From the multivariable generalization of (2), we have, using
(10),
,
where `' is for right-going and `' is for left-going. Thus,
following the classical definition for the scalar case, the wave impedance
is defined by
More generally, when there is a loss represented by a diagonal matrix
,
we have, in the continuous-time case,
A linear propagation medium in the discrete-time case is completely determined by its wave impedance (generalized here to permit frequency-dependent and spatially varying wave impedances). A waveguide is defined for purposes of this paper as a length of medium in which the wave impedance is either constant with respect to spatial position , or else it varies smoothly with in such a way that there is no scattering (as in the conical acoustic tube). For simplicity, we will suppress the possible spatial dependence and write only .3