Relation of DWNs to FDNs

When ${\bf U}$ in the proof is diagonal and positive, a physical waveguide interpretation always exists with ${\bf U}=\mbox{diag}({\bf {{\tilde \Gamma}}})$. A generalized waveguide interpretation exists for all ${\bf U}$ via vector transformers [28, p. 55 sec. 4] in which ${\bf U}$ acts as an ideal transformer (in the classical network theory sense) on the vector of all $N$ waveguide variables. If $\bf {p}=\bf {p}^++\bf {p}^-$ denotes the vector of physical pressures at the junction and $\bf {p}=\bf {p}^++\bf {p}^-$ denotes the physical volume velocities, then we have that the junction power, defined as ${\cal P}_J = \bf {p}^\ast \bf {p}$ is invariant with respect to insertion of a vector transformer (similarity transformation applied to the scattering matrix).

It can be quickly verified³ that all scattering matrices arising from the parallel intersection of $N$ physical waveguides possess one eigenvalue equal to $1$ and $N-1$ eigenvalues equal to $-1$ [28]. In the case of physical waveguides of equal impedances, the eigenvector associated with the eigenvalue $1$ corresponds to equal incoming waves, while an eigenvector associated with the eigenvalue $-1$ corresponds to equal incoming waves on $N-1$ branches, and a large opposite wave on the remaining branch which pulls the junction pressure to zero. Adding a vector transformer to the parallel $N$-branch scattering junction gives scattering matrices of the form ${\bf A}= {\bf T}^{-1}{\bf D}{\bf T}$ with ${\bf D}= \mbox{diag}(1,-1,\ldots,-1)$. To obtain more general eigenvalue signatures ${\bf D}= \mbox{diag}(\pm 1)$, a combination of series and parallel junctions must be used. Finally, to reach the most general complex case, we must admit complex eigenvalues.

We now consider the relation between unitary FDN feedback matrices and waveguide scattering junctions. As can be seen from comparing (27) to ${\tilde {\bf A}}^\ast {\tilde {\bf A}}={\bf I}$ (true for any unitary matrix), we see that ${\bf A}$ unitary corresponds to a scattering junction in which the total complex power is given by the ordinary $L_2$ norm of the incoming or outgoing traveling waves. Since the physical power associated with an incoming wave vector $\bf {p}^+$ is ${\cal P}_J = {\bf {p}^+}^\ast {\bf\Gamma}\bf {p}^+$, where in the absence of a vector transformer ${\bf\Gamma}=\mbox{diag}{\Gamma_1,\ldots,\Gamma_N}$, we see that ${\bf A}$ unitary corresponds to a scattering junction joining waveguides of equal wave impedance, i.e., ${\bf\Gamma}=\mbox{diag}{1,\ldots,1}$. Since Householder reflections comprise only a subset unitary matrices, we see that a unitary FDN matrix corresponds to a transformer-coupled parallel/series waveguide junction in which all branch admittances are the same. In the more general (unnormalized) case in which the branch impedances are different, i.e., ${\bf\Gamma}=\mbox{diag}(\Gamma_1,\ldots,\Gamma_N)$, we obtain (using a vector transformer) the larger class of scattering matrices which preserve an elliptic norm as induced by a positive-definite (or Hermitian) generalized junction impedance.

Since, as discussed above, only a subset of all $N$-by-$N$ unitary matrices is given by a physical junction of $N$ waveguides, the unitary FDN point of view yields lossless systems outside the scope of those suggested by multiport scattering theory. On the other hand, since only normalized waveguide junctions exhibit unitary scattering matrices, the DWN approach gives rise to new classes of FDNs. Moreover, by considering more than one scattering junction, the DWN approach suggests new classes of network topologies following physical analogies. Similarly, FDN matrices can be partitioned to embed several FDN subsystems into larger FDN systems.

Formally, every DWN can be expressed as an FDN by collecting all of its delay lines into a diagonal delay matrix ${\bf D}(z)$ as in (4), and finding the matrix ${\bf A}$ which computes the delay-line inputs from the delay-line outputs. Therefore, every waveguide network yields a feedback matrix for consideration in the FDN framework. Conversely, every real FDN can be expressed as a single-junction waveguide network using an ideal vector transformer at the junction.

Theorem 1 characterizes lossless FDN feedback matrices ${\bf A}$ as those having eigenvalues on the unit circle, where the definition of losslessness was given by (27). It remains to be shown that ${\bf A}$ satisfying (27) implies that the poles of the corresponding FDN are all on the unit circle. To this end, recall the form of the state-transition matrix (8), and define the extended generalized admittance

$$\Gamma^\dagger = \left[ \begin{array}{ll} {\bf I}& {\bf0}\\ {\bf0} & {\bf\Gamma} \end{array} \right]$$ (28)

By analogy to the derivation of (18), we get

$${{\bf A}^\dagger}^T \Gamma^\dagger {\bf A}^\dagger =\mbox{di... ...1}, \dots {\bf I}_{m_N - 1}, {\bf A}^T \Gamma {\bf A} \right)$$ (29)

This equation shows that ${\bf A}$ is lossless if and only if ${{\bf A}^\dagger}$ is lossless, and its eigenvalues are on the unit circle by Theorem 1. But from (17), we have that the eigenvalues of ${{\bf A}^\dagger}$ are the poles of the corresponding FDN. Therefore, the FDN is lossless and its impulse response consists only of non-increasing and non-decaying modes. Desired decay characteristics versus frequency for obtaining a specific artificial reverberator can now be controlled separately by means of attenuation coefficients and filters, as indicated in section II.