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[[File:Window function and frequency response - Hann.svg|thumb|500px|right|Hann function with ''B'' = 1.50 (left), and its frequency response (right)]]
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The '''Hann function''', named after the Austrian meteorologist [[Julius von Hann]], is a [[discrete_signal|discrete]] [[window function]] given by
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:<math>w(n)= 0.5\; \left(1 - \cos \left ( \frac{2 \pi n}{N-1} \right) \right)</math>
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or
 
:<math>w(n)= \sin^2 \left ( \frac{ \pi n}{N-1} \right) </math>
 
or, in terms of the [[haversine]] function,
 
:<math>w(n)=\operatorname{haversin}\left(\frac {2 \pi n} {N-1} \right).</math>
 
== Spectrum ==
 
The Hann window is a linear combination of modulated [[Rectangular_function|rectangular windows]] <math>w_r = \mathbf{1}_{[0,N-1]}</math>. Thanks to the [[Euler formula]]
:<math>w(n)= \frac{1}{2} \,w_r(n) -\frac{1}{4} e^{\mathrm{i}2\pi \frac{n}{N-1}} w_r(n) - \frac{1}{4}e^{-\mathrm{i}2\pi \frac{n}{N-1}} w_r(n)</math>
Thanks to the basic properties of the [[Fourier transform]], its spectrum is
:<math>\hat{w} (\omega) = \frac{1}{2} \hat{w}_r (\omega) - \frac{1}{4} \hat{w}_r \left(\omega + \frac{2\pi}{N-1}\right) - \frac{1}{4} \hat{w}_r \left(\omega - \frac{2\pi}{N-1}\right) </math>
with the spectrum of the rectangular window
:<math>\hat{w}_r (\omega) = e^{-\mathrm{i} \omega \frac{N-1}{2}} \frac{\sin(N\omega/2)}{\sin(\omega/2)}</math>
(the modulation factor vanished if windows are time-shifted around 0)
 
== Name ==
 
Hann function is the original name, in honour of von Hann; however, the erroneous 'Hanning' function is also heard of on occasion, derived from the paper in which it was named, where the term "hanning a signal" was used to mean applying the Hann window to it.  The confusion arose from the similar [[Hamming function]], named after [[Richard Hamming]].
 
== Use ==
 
The Hann function is typically used as a [[window function]] in [[digital signal processing]] to select a subset of a series of samples in order to perform a [[Fourier transform]] or other calculations.
 
i.e. (using continuous version to illustrate)
:<math>S(\tau)= \int w(t+\tau)f(t) \, dt </math>
 
The advantage of the Hann window is very low [[aliasing]], and the tradeoff is slightly decreased resolution (widening of the main lobe). If the Hann window is used to sample a signal in order to convert to frequency domain, it is complex to reconvert to the time domain without adding distortions.
 
==See also==
* [[Apodization]]
* [[Raised cosine distribution]]
* [[Window function]]
 
==References==
*{{cite doi|10.1109/PROC.1978.10837}}
 
== External links ==
* [http://mathworld.wolfram.com/HanningFunction.html Hann function] at [[MathWorld]]
 
[[Category:Signal processing]]
 
 
{{mathapplied-stub}}

Latest revision as of 19:15, 3 November 2014

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