)\ &= w\left[ n+\tfrac{N}{2}\right]\\
&= a_0
\end{align}
</math>
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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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==== DPSS or Slepian window ====
The DPSS (discrete prolate spheroidal sequence) or [[Slepian function]], taper, or window [[Spectral concentration problem|maximizes the energy concentration in the main lobe]],<ref name=Slepian/> and is used in [[multitaper]] spectral analysis, which averages out noise in the spectrum and reduces information loss at the edges of the window.
The main lobe ends at a frequency bin given by the parameter ''α''.<ref name=KaiserDPSS/>
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where <math>C^{\mu}_{N}</math> is the [[Ultraspherical polynomial]] of degree N, and <math>x_0</math> and <math>\mu</math> control the side-lobe patterns.<ref name=Deczky/>
Certain specific values of <math>\mu</math> yield other well-known windows: <math>\mu=0</math> and <math>\mu=1</math> give the Dolph–Chebyshev and [[Tapio Saramäki|Saramäki]] windows respectively.<ref name=Kabal/> See [
{{clear}}
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