A method of separation of exponentials and its relationship to time domain synthesis of a finite lumped-parameter relaxation system

Abstract

In this paper, the general mathematical theory of linear passive one-ports and the class of positive real functions are briefly reviewed as background material. Then a time domain method for synthesis of a finite lumped RC system is given, which involves breaking down the given system into n subsystems. Finally, it is shown that one method of separation of a finite number of exponentials from a given completely monotonic curve over a finite interval is related to the above mentioned method of time domain synthesis. Here the given curve is treated as the impulse response of a finite lumped RC system. It is shown that the parameters being sought are related to the eigenvalues of the system matrix and the first components of the orthonormal eigenvectors corresponding to those eigenvalues, and it is proved that the system matrix has distinct positive eigenvalues.