Passive Reflectances

From (J.18), we have that the reflectance seen at a continuous-time impedance $R(s)$ is given for force waves by

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

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.,

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

Any stable $S(s)$ satisfying (K.1) is called a passive reflectance.

Solving for $R(s)$, we can characterize every passive impedance in terms of its corresponding reflectance:

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

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:

$$S(z) \isdef \frac{F^{-}(z)}{F^{+}(z)} = \frac{R(z)-R_0}{R(z)+R_0}$$ (K.1)

$$R(z) = R_0\frac{1+S(z)}{1-S(z)}$$ (K.2)

$$\Gamma(z) = \Gamma _0\frac{1-S(z)}{1+S(z)}$$ (K.3)

and

$$\left\vert S(z)\right\vert \leq 1, \quad \left\vert z\right\vert \leq 1$$

Stable functions $S(z)$ satisfying (K.1) are also known in the mathematics literature as Schur functions. In the limit as damping goes to zero (all poles of $R(z)$ converge to the unit circle, with interlacing zeros as required to remain positive real), the reflectance approaches the transfer function of an allpass filter. Thus, the Schur function is a generalization of an allpass transfer function to allow for loss. Recalling that a lossless impedance is called a reactance, we can say that every reactance gives rise to an allpass reflectance. Thus, for example, waves reflecting off a mass at the end of the string will be allpass filtered. It is intuitively very straightforward that the reflection magnitude cannot exceed $1$ at any frequency when reflecting from a passive impedance back into a passive medium at impedance $R_0$. It is also intuitively satisfying that lossless reflection involves only a phase shift and no amplification or attenuation at any frequency.