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 < beta <= 1 is presented by using Havriliak-Negami-like functions. Mathematical expressions to fit their parameters alpha, gamma, and tau, as functions of beta (0 < beta <= 1 and 1 < beta < 2) are given, which allows a quick identification in the frequency domain of the corresponding shape factor 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 beta -> 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.