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
communications and the resolution of spectrometers
.
The convention of "width" meaning "half maximum" is also widely used in half-power bandwidth
.
When half-power point is applied to antenna half-power beam width
.
Specific distributions
Normal distribution
If the considered function is the density of a normal distribution of the form
where σ is the standard deviation and x0 is the expected value, then the relationship between FWHM and the standard deviation is[1]
![{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left[-{\frac {(x-x_{0})^{2}}{2\sigma ^{2}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d250fe5233cdc17d74ac5998d66fdcb23234ea47)
The corresponding area within this FWHM accounts to approximately 76%.
The width does not depend on the expected value x0; it is invariant under translations.
If the FWHM of a Gaussian function is known, then it can be integrated by simple multiplication.
Other distributions
In spectroscopy half the width at half maximum (here γ), HWHM, is in common use. For example, a Lorentzian/Cauchy distribution of height 1/πγ can be defined by
![{\displaystyle 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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eabd6080f9c6ed91bfcbaaf319a6f8b9900b931)
Another important distribution function, related to solitons in optics, is the hyperbolic secant:
Any translating element was omitted, since it does not affect the FWHM. For this impulse we have:
where arcsch is the
inverse hyperbolic secant
.
See also
References
This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22.
External links
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