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

and we have

Thus, the wave impedance is the factor of

More generally, when there is a loss represented by a diagonal matrix
,
we have, in the continuous-time case,

For the discrete-time case, we may map from the plane to the plane via the bilinear transform [65], or we may sample the inverse Laplace transform of and take its transform to obtain .

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}

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