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
![$ \mathbf{p}^+$](http://www.dsprelated.com/josimages_new/pasp/img4056.png)
![$ n$](http://www.dsprelated.com/josimages_new/pasp/img146.png)
![$ \mathbf{p}^-$](http://www.dsprelated.com/josimages_new/pasp/img4057.png)
![$ n$](http://www.dsprelated.com/josimages_new/pasp/img146.png)
![$ \mathbf{A}$](http://www.dsprelated.com/josimages_new/pasp/img569.png)
The junction of physical waveguides determines the structure of the
matrix
according to the basic principles of physics.
Considering the parallel junction of lossless acoustic tubes, each
having characteristic admittance
, the continuity of pressure and
conservation of volume velocity at the junction give us the following
scattering matrix for the pressure waves [433]:
where
![]() |
(C.121) |
Equation (C.121) can be derived by first writing the volume velocity at the
![$ j$](http://www.dsprelated.com/josimages_new/pasp/img664.png)
![$ v_j = (p_j^+ - p_j^-)\Gamma_j$](http://www.dsprelated.com/josimages_new/pasp/img4061.png)
![$\displaystyle p = 2 \sum_{i=1}^{N}\Gamma_{i} p_i^+ / \Gamma_J
$](http://www.dsprelated.com/josimages_new/pasp/img4062.png)
Normalized Scattering
For ideal numerical scaling in the sense, we may choose to propagate
normalized waves which lead to normalized scattering junctions
analogous to those encountered in normalized ladder filters [297].
Normalized waves may be either normalized pressure
or normalized velocity
. Since the signal power associated with a traveling
wave is simply
,
they may also be called root-power waves [432].
Appendix C develops this topic in more detail.
The scattering matrix for normalized pressure waves is given by
The normalized scattering matrix can be expressed as a negative Householder reflection
where
![$ \tilde{{\bm \Gamma}}^T= [\sqrt{\Gamma_1},\ldots,\sqrt{\Gamma_N}]$](http://www.dsprelated.com/josimages_new/pasp/img4068.png)
![$ \Gamma_i$](http://www.dsprelated.com/josimages_new/pasp/img4069.png)
![$ i$](http://www.dsprelated.com/josimages_new/pasp/img314.png)
![$ \tilde{{\bm \Gamma}}$](http://www.dsprelated.com/josimages_new/pasp/img4070.png)
![$ \mathbf{A}= 2\mathbf{1}{\bm \Gamma}^T/\left<\mathbf{1},{{\bm \Gamma}}\right>-\mathbf{I}$](http://www.dsprelated.com/josimages_new/pasp/img4071.png)
![$ \mathbf{1}^T=[1,\ldots,1]$](http://www.dsprelated.com/josimages_new/pasp/img4072.png)
![$ {\bm \Gamma}^T= [\Gamma_1,\ldots,\Gamma_N]$](http://www.dsprelated.com/josimages_new/pasp/img4073.png)
General Conditions for Losslessness
The scattering matrices for lossless physical waveguide junctions give an apparently unexplored class of lossless FDN prototypes. However, this is just a subset of all possible lossless feedback matrices. We are therefore interested in the most general conditions for losslessness of an FDN feedback matrix. The results below are adapted from [463,385].
Consider the general case in which
is allowed to be any
scattering matrix, i.e., it is associated with a
not-necessarily-physical junction of
physical waveguides.
Following the definition of losslessness in classical network theory,
we may say that a waveguide scattering matrix
is said to be
lossless if the total complex power
[35] at the junction is scattering invariant, i.e.,
where
![$ {\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]).
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Waveguide Transformers and Gyrators
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Digital Waveguide Mesh