Quantum reflection - Revision history

en>Rjwilmsi: /* Single-dimensional approximation */Journal cites, added 1 PMID using AWB (10484)

2014-11-03T17:15:04Z

Single-dimensional approximation: Journal cites, added 1 PMID using AWB (10484)

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-[[File:Window function and frequency response - Hann.svg|thumb|500px|right|Hann function with ''B''&nbsp;=&nbsp;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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en>Rjwilmsi: /* Casimir and van der Waals attraction */page/pages cleanup of Category:Pages with citations using conflicting page specifications, using AWB (9085)

2013-04-16T06:07:15Z

Casimir and van der Waals attraction: page/pages cleanup of Category:Pages with citations using conflicting page specifications, using AWB (9085)

← Older revision		Revision as of 08:07, 16 April 2013
Line 1:		Line 1:
	~~Marvella~~ is ~~what you can contact her but it's not~~ the ~~most female name out there~~. ~~It's not~~ a ~~common thing but what she likes doing~~ is ~~foundation leaping and now she~~ is ~~attempting~~ to ~~earn money with~~ it. ~~For years I've been operating~~ as a ~~payroll clerk~~. ~~California is where I've usually been living and I love each day living right here~~.<br><br>~~Also visit my web blog~~ ... [http://~~www~~.~~hotporn123~~.com/~~blog/154369 std testing~~ at ~~home~~]		[[File:Window function and frequency response - Hann.svg\|thumb\|500px\|right\|Hann function with ''B'' = 1.50 (left), and its frequency response (right)]]

			The '''Hann function''', named after the Austrian meteorologist [[Julius von Hann]], is a [[discrete_signal\|discrete]] [[window function]] given by

			:<math>w(n)= 0.5\; \left(1 - \cos \left ( \frac{2 \pi n}{N-1} \right) \right)</math>

			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}}

96.35.172.150: /* Definition */

2012-05-31T19:12:13Z

Definition

New page

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