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 [463,385].
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
![$ {\bm \Gamma}$](http://www.dsprelated.com/josimages_new/pasp/img530.png)
![$ x^\ast {\bm \Gamma}
x$](http://www.dsprelated.com/josimages_new/pasp/img4078.png)
![$ x$](http://www.dsprelated.com/josimages_new/pasp/img179.png)
![$ {\bm \Gamma}$](http://www.dsprelated.com/josimages_new/pasp/img530.png)
![$ \vert\vert\,x\,\vert\vert _{\bm \Gamma}^2 = x^\ast {\bm \Gamma}x$](http://www.dsprelated.com/josimages_new/pasp/img4079.png)
![$ {\bm \Gamma}=\mathbf{I}$](http://www.dsprelated.com/josimages_new/pasp/img4080.png)
![$ \mathbf{A}$](http://www.dsprelated.com/josimages_new/pasp/img569.png)
The following theorem gives a general characterization of lossless scattering:
Theorem: A scattering matrix (FDN feedback matrix)
is
lossless if and only if its eigenvalues lie on the unit circle and its
eigenvectors are linearly independent.
Proof: Since
is positive definite, it can be factored (by
the Cholesky factorization) into the form
, where
is an upper triangular matrix, and
denotes the Hermitian
transpose of
, i.e.,
. Since
is
positive definite,
is nonsingular and can be used as a
similarity transformation matrix. Applying the Cholesky decomposition
in Eq.
(C.125) yields
![\begin{eqnarray*}
& & \mathbf{A}^\ast {\bm \Gamma}\mathbf{A}= {\bm \Gamma}\\
&\...
...\implies&
\tilde{\mathbf{A}}^\ast \tilde{\mathbf{A}}= \mathbf{I}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4084.png)
where
, and
![$\displaystyle \tilde{\mathbf{A}}\isdef \mathbf{U}\mathbf{A}\mathbf{U}^{-1}
$](http://www.dsprelated.com/josimages_new/pasp/img4086.png)
![$ \mathbf{A}$](http://www.dsprelated.com/josimages_new/pasp/img569.png)
![$ \mathbf{U}^{-1}$](http://www.dsprelated.com/josimages_new/pasp/img4087.png)
![$ \tilde{\mathbf{A}}$](http://www.dsprelated.com/josimages_new/pasp/img4088.png)
![$ z$](http://www.dsprelated.com/josimages_new/pasp/img76.png)
![$ \tilde{\mathbf{A}}$](http://www.dsprelated.com/josimages_new/pasp/img4088.png)
![$ \mathbf{A}$](http://www.dsprelated.com/josimages_new/pasp/img569.png)
![$ N$](http://www.dsprelated.com/josimages_new/pasp/img20.png)
![$ \tilde{\mathbf{A}}$](http://www.dsprelated.com/josimages_new/pasp/img4088.png)
![$ \mathbf{A}\tilde{\mathbf{A}}= \tilde{\mathbf{A}}\mathbf{A}$](http://www.dsprelated.com/josimages_new/pasp/img4089.png)
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.125) 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
[433]).
Next Section:
Gyrators
Previous Section:
Normalized Scattering