Analysis of signals from superposed relaxation processes

Abstract

System functions of many linear physical systems or autocorrelation functions of their output signals are often a sum of relaxator terms with different relaxation times and amplitudes. In the time domain a superposition of exponential decays e−γt is observed, while in the frequency domain a sum of relaxator terms of structure 1/(γ+iω) is measured. Thus, the response is either the Laplace transform of the system function or its Stieltjes transform, respectively. In both cases it is the task of an analyst to gain the relaxation times and weights from the measured signal. An exact reconstruction of the system function is limited by the noise, measuring time, and number of points measured. In this paper procedures for the approximate reconstruction of the system function are introduced. The equivalence of most of them is shown and their properties are discussed. An expression for the limit of resolution is derived for a given signal-to-noise ratio. The results are applicable to experimental data from various physical systems. For illustration the autocorrelation function of the light scattered from polymer solutions and the response of photoconductors are used.