Bounded Real Scattering Matrices and the Foundations of Linear Passive Network Theory

Abstract

In this paper the most general linear, passive, time-invariantn-port (e.g., networks which may be both distributed and non-reciprocal) is studied from an axiomatic point of view, and a completely rigorous theory is constructed by the systematic use of theorems of Bochner and Wiener. Ann-port\Phiis defined to be an operator inH_n, the space of alln-vectors whose components are measurable functions of a real variablet, (- \infty < t < \infty)(and as such need not be single-valued). Under very weak conditions on the domain of\Phi, it is shown that linearity and passivity imply causality. In every case,\Phi_a, then-port corresponding to\phiaugmented bynseries resistors is always causal (\Phiis the "augmented network," Fig. 2). Under the further assumptions that the domain of\Phi_ais dense in Hilbert space and\phiis time-invariant, it is proved that\Phipossesses a frequency response and defines ann \times nmatrixS(z)(the scattering matrix) of a complex variablez = \omega + i\betawith the following properties: 1)S(z)is analytic in Imz > 0; 2)Q(z) = I_n - S^{\ast}(z)S(z)is the matrix of a non-negative quadratic form for allzin the strict upper half-plane and almost all\omega. Conversely, it is also established that any such matrix represents the scattering description of a linear, passive, time-invariantn-port\Phisuch that the domain of\Phi_acontains all of Hilbert space. Such matrices are termed "bounded real scattering matrices" and are a generalization of the familiar positive-real immittance matrices. When\Phiand\Phi^{-1}are single-valued, it is possible to define two auxiliary positive-real matricesY(z)andZ(z), the admittance and impedance matrices of\Phi, respectively, which either exist for allzin Imz > 0and almost all\omegaor nowhere. The necessary and sufficient conditions for anm>n \times n matrix A_{n}(z)to represent either the scattering or immittance description of a linear, passive, time-invariantn-port\Phiare derived in terms of the real frequency behavior ofA_{n}(\omega). Necessary and sufficient conditions for\Phi_ato admit the representationi(t) = \int_{-\infty}^{\infty} dW_{n}(\tau)e(t - {\tau})for all integrablee(t)in its domain are given in terms ofS(z). The last section concludes with a discussion concerning the nature of the singularities ofS(z)and the possible extension of the theory to active networks.