The Transformation of Time-Domain Relaxation Data into the Frequency Domain

A numerical technique is developed for computing the Laplace transform of the time-domain behavior of a dielectric in order to obtain its frequencydomain behavior. The method is based on fitting a cubic spline to the original data and using the spline to define the integration. The error in the computation is investigated for data uniformly spaced on a logarithmic time scale. It is shown that the error is much smaller than with previous methods, is computationally stable, and converges as the fourth power of the sample density. For an error of 10-4 or less, only 10 points per decade are required for all frequencies that correspond to the time window of the measurement. It is also shown that it is possible to estimate those parts of the integrals that lie outside the measurement window from the data inside the window, so that the errors from the unknown parts are kept small and affect only the extremes of the frequency range.