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Passive Reflectances

From Eq.$ \,$ (C.75), we have that the reflectance seen at a continuous-time impedance $ R(s)$ is given for force waves by

$\displaystyle \hat{\rho}_f(s) \isdefs \frac{F^{-}(s)}{F^{+}(s)} \eqsp \frac{R(s)-R_0}{R(s)+R_0} \protect$ (C.76)

where $ R_0$ is the wave impedance connected to the impedance $ R(s)$ , and the corresponding velocity reflectance is $ \hat{\rho}_v(s)= -\hat{\rho}_f(s)$ . As mentioned above, all passive impedances are positive real. As shown in §C.11.2, $ R(s)$ is positive real if and only if $ \hat{\rho}_f(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.,

$\displaystyle \left\vert\hat{\rho}_f(s)\right\vert \leq 1,$   re$\displaystyle \left\{s\right\} \leq 0. \protect$ (C.77)

In particular, $ \left\vert\hat{\rho}_f(j\omega)\right\vert \leq 1$ for all radian frequencies $ \omega\in(-\infty,\infty)$ . Any stable $ \hat{\rho}_f(s)$ satisfying Eq.$ \,$ (C.77) may be called a passive reflectance.

If the impedance $ R(s)$ goes to infinity (becomes rigid), then $ \hat{\rho}_f(s)$ approaches $ 1$ , a result which agrees with an analysis of rigid string terminations (p. [*]). Similarly, when the impedance goes to zero, $ \hat{\rho}_f(s)$ becomes $ -1$ , which agrees with the physics of a string with a free end. In acoustic stringed instruments, bridges are typically quite rigid, so that $ \hat{\rho}_f(j\omega)\approx 1$ for all $ \omega $ . If a body resonance is strongly coupled through the bridge, $ \vert\hat{\rho}_f(j\omega_c)\vert$ can be significantly smaller than 1 at the resonant frequency $ \omega_c$ .

Solving for $ R(s)$ in Eq.$ \,$ (C.77), we can characterize every impedance in terms of its reflectance:

$\displaystyle R(s) = R_0\frac{1+\hat{\rho}_f(s)}{1-\hat{\rho}_f(s)}

Rewriting Eq.$ \,$ (C.76) in the form

$\displaystyle \hat{\rho}_f(s) \eqsp \frac{\dfrac{R(s)}{R_0}-1}{\dfrac{R(s)}{R_0}+1},

we see that the reflectance is determined by the ratio of the ``new impedance'' $ R(s)$ to the ``old'' impedance $ R_0$ in which the incoming waves travel. In other words, the incoming waves see the wave impedance ``step'' from $ R_0$ to $ R(s)$ , which results in a ``scattering'' of the incident wave into reflected and transmitted components, as discussed in §C.8. The reflection and transmission coefficients depend on frequency when $ R(j\omega)$ is not constant with respect to $ \omega $ .

In the discrete-time case, which may be related to the continuous-time case by the bilinear transform7.3.2), we have the same basic relations, but in the $ z$ plane:

$\displaystyle \hat{\rho}_f(z)$ $\displaystyle \isdef$ $\displaystyle \frac{F^{-}(z)}{F^{+}(z)}
\eqsp \frac{R(z)-R_0}{R(z)+R_0}$  
$\displaystyle R(z)$ $\displaystyle =$ $\displaystyle R_0\frac{1+\hat{\rho}_f(z)}{1-\hat{\rho}_f(z)}$  
$\displaystyle \Gamma(z)$ $\displaystyle =$ $\displaystyle \Gamma _0\frac{1-\hat{\rho}_f(z)}{1+\hat{\rho}_f(z)}
\protect$ (C.78)

where $ \Gamma\isdef 1/R$ denotes admittance, with

$\displaystyle \left\vert\hat{\rho}_f(z)\right\vert \leq 1, \quad \left\vert z\right\vert \leq 1. \protect$ (C.79)

Mathematically, any stable transfer function having these properties may be called a Schur function. Thus, the discrete-time reflectance $ \hat{\rho}_f(z)$ of an impedance $ R(z)$ is a Schur function if and only if the impedance is passive (positive real).

Note that Eq.$ \,$ (C.79) may be obtained from the general formula for scattering at a loaded waveguide junction for the case of a single waveguide ($ N=1$ ) terminated by a lumped load (§C.12).

In the limit as damping goes to zero (all poles of $ R(z)$ converge to the unit circle), the reflectance $ \hat{\rho}_f(z)$ becomes a digital allpass filter. Similarly, $ \hat{\rho}_f(s)$ becomes a continuous-time allpass filter as the poles of $ R(s)$ approach the $ j\omega $ axis.

Recalling that a lossless impedance is called a reactance7.1), we can say that every reactance gives rise to an allpass reflectance. Thus, for example, waves reflecting off a mass at the end of a vibrating string will be allpass filtered, because the driving-point impedance of a mass ($ R(s)=ms$ ) is a pure reactance. In particular, the force-wave reflectance of a mass $ m$ terminating an ideal string having wave impedance $ R_0$ is $ \hat{\rho}_f(s)=
(ms-R_0)/(ms+R_0)$ , which is a continuous-time allpass filter having a pole at $ s=-R_0/m$ and a zero at $ s=R_0/m$ .

It is intuitively reasonable that a passive reflection gain cannot exceed $ 1$ at any frequency (i.e., the reflectance is a Schur filter, as defined in Eq.$ \,$ (C.79)). It is also reasonable that lossless reflection would have a gain of 1 (i.e., it is allpass).

Note that reflection filters always have an equal number of poles and zeros, as can be seen from Eq.$ \,$ (C.76) above. This property is preserved by the bilinear transform, so it holds in both the continuous- and discrete-time cases.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2018-01-17 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University