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Medium Passivity

Following the classical definition of passivity [4,133], a medium is said to be passive if

{\cal P}^++ {\cal P}^-\geq 0
\end{displaymath} (17)

for $\vert z\vert \geq 1$. Thus, a sufficient condition for ensuring passivity in a medium is that each traveling active-power component be real and non-negative.

To derive a definition of passivity in terms of the wave impedance, consider a perfectly reflecting interruption in the transmission line, such that ${\mbox{\boldmath$u$}}^-= {\mbox{\boldmath$u$}}^+$. For a passive medium, using (16), (17) becomes

{\mbox{\boldmath$R$}}(z) + {\mbox{\boldmath$R$}}^*(1/z^*) \geq 0
\end{displaymath} (18)

for $\vert z\vert \geq 1$. The wave impedance ${\mbox{\boldmath$R$}}(z)$ is an $m$-by-$m$ function of the complex variable $z$. Condition (18) is essentially the same thing as saying ${\mbox{\boldmath$R$}}(z)$ is positive real [128], except that it is allowed to be complex (but Hermitian), even for real $z$.5The matrix ${\mbox{\boldmath$R$}}^*(1/z^*)$ is the paraconjugate of ${\mbox{\boldmath$R$}}$, i.e., the unique analytic continuation (when it exists) of the Hermitian transpose of ${\mbox{\boldmath$R$}}$ from the unit circle to the complex $z$ plane [110]. Since the inverse of a positive-real function is positive real, the corresponding generalized wave admittance ${\mbox{\boldmath$\Gamma$}}(z) =
{\mbox{\boldmath$R$}}^{-1}(z)$ is positive real (and hence analytic) in $\vert z\vert \geq 1$. In other terms, the sum of the wave impedance and its paraconjugate is positive semidefinite.

We say that wave propagation in the medium is lossless if the wave impedance matrix satisfies

\end{displaymath} (19)

i.e., if ${\mbox{\boldmath$R$}}(z)$ is para-Hermitian (which implies its inverse ${\mbox{\boldmath$\Gamma$}}(z)$ is also).

Most applications in waveguide modeling are concerned with nearly lossless propagation in passive media. In this paper, we will state results for ${\mbox{\boldmath$R$}}(z)$ in the more general case when applicable, while considering applications only for constant and diagonal impedance matrices ${\mbox{\boldmath$R$}}$. As shown in Section 2.3, coupling in the wave equation (5) implies a non-diagonal impedance matrix, since there is usually a proportionality between the speed of propagation ${\mbox{\boldmath$C$}}$ and the impedance ${\mbox{\boldmath$R$}}$ through the non-diagonal matrix ${\mbox{\boldmath$M$}}$.

The wave components of equations (11) travel undisturbed along each axis. This propagation is implemented digitally using $m$ bidirectional delay lines, as depicted in Fig. 1. We call such a collection of delay lines an $m$-variable waveguide section. Waveguide sections are then joined at their endpoints via scattering junctions (discussed further in following sections) to form a DWN.

Figure: An $m$-variable waveguide section.
\begin{figure}\centerline{\epsfxsize=5in \epsfbox{figure/DelayLines.eps}}\end{figure}

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``Aspects of Digital Waveguide Networks for Acoustic Modeling Applications'', by Julius O. Smith III and Davide Rocchesso , December 19, 1997, Web published at
Copyright © 2007-02-07 by Julius O. Smith III and Davide Rocchesso
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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