Vector Waveguides and Scattering Junctions

It is also possible to extend a DWN to the vector case [167,169]; this has also been done in the WDF framework in [46,131], as discussed in §2.3.7. We briefly introduce vector waveguides, because it will be necessary to apply them when simulating the behavior of stiff systems and elastic solids; we will examine this problem in depth in Chapter 5.

A vector waveguide accepts two incoming signals $\hat{{\bf U}}_{1}^{-}$ and $\hat{{\bf U}}_{2}^{-}$ and outputs $\hat{{\bf U}}_{1}^{+}$ and $\hat{{\bf U}}_{2}^{+}$; all are assumed to be $q\times 1$ vectors (note that we have used $z$-transformed quantities here). The waveguide itself, like its scalar counterpart, is described by two parameters: its impedance ${\bf Z}$, a $q\times q$ matrix, which we will assume to be constant and symmetric positive definite (though it may be generalized to a para-Hermitian matrix function of the unit delay $z^{-1}$ [169]) and its generalized delay, ${\bf H}(z^{-1})$, a $2q\times 2q$ matrix function of the unit delay, which we assume to be para-unitary (lossless) [193]. The input and output voltage waves are related by

$$\begin{bmatrix}\hat{{\bf U}}_{1}^{+}(z^{-1})\\ \hat{{\bf U}}_{2}^... ...x}\hat{{\bf U}}_{2}^{-}(z^{-1})\\ \hat{{\bf U}}_{1}^{-}(z^{-1})\\ \end{bmatrix}$$

(In applications in Chapter 5, where bidirectional delay lines are extracted from an MDKC, we will always set ${\bf H}(z^{-1})$ to be a multiple of the $2q\times 2q$ identity matrix.) We can define instantaneous current wave vectors ${\bf I}_{j}^{+}$ and ${\bf I}_{j}^{-}$, for $j=1,2$, and Ohm's Law becomes, in the vector case,

$${\bf U}_{j}^{+} = {\bf ZI}_{j}^{+}\hspace{0.5in}{\bf U}_{j}^{-} = -{\bf ZI}_{j}^{-}\hspace{0.5in}$$

The scattering equations at a series or parallel junction of $k$ waveguides generalize in a straightforward way to the vector case--we have

for $k=1,\hdots,M$, where ${\bf Y}_{j} = {\bf Z}_{j}^{-1}$ is the admittance of the $j$th waveguide, which must exist because ${\bf Z}_{j}$ is assumed positive definite (for passivity) [167]. The vector junction admittance and impedance are defined by

$${\bf Z}_{J} \triangleq \sum_{j=1}^{M}{\bf Z}_{j}$$ Series junction

$${\bf Y}_{J} \triangleq \sum_{j=1}^{M}{\bf Y}_{j}$$ Parallel junction

By virtue of the fact that they are sums of positive definite matrices, they will also be positive definite, and thus their inverses, used in the scattering equations (4.16), must exist. Vector waveguides were explored extensively in [169] in the context of artificial reverberation. Power normalization may also be applied by scaling the wave variables by a square root of the impedance (which is non-unique) [167]. Vector junction passivity has been shown to hold in the fixed word-length case in [167].