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

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-portPhiis 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 ofPhi, it is shown that linearity and passivity imply causality. In every case,Phi_a, then-port corresponding tophiaugmented bynseries resistors is always causal (Phiis the "augmented network," Fig. 2). Under the further assumptions that the domain ofPhi_ais dense in Hilbert space andphiis time-invariant, it is proved thatPhipossesses a frequency response and defines ann times nmatrixS(z)(the scattering matrix) of a complex variablez = omega + ibetawith 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 allomega. Conversely, it is also established that any such matrix represents the scattering description of a linear, passive, time-invariantn-portPhisuch that the domain ofPhi_acontains all of Hilbert space. Such matrices are termed "bounded real scattering matrices" and are a generalization of the familiar positive-real immittance matrices. WhenPhiandPhi^{-1}are single-valued, it is possible to define two auxiliary positive-real matricesY(z)andZ(z), the admittance and impedance matrices ofPhi, respectively, which either exist for allzin Imz > 0and almost allomegaor nowhere. The necessary and sufficient conditions for anm>n times n matrix A_{n}(z)to r- epresent either the scattering or immittance description of a linear, passive, time-invariantn-portPhiare derived in terms of the real frequency behavior ofA_{n}(omega). Necessary and sufficient conditions forPhi_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.