### 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

*positive real*. As shown in §C.11.2, is positive real if and only if is stable and has magnitude less than or equal to on the axis (and hence over the entire left-half plane, by the maximum modulus theorem),

*i.e.*,

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

*passive reflectance*.

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:

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

*Schur function*. Thus, the discrete-time reflectance of an impedance 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 () 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

Consider the special case of a reflection and transmission at a
*yielding termination*, or ``bridge'', of an ideal vibrating
string on its right end, as shown in Fig.C.28. Denote the
incident and reflected velocity waves by and ,
respectively, and similarly denote the force-wave components by
and
. Finally, denote the velocity of the
termination itself by
, and its force-wave
reflectance by

The bridge velocity is given by

*bridge velocity transmittance*is given by

*force transmittance*is given by

#### Power-Complementary Reflection and Transmission

We can show that the reflectance and transmittance of the yielding
termination are *power complementary*. That is, the reflected
and transmitted signal-power sum to yield the incident signal-power.

The average power incident at the bridge at frequency can be
expressed in the frequency domain as
.
The reflected power is then
. Removing the minus sign, which can be
associated with reversed direction of travel, we obtain that the
*power reflection frequency response* is
, which
generalizes by analytic continuation to
. The power
transmittance is given by

**Next Section:**

Positive Real Functions

**Previous Section:**

Reflectance of an Impedance