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

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