### 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.,

 (C.124)

where is any Hermitian, positive-definite matrix (which has an interpretation as a generalized junction admittance). The form is by definition the square of the elliptic norm of induced by , or . Setting , we obtain that must be unitary. This is the case commonly used in current FDN practice.

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

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 [346].

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]).

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