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

Abstract
This paper develops a realizability theory forn-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 then-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) Then-port has a convolution representation; 2) Then-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 then-port to exist and be positive-real. The last statement is the main conclusion of this paper. The concept of ann-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 passiven-ports having convolutions representations; this representation is shown to encompass the 1-port representations obtained by König and Meixner [4].

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