Passive Reflectances

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From (L.19), we have that the reflectance seen at a continuous-time impedance is given for force waves by

$\displaystyle S(s) \isdef \frac{F^{-}(s)}{F^{+}(s)} = \frac{R(s)-R_0}{R(s)+R_0}$

where

is the wave impedance connected to the impedance

. As discussed earlier, all passive impedances are positive real. It can be easily verified that

positive real implies that

is stable and has magnitude less than or equal to

on the $j\omega$ axis (and hence over the entire left-half plane, by the maximum modulus theorem), i.e.,

$\displaystyle \left\vert S(s)\right\vert \leq 1,$ re $\displaystyle \left\{s\right\} \leq 0$

Any stable

satisfying (M.1) is called a passive reflectance.

Solving for , we can characterize every passive impedance in terms of its corresponding reflectance:

$\displaystyle R(s) = R_0\frac{1+S(s)}{1-S(s)}$

The reflectance is always defined relative to an impedance

which is the impedance attached to

to create an impedance discontinuity and thereby generate (frequency-dependent) reflections.

In the discrete-time case, we have the same basic relations, but in the plane:

$\displaystyle S(z)$

$\displaystyle \isdef$

$\displaystyle \frac{F^{-}(z)}{F^{+}(z)} = \frac{R(z)-R_0}{R(z)+R_0}$

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