< List of probability distributions < Voigt distribution

## What is the Voigt distribution?

The **Voigt distribution **is a convolution of two distributions that are commonly used in spectroscopy and diffraction—the normal (Gaussian) distribution and the Lorentzian (Cauchy) distribution [1]. It was originally developed in 1902 by Woldemar Voigt for modeling line broadening in spectroscopy, though it has since been applied to many different fields.

No analytical solution exists for the Gaussian/Lorentzian convolution integral, but it’s cumulative distribution function (CDF) can be expressed as

where erf(z) is the error function.

The Voigt distribution can be represented as a single parametrized function with two parameters, *H *and *w*. The shape of this function changes depending on *w*, which refers to the ratio between the Gaussian (normal) standard deviation and Lorentzian half-width at half-maximum (HWHM).

- When
*w*equals 0, the shape of the curve becomes purely Gaussian (normal); - when
*w*equals 1, its shape is purely Lorentzian; - when
*w*lies between 0 and 1, its shape is an intermediate form between these two distributions.

## Applications of The Voigt Distribution

Today, the Voigt distribution has numerous applications across various disciplines. In atmospheric sciences and telecommunications related to molecular spectroscopy or radiative transfer (e.g., infrared radiation), it can be used to model line broadening from pressure effects or Doppler shifting from thermal motion of molecules. It also has applications in electrical engineering for signal processing tasks such as filter design or modulation/demodulation techniques for digital transmission systems.

In mathematics and statistical mechanics, it is used for solving variational problems with constraints through Lagrange multipliers or finding solutions to Fredholm integral equations of first kind. Additionally, its analytical properties have been applied in other areas such as medical imaging and finance theory.

## References

[1] Victoria University Nano-optics and plasmonics. The Voigt function, corresponding to the convolution of a lorentzian and a gaussian distribution: http://nano-optics.ac.nz/rcppfaddeeva/reference/Voigt.html

[2] Dinnebier, R. (2018). Rietveld Refinement, Practical Powder Diffraction Pattern Analysis Using TOPAS. De Gruyter.

[3] Rocco, H. The Voigt Profile as a Sum of a Gaussian and a Lorentzian Functions, when the Weight Coefficient Depends Only on the Widths Ratio. ACTA PHYSICA POLONICA A, Vol 122 (2012)