General Conditions for Losslessness

The scattering matrices for lossless physical waveguide junctions give an apparently unexplored class of lossless FDN prototypes. However, this is just a subset of all possible lossless feedback matrices. We are therefore interested in the most general conditions for losslessness of an FDN feedback matrix. The results below are adapted from [465,388].

Consider the general case in which
is allowed to be any
scattering matrix, *i.e.*, it is associated with a
not-necessarily-physical junction of
physical waveguides.
Following the definition of losslessness in classical network theory,
we may say that a waveguide scattering matrix
is said to be
*lossless* if the *total complex power*
[35] at the junction is scattering invariant, *i.e.*,

where is any Hermitian, positive-definite

The following theorem gives a general characterization of lossless scattering:

**Theorem**: A scattering matrix (FDN feedback matrix)
is
lossless if and only if its eigenvalues lie on the unit circle and its
eigenvectors are linearly independent.

*Proof*: Since
is positive definite, it can be factored (by
the Cholesky factorization) into the form
, where
is an upper triangular matrix, and
denotes the Hermitian
transpose of
, *i.e.*,
. Since
is
positive definite,
is nonsingular and can be used as a
similarity transformation matrix. Applying the Cholesky decomposition
in Eq.
(C.122) yields

where , and

is similar to using as the similarity transform matrix. Since is unitary, its eigenvalues have modulus 1. Hence, the eigenvalues of every lossless scattering matrix lie on the unit circle in the plane. It readily follows from similarity to that admits linearly independent eigenvectors. In fact, is a normal matrix ( ), since every unitary matrix is normal, and normal matrices admit a basis of linearly independent eigenvectors [349].

Conversely, assume
for each eigenvalue of
, and
that there exists a matrix
of linearly independent
eigenvectors of
. The matrix
diagonalizes
to give
, where
diag
. Taking the Hermitian transform of
this equation gives
. Multiplying, we
obtain
. Thus, (C.122) is satisfied for
which is Hermitian and positive
definite.

Thus, lossless scattering matrices may be fully parametrized as , where is any unit-modulus diagonal matrix, and is any invertible matrix. In the real case, we have diag and .

Note that not all lossless scattering matrices have a simple
*physical* interpretation as a scattering matrix for an
intersection of
lossless reflectively terminated waveguides. In
addition to these cases (generated by all non-negative branch
impedances), there are additional cases corresponding to sign flips
and branch permutations at the junction. In terms of classical
network theory [35], such additional cases can be seen as
arising from the use of ``gyrators'' and/or ``circulators'' at the
scattering junction
[437]).

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