... filters,¹

Following previous usage conventions, we use the term ``ladder filter'' when the graph of the filter is planar, and ``lattice filter'' when the graph is non-planar.

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... tube,²

``Stiffness'' is defined here for air as the reciprocal of the adiabatic compressibility of the gas [61, p. 230]. This definition helps to unify the scattering formalism for acoustic tubes with that of mechanical systems such as vibrating strings.

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....³

There are no tube shapes supporting exact traveling waves other than cylindrical and conical (or conical wedge, which is a hybrid) [69]. However, the ``Salmon horn family'' (see, e.g., [60,96]) characterizes a larger class of approximate ``one-parameter traveling waves.'' In the cone, the wave equation is solved for pressure $p(x,t)$ using a change of variables $p^\prime = p x$, where $x$ is the distance from the apex of the cone, causing the wave equation for the cone to reduce to that of the cylindrical case [2].

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... power⁴

Note that $\left\vert z\right\vert=1$ corresponds to the average physical power at frequency $\omega$, where $z=\exp(j\omega T)$, and the wave variable magnitudes on the unit circle may be interpreted as RMS levels [4, p. 48]. For $\vert z\vert>1$, we may interpret the power ${\cal P}(z)={\mbox{\boldmath$u$}}^*(1/z^*) {\mbox{\boldmath$p$}}(z)$ as the steady state power obtained when exponential damping is introduced into the waveguide giving decay time-constant $\tau$, where $z = \exp(-T/\tau)\exp(j\omega T)$ [4, p. 48].

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....⁵

A complex-valued function of a complex variable $f(z)$ is said to be positive real if

1)

$z\, \hbox{real}\; \Rightarrow\; f(z)\, \hbox{real}$

2)

$\vert z\vert \geq 1 \; \Rightarrow \; \hbox{Re}\{f(z)\} \geq 0$

Positive real functions characterize passive impedances in classical network theory [129].

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...$\vert z\vert>1$⁶

Note that by the maximum modulus theorem, if a matrix of meromorphic functions is positive definite on the unit circle and analytic outside the unit circle, it is necessarily positive definite everywhere outside the unit circle.

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... velocity⁷

The symbols for the variables velocity and force have been chosen to maintain consistency with the analogous acoustical quantities.

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... sample,⁸

Strictly speaking, a squared wave variable in a normalized waveguide is proportional to signal power [97], and we should multiply by the sampling interval $T$ to obtain units of energy. However, such a constant scale factor is inconsequential.

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...WAVCAS.⁹

Altering a ladder filter to convert it to a DWN can also be accomplished using the cut-set method of Kung [51,12].

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