Exponential analysis in physical phenomena

Many physical phenomena are described by first-order differential equations whose solution is an exponential decay. Determining the time constants and amplitudes of exponential decays from the experimental data is a common task in semiconductor physics (deep level transient spectroscopy), biophysics (fluorescence decay analysis), nuclear physics and chemistry (radioactive decays, nuclear magnetic resonance), chemistry and electrochemistry (reaction kinetics) and medical imaging. This review article discusses the fundamental mathematical limitations of exponential analysis, outlines the critical aspects of acquisition of exponential transients for subsequent analysis, and gives a comprehensive overview of numerical algorithms used in exponential analysis. In the first part of the article the resolution of exponential analysis as a function of noise in input decays is discussed. It is shown that two exponential decays can be resolved in a transient only if the ratio of their time constants is greater than the resolution limit, which can be explicitly calculated from the signal-to-noise ratio in the transient. Although the signal-to-noise ratio is generally limited by the sensitivity of the equipment, it is shown that digitalization of the decays may be a major source of noise. The requirements for type of analog-to-digital converter, number of digitized data points and duration of digitized transients, which must be met to obtain the theoretical resolution limit and to improve stability of the exponential analysis, are formulated. The second part of the review article gives an overview and comparison of major numerical techniques of exponential analysis, such as the nonlinear least squares fit, the Prony method, the method of modulating functions, the method of moments, the Laplace–Padé approximation, the Tikhonov regularization method, the Gardner transformation, the method of maximum entropy and others.