Complexity

Digital Waveguide Networks

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.

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*
[1] 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** 1:
A scattering matrix (FDN feedback matrix) is lossless
if and only if its eigenvalues lie on the unit circle and it admits a basis of
linearly independent eigenvectors.
*Proof*:

In general, the Cholesky factorization
gives an upper
triangular matrix which converts to a unitary matrix via
similarity transformation:
, where
.
Hence, the eigenvalues of every lossless scattering matrix lie on the unit
circle. It readily follows from similarity to
that admits
linearly independent eigenvectors. In fact,
is a normal matrix
(since it is unitary), and normal matrices admit a basis of linearly
independent eigenvectors [21].

Conversely, assume for each eigenvalue of , and that
there exists a matrix of linearly independent eigenvectors of .
Then the matrix diagonalizes to give
, where
.
Multiplying, we obtain
. Thus, (27) 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 and .

Relation of DWNs to FDNs

Complexity

Digital Waveguide Networks

``Circulant and Elliptic Feedback Delay Networks for Artificial Reverberation'', by Davide Rocchesso and Julius O. Smith III, preprint of version in

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