Editing Full width at half maximum

{{Short description|Concept in statistics and wave theory}}

[[Image:FWHM.svg|thumb|250px|right|Full width at half maximum]]

In a distribution, '''full width at half maximum''' ('''FWHM''') is the difference between the two values of the [[independent variable]] at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve measured between those points on the ''y''-axis which are half the maximum amplitude.
'''Half width at half maximum''' ('''HWHM''') is half of the FWHM if the function is symmetric.
The term '''full duration at half maximum''' (FDHM) is preferred when the independent variable is [[time]].

FWHM is applied to such phenomena as the duration of [[pulse (signal processing)|pulse]] waveforms and the [[spectral width]] of sources used for [[optical communication]]s and the resolution of [[spectrometer]]s.
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 −3&nbsp;[[decibel|dB]] of attenuation, called ''half-power point'' or, more specifically, ''[[half-power bandwidth]]''.
When half-power point is applied to antenna [[beam width]], it is called ''[[half-power beam width]]''.

==Specific distributions==

===Normal distribution===
{{see also|Gaussian beam#Beam waist}}

If the considered function is the density of a [[normal distribution]] of the form
<math display="block">f(x) = \frac{1}{\sigma \sqrt{2 \pi} } \exp \left[ -\frac{(x-x_0)^2}{2 \sigma^2} \right]</math>
where ''σ'' is the [[standard deviation]] and ''x''<sub>0</sub> is the [[expected value]], 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 display="block"> \mathrm{FWHM} =   2\sqrt{2 \ln 2 } \; \sigma \approx 2.355 \; \sigma.</math>
The FWHM does not depend on the expected value ''x''<sub>0</sub>; it is invariant under translations.
The area within this FWHM is approximately 76% of the total area under the function.

===Other distributions===
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 {{sfrac|1|''πγ''}} can be defined by
<math display="block">f(x) = \frac{1}{\pi\gamma \left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]} \quad \text{ and } \quad \mathrm{FWHM} = 2 \gamma. </math>

Another important distribution function, related to [[soliton]]s in [[optics]], is the [[hyperbolic secant distribution|hyperbolic secant]]:
<math display="block">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 display="block">\mathrm{FWHM} = 2 \operatorname{arcsch} \left(\tfrac{1}{2}\right) X = 2 \ln (2 + \sqrt{3}) \; X \approx 2.634 \; X </math>
where {{math|arcsch}} is the [[Inverse hyperbolic function|inverse hyperbolic secant]].

== See also ==
*[[Gaussian function]]
*[[Cutoff frequency]]
*[[Spatial resolution]]

== References ==
*{{FS1037C}}
{{reflist}}

== External links ==
*[http://mathworld.wolfram.com/FullWidthatHalfMaximum.html FWHM at Wolfram Mathworld]

[[Category:Statistical deviation and dispersion]]
[[Category:Telecommunication theory]]
[[Category:Waves]]