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[[Image:FWHM.svg|thumb|250px|right|full width at half maximum]]
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'''Full width at half maximum''' ('''FWHM''') is an expression of the extent of a function, given by the difference between the two extreme values of the [[independent variable]] at which the dependent variable is equal to half of its maximum value.


FWHM is applied to such phenomena as the duration of [[pulse (signal processing)|pulse]] waveforms and the [[spectral width]] of sources used for optical [[telecommunication|communications]] and the resolution of [[spectrometer]]s.
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The term ''full duration at half maximum'' (FDHM) is preferred when the independent variable is [[time]].
 
The convention of "width" meaning "half maximum" is also widely used in [[signal processing]] to define [[bandwidth (signal processing)|bandwidth]] as "width of frequency range where less than half the signal's power is attenuated", i.e., the power is at least half the maximum. In signal processing terms, this is at most &minus;3 [[decibel|dB]] of attenuation, called "[[half power point]]".
 
If the considered function is the [[normal distribution]] of the form
 
:<math>f(x) = \frac{1}{\sigma \sqrt{2 \pi} } \exp \left[ -\frac{(x-x_0)^2}{2 \sigma^2} \right]</math>
 
 
where <math>\sigma</math> is the [[standard deviation]] and <math>x_0</math> can be any value (the width of the function does not depend on translation), then the relationship between FWHM and the [[standard deviation]] is<ref>[http://mathworld.wolfram.com/GaussianFunction.html Gaussian Function - from Wolfram MathWorld<!-- Bot generated title -->]</ref>
 
:<math> \mathrm{FWHM} =  2\sqrt{2 \ln 2 } \; \sigma \approx 2.355 \; \sigma.</math>
 
In [[spectroscopy]] half the width at half maximum (here γ), HWHM, is in common use. For example, a [[cauchy distribution|Lorentzian/Cauchy distribution]] of height (1/πγ) can be defined by
:<math>f(x) = \frac{1}{\pi\gamma \left[1 + \left(\frac{x  - x_0}{\gamma}\right)^2\right]} </math> and <math>\mathrm{FWHM} =  2 \gamma </math>
 
Another important distribution function, related to [[soliton]]s in [[optics]], is the [[hyperbolic secant distribution|hyperbolic secant]]:
 
:<math>f(x)=\operatorname{sech} \left( \frac{x}{X} \right).</math>
 
Any translating element was omitted, since it does not affect the FWHM. For this impulse we have:
 
:<math>\mathrm{FWHM} =  2 \; \operatorname{arsech} \left( \frac{1}{2} \right) X = 2 \ln (2 + \sqrt{3}) \; X \approx 2.634 \; X </math>
 
where ''arsech'' is the [[Inverse hyperbolic function|inverse hyperbolic secant]].
 
== References ==
*{{FS1037C}}
{{reflist}}
 
== See also ==
*[[Gaussian function]]
*[[Cutoff frequency]]
 
== External links ==
*[http://mathworld.wolfram.com/FullWidthatHalfMaximum.html FWHM at Wolfram Mathworld]
 
[[Category:Statistical deviation and dispersion]]
[[Category:Telecommunication theory]]
[[Category:Waves]]
 
{{Mathapplied-stub}}

Revision as of 04:13, 20 February 2014

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