Passive Reflectances

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

where is the wave impedance connected to the impedance , and the corresponding velocity reflectance is . As mentioned above, all passive impedances are

In particular, for all radian frequencies . Any stable satisfying Eq. (C.77) may be called a

If the impedance goes to infinity (becomes rigid), then approaches , a result which agrees with an analysis of rigid string terminations (p. ). Similarly, when the impedance goes to zero, becomes , which agrees with the physics of a string with a free end. In acoustic stringed instruments, bridges are typically quite rigid, so that for all . If a body resonance is strongly coupled through the bridge, can be significantly smaller than 1 at the resonant frequency .

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

Rewriting Eq. (C.76) in the form

we see that the reflectance is determined by the ratio of the ``new impedance'' to the ``old'' impedance in which the incoming waves travel. In other words, the incoming waves see the wave impedance ``step'' from to , 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 is not constant with respect to .

In the discrete-time case, which may be related to the continuous-time
case by the bilinear transform (§7.3.2), we have the same basic
relations, but in the
plane:

where denotes admittance, with

Mathematically, any stable transfer function having these properties may be called a

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 ( ) terminated by a lumped load (§C.12).

In the limit as damping goes to zero (all poles of
converge to
the unit circle),
the reflectance
becomes a digital *allpass filter*. Similarly,
becomes a continuous-time allpass filter as the poles of
approach the
axis.

Recalling that a lossless impedance is called a *reactance*
(§7.1), 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 (
) is a pure
reactance. In particular, the force-wave reflectance of a mass
terminating an ideal string having wave impedance
is
, which is a continuous-time allpass filter having
a pole at
and a zero at
.

It is intuitively reasonable that a passive reflection gain cannot
exceed
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.

- Reflectance and Transmittance of a Yielding String Termination
- Power-Complementary Reflection and Transmission

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