In this paper the most general linear, passive, time-invariant n -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. An n -port \Phi is defined to be an operator in H_n , the space of all n -vectors whose components are measurable functions of a real variable t, (- \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 , the n -port corresponding to \phi augmented by n series resistors is always causal ( \Phi is the "augmented network," Fig. 2). Under the further assumptions that the domain of \Phi_a is dense in Hilbert space and \phi is time-invariant, it is proved that \Phi possesses a frequency response and defines an n \times n matrix S(z) (the scattering matrix) of a complex variable z = \omega + i\beta with the following properties: 1) S(z) is analytic in Im z > 0 ; 2) Q(z) = I_n - S^{\ast}(z)S(z) is the matrix of a non-negative quadratic form for all z in 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-invariant n -port \Phi such that the domain of \Phi_a contains all of Hilbert space. Such matrices are termed "bounded real scattering matrices" and are a generalization of the familiar positive-real immittance matrices. When \Phi and \Phi^{-1} are single-valued, it is possible to define two auxiliary positive-real matrices Y(z) and Z(z) , the admittance and impedance matrices of \Phi , respectively, which either exist for all z in Im z > 0 and almost all \omega or nowhere. The necessary and sufficient conditions for an m>n \times n matrix A_{n}(z) to represent either the scattering or immittance description of a linear, passive, time-invariant n -port \Phi are derived in terms of the real frequency behavior of A_{n}(\omega) . Necessary and sufficient conditions for \Phi_a to admit the representation i(t) = \int_{-\infty}^{\infty} dW_{n}(\tau)e(t - {\tau}) for all integrable e(t) in its domain are given in terms of S(z) . The last section concludes with a discussion concerning the nature of the singularities of S(z) and the possible extension of the theory to active networks.

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