Performing the dielectric circle with numerical transformations

In 1953 Bernhard Gross developed for relaxational, linear, and time invariant systems a scheme of integral transformations between the frequency domain, the time domain, and the distribution domain. Applying this scheme to dielectrics we call it the dielectric circle. The transformations are a Laplace transform, a Fourier transform, and a Hilbert transform. We derived numerical iterative methods to carry out the transformations and their inversions for experimental data with inherent noise. The spectrum of the data can be as wide as twelve decades and more. Our system also allows to perform the Kramers-Kronig relations between the real and the imaginary part of the susceptibility. In the numerical transformations the distribution domain as a basis is employed. A presumed distribution function is altered and adjusted iteratively by comparing data calculated from the presumed spectrum with the measured data. To gain stability against noise, which is a main problem in the transformations, smoothing procedures are carried out. Special attention is laid on the suppression of unphysical oscillations at the edges of the spectrum. To avoid these oscillations the spectrum is enlarged by artificially added data, which are erased after the completed transformation. The applicability of the method is demonstrated for a model function as well as for dielectric data recorded using silicon oxide.