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 waveforms and the spectral width of sources used for optical communications and the resolution of spectrometers. The convention of "width" meaning "half maximum" is also widely used in signal processing to define 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 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
If the considered function is the density of a normal distribution of the form
f ( x ) = 1 σ 2 π exp [ − ( x − x 0 ) 2 2 σ 2 ] {\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left[-{\frac {(x-x_{0})^{2}}{2\sigma ^{2}}}\right]}
where σ is the standard deviation and x0 is the expected value, then the relationship between FWHM and the standard deviation is
F W H M = 2 2 ln 2 σ ≈ 2.355 σ . {\displaystyle \mathrm {FWHM} =2{\sqrt {2\ln 2}}\;\sigma \approx 2.355\;\sigma .}
The FWHM does not depend on the expected value x0; it is invariant under translations. The area within this FWHM is approximately 76% of the total area under the function.
Gamma distribution The density of a gamma distribution is given by
f ( x ) = 1 Γ ( α ) θ α x α − 1 e − x / θ {\displaystyle f(x)={\frac {1}{\Gamma (\alpha )\theta ^{\alpha }}}x^{\alpha -1}e^{-x/\theta }} with shape parameter α {\displaystyle \alpha } and scale parameter θ. For α > 1 {\displaystyle \alpha >1} , the FWHM is given by
F W H M = ( α − 1 ) θ [ W 0 ( − 2 − 1 / ( α − 1 ) e ) − W − 1 ( − 2 − 1 / ( α − 1 ) e ) ] {\displaystyle \mathrm {FWHM} =(\alpha -1)\theta \left[W_{0}\left(-{\frac {2^{-1/(\alpha -1)}}{e}}\right)-W_{-1}\left(-{\frac {2^{-1/(\alpha -1)}}{e}}\right)\right]} where W k {\displaystyle W_{k}} denotes the k {\displaystyle k} th branch of the Lambert W function. When α = 1 {\displaystyle \alpha =1} , the gamma distribution reduces to an exponential distribution, whose FWHM is
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