## FDNs as Digital Waveguide Networks

This section supplements §2.7 on *Feedback Delay Networks*
in the context of digital waveguide theory. Specifically, we review
the interpretation of an FDN as a special case of a digital waveguide
network, summarizing [463,464,385].

Figure C.36 illustrates an -branch DWN. It consists of a single
scattering junction, indicated by a white circle, to which
branches are connected. The far end of each branch is terminated by an
ideal non-inverting reflection (black circle). The waves traveling
into the junction are associated with the FDN delay line outputs
, and the length of each waveguide is
*half* the length of the corresponding FDN delay line
(since a traveling wave must traverse the branch twice to complete a
round trip from the junction to the termination and back). When
is odd, we may replace the reflecting termination by a unit-sample
delay.

### Lossless Scattering

The delay-line inputs (outgoing traveling waves) are computed by multiplying the delay-line outputs (incoming traveling waves) by the feedback matrix (scattering matrix) . By defining , , we obtain the more usual DWN notation

where is the vector of incoming traveling-wave samples arriving at the junction at time , is the vector of outgoing traveling-wave samples leaving the junction at time , and is the

*scattering matrix*associated with the waveguide junction.

The junction of physical waveguides determines the structure of the matrix according to the basic principles of physics.

Considering the parallel junction of lossless acoustic tubes, each having characteristic admittance , the continuity of pressure and conservation of volume velocity at the junction give us the following scattering matrix for the pressure waves [433]:

where

(C.121) |

Equation (C.121) can be derived by first writing the volume velocity at the -th tube in terms of pressure waves as . Applying the conservation of velocity we can find the expression

### Normalized Scattering

For ideal numerical scaling in the sense, we may choose to propagate
*normalized waves* which lead to normalized scattering junctions
analogous to those encountered in normalized ladder filters [297].
Normalized waves may be either normalized pressure
or normalized velocity
. Since the signal power associated with a traveling
wave is simply
,
they may also be called *root-power waves* [432].
Appendix C develops this topic in more detail.

The scattering matrix for normalized pressure waves is given by

The normalized scattering matrix can be expressed as a negative Householder reflection

where , and is the wave admittance in the th waveguide branch. To eliminate the sign inversion, the reflections at the far end of each waveguide can be chosen as -1 instead of 1. The geometric interpretation of (C.124) is that the incoming pressure waves are reflected about the vector . Unnormalized scattering junctions can be expressed in the form of an ``oblique'' Householder reflection , where and .

### 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 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

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:**

Waveguide Transformers and Gyrators

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Digital Waveguide Mesh