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

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.