Stability of a numerical Laplace transform for dielectric measurements

Abstract

The stability of a numerical Laplace transform used to convert time-domain dielectric loss data into the frequency domain is examined. It is shown that for a transform using cubic spline integration, cubic least squares interpolation over piecewise linearly sampled data and proper endpoint continuations, the uncertainty in the data transformed into the frequency domain is comparable to that of the original data. Specific topics covered include the effect of finite numeric precision of the data, noise spikes and data extrapolation. An analytic expression in terms of modified Bessel functions is developed to estimate the degree of polynomial needed to fit an exponential over a finite range in time. This last development is used to show that a low polynomial degree is needed for a ratio of final to starting time of less than two. >