The Voigt Profile as a Sum of a Gaussian and a Lorentzian Functions, when the Weight Coefficient Depends on the Widths Ratio and the Independent Variable

Abstract

Assuming that V (x) ≈ (1 − µ) G1(x) + µL1(x) is a very good approximation of the Voigt function, in this work we analytically nd µ from mathematical properties of V (x). G1(x) and L1(x) represent a Gaussian and a Lorentzian function, respectively, with the same height and HWHM as V (x), the Voigt function, x being the distance from the function center. In this paper we extend the analysis that we have done in a previous paper, where µ is only a function of a; a being the ratio of the Lorentz width to the Gaussian width. Using one of the dierential equation that V (x) satises, in the present paper we obtain µ as a function, not only of a, but also of x. Kielkopf rst proposed µ(a, x) based on numerical arguments. We nd that the Voigt function calculated with the expression µ(a, x) we have obtained in this paper, deviates from the exact value less than µ(a) does, specially for high |x| values.

Robust Frequency Estimation Under Additive Symmetric α-Stable Gaussian Mixture Noise

“…where 0 nðaÞ 1 is the mixed coefficient, f 1 ðejgÞ and f 2 ðejkÞ denote the unnormalized Cauchy and Gaussian processes, with the dispersion η and variance λ 2 [38], respectively. For the sake of the analytical form of nðaÞ, f 1 ðejgÞ and f 2 ðejk 2 Þ, previous works [45][46][47] are developed, among which the most accurate expression is…”

Optimum linear regression in additive Cauchy–Gaussian noise

“…This motivates us to rewrite the Voigt profile as the sum of two new Cauchy and Gaussian functions, which is called pseudo-Voigt function [20][21][22][23]:…”

“…However, the analytical form of μ a ðe n ; γ; σ 2 Þ is very complicated to calculate [23]. For the sake of computational simplicity, previous works [20][21][22] focus on μ a ðγ; σ 2 Þ instead of μ a ðe n ; γ; σ 2 Þ.…”