Frequency domain description of Kohlrausch response through a pair of Havriliak-Negami-type functions: An analysis of functional proximity

Abstract

An approximation to the Fourier transform (FT) of the Kohlrausch function (stretched exponential) with shape parameter $0<\ensuremath{\beta}\ensuremath{\leqslant}1$ is presented by using Havriliak-Negami-like functions. Mathematical expressions to fit their parameters $\ensuremath{\alpha}$, $\ensuremath{\gamma}$, and $\ensuremath{\tau}$, as functions of $\ensuremath{\beta}$ ($0<\ensuremath{\beta}\ensuremath{\leqslant}1$ and $1<\ensuremath{\beta}<2$) are given, which allows a quick identification in the frequency domain of the corresponding shape factor $\ensuremath{\beta}$. Reconstruction via fast Fourier transform of frequency approximants to time domain are shown as good substitutes in short times though biased in long ones (increasing discrepancies as $\ensuremath{\beta}\ensuremath{\rightarrow}1$). The method is proposed as a template to commute time and frequency domains when analyzing complex data. Such a strategy facilitates intensive algorithmic search of parameters while adjusting the data of one or several Kohlrausch-Williams-Watts relaxations.