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.