Passivity of a Finite-Difference Scheme

A condition stronger than stability as defined above is
*passivity*. Passivity is not a traditional metric for
finite-difference scheme analysis, but it arises naturally in special
cases such as wave digital filters (§F.1) and digital waveguide
networks [55,35]. In such modeling frameworks, all
signals have a *physical interpretation* as wave variables, and
therefore a physical energy can be associated with them. Moreover,
each delay element can be associated with some real *wave
impedance*. In such situations, passivity can be defined as the case
in which all impedances are nonnegative. When complex, they must be
*positive* real (see §C.11.2).

To define passivity for all linear, shift-invariant finite difference schemes, irrespective of whether or not they are based on an impedance description, we will say that a finite-difference scheme is passive if all of its internal modes are stable. Thus, passivity is sufficient, but not necessary, for stability. In other words, there are finite difference schemes which are stable but not passive [55]. A stable FDS can have internal unstable modes which are not excited by initial conditions, or which always cancel out in pairs. A passive FDS cannot have such ``hidden'' unstable modes.

The absence of hidden modes can be ascertained by converting the FDS
to a state-space model and checking that it is *controllable*
(from initial conditions and/or excitations) and *observable*
[452]. When the initial conditions can set the entire initial
state of the FDS, it is then controllable from initial conditions, and
only observability needs to be checked. A simple example of an
unobservable mode is the second harmonic of an ideal string (and all
even-numbered harmonics) when the only output observation is the
midpoint of the string.

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