Distribution Theory as the Basis of Generalized Passive-Network Analysis

The properties of linear time-invariant passive systems are described in the context of generalized function (distribution) theory. The postulatory description of these systems is phrased in both the scattering or wave formulation, and the impedance or voltage-current basis. One conclusion of this paper is that a specific definition of passivity (attributed to G. Raisbeck) leads to a complete description of the system only when the postulate of causality is separately invoked; i.e., this definition of passivity does not imply causality and it is therefore a more fundamental assumption. Another conclusion is that the two seemingly different sets of postulates (scattering and impedance) are in many instances identical; the differences arise solely in the assumptions made concerning the domains of the two operators. Distribution theory is also used in an essential way to obtain new existence theorems, for both formulations, which are stated entirely on the real-frequency axis
(p = j\omega)
. The language herein is deliberately that of generalized function theory, so that the class of admissible systems and the inputs to these systems are broadened. In particular, the results -can be applied directly to systems which are described in terms of elementary functions.