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Passive Reflectances

From (L.19), we have that the reflectance seen at a continuous-time impedance $ R(s)$ 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 $ R_0$ is the wave impedance connected to the impedance $ R(s)$. As discussed earlier, all passive impedances are positive real. It can be easily verified that $ R(s)$ positive real implies that $ S(s)$ is stable and has magnitude less than or equal to $ 1$ 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 $ S(s)$ satisfying (M.1) is called a passive reflectance.

Solving for $ R(s)$, 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 $ R_0$ which is the impedance attached to $ R(s)$ 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 $ z$ plane:

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