An N-Port Realizability Theory Based on the Theory of Distributions

This paper develops a realizability theory for -ports having immittance matrices through the use of Schwartz\´s theory of distributions [1]. The development given here differs from other such theories, which relate the passivity of the -port to the positivereality of the immittance matrix, in that the use of distribution theory produces a number of simplifications and leads to a comparatively concise yet rigorous realizability theory. This theory is based on but two postulates: 1) The -port has a convolution representation; 2) The -port is passive. Taken together, they are entirely equivalent to the properties of single-valuedness, linearity, time-invariance, continuity, passivity and causality. These two postulates are also necessary and sufficient for the immittance matrix of the -port to exist and be positive-real. The last statement is the main conclusion of this paper. The concept of an -port is extended here in that its driving and responding port variables may now be distributions as well as ordinary functions, this extension being made in a rigorous way. Also, a representation for positive-real matrices that is due to Youla [5] is exploited to obtain an explicit time-domain representation for passive -ports having convolutions representations; this representation is shown to encompass the 1-port representations obtained by König and Meixner [4].