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.
The term half bandwidth is sometimes used to characterize spectroradiometers. Half bandwidth has the meaning of HWHM; it should not be conflated with the half-width of a band, which is synonymous with FWHM.
Contents
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
=
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
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
f
(
x
)
=
1
π
γ
[
1
+
(
x
−
x
0
γ
)
2
]
and
F
W
H
M
=
2
γ
.
{\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 .}
Another important distribution function, related to solitons in optics, is the hyperbolic secant: