Passive Reflectances
From Eq.(C.75),
we have that the reflectance seen at a continuous-time impedance
is given for force waves by
where
![$ R_0$](http://www.dsprelated.com/josimages_new/pasp/img141.png)
![$ R(s)$](http://www.dsprelated.com/josimages_new/pasp/img153.png)
![$ \hat{\rho}_v(s)= -\hat{\rho}_f(s)$](http://www.dsprelated.com/josimages_new/pasp/img3725.png)
![$ R(s)$](http://www.dsprelated.com/josimages_new/pasp/img153.png)
![$ \hat{\rho}_f(s)$](http://www.dsprelated.com/josimages_new/pasp/img2167.png)
![$ 1$](http://www.dsprelated.com/josimages_new/pasp/img138.png)
![$ j\omega $](http://www.dsprelated.com/josimages_new/pasp/img71.png)
In particular,
![$ \left\vert\hat{\rho}_f(j\omega)\right\vert \leq 1$](http://www.dsprelated.com/josimages_new/pasp/img3728.png)
![$ \omega\in(-\infty,\infty)$](http://www.dsprelated.com/josimages_new/pasp/img3729.png)
![$ \hat{\rho}_f(s)$](http://www.dsprelated.com/josimages_new/pasp/img2167.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
If the impedance goes to infinity (becomes rigid), then
approaches
, a result which agrees with an analysis of
rigid string terminations (p.
). Similarly, when the
impedance goes to zero,
becomes
, which agrees with
the physics of a string with a free end. In acoustic stringed
instruments, bridges are typically quite rigid, so that
for all
. If a body resonance is
strongly coupled through the bridge,
can be
significantly smaller than 1 at the resonant frequency
.
Solving for in Eq.
(C.77), we can characterize every
impedance in terms of its reflectance:
![$\displaystyle R(s) = R_0\frac{1+\hat{\rho}_f(s)}{1-\hat{\rho}_f(s)}
$](http://www.dsprelated.com/josimages_new/pasp/img3732.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![$\displaystyle \hat{\rho}_f(s) \eqsp \frac{\dfrac{R(s)}{R_0}-1}{\dfrac{R(s)}{R_0}+1},
$](http://www.dsprelated.com/josimages_new/pasp/img3733.png)
![$ R(s)$](http://www.dsprelated.com/josimages_new/pasp/img153.png)
![$ R_0$](http://www.dsprelated.com/josimages_new/pasp/img141.png)
![$ R_0$](http://www.dsprelated.com/josimages_new/pasp/img141.png)
![$ R(s)$](http://www.dsprelated.com/josimages_new/pasp/img153.png)
![$ R(j\omega)$](http://www.dsprelated.com/josimages_new/pasp/img3723.png)
![$ \omega $](http://www.dsprelated.com/josimages_new/pasp/img15.png)
In the discrete-time case, which may be related to the continuous-time
case by the bilinear transform (§7.3.2), we have the same basic
relations, but in the plane:
where
![$ \Gamma\isdef 1/R$](http://www.dsprelated.com/josimages_new/pasp/img3740.png)
Mathematically, any stable transfer function having these properties may be called a Schur function. Thus, the discrete-time reflectance
![$ \hat{\rho}_f(z)$](http://www.dsprelated.com/josimages_new/pasp/img3742.png)
![$ R(z)$](http://www.dsprelated.com/josimages_new/pasp/img3743.png)
Note that Eq.(C.79) may be obtained from the general formula for
scattering at a loaded waveguide junction for the case of a single
waveguide (
) terminated by a lumped load (§C.12).
In the limit as damping goes to zero (all poles of converge to
the unit circle),
the reflectance
becomes a digital allpass filter. Similarly,
becomes a continuous-time allpass filter as the poles of
approach the
axis.
Recalling that a lossless impedance is called a reactance
(§7.1), we can say that every reactance gives rise to an
allpass reflectance. Thus, for example, waves reflecting off a
mass at the end of a vibrating string will be allpass filtered,
because the driving-point impedance of a mass () is a pure
reactance. In particular, the force-wave reflectance of a mass
terminating an ideal string having wave impedance
is
, which is a continuous-time allpass filter having
a pole at
and a zero at
.
It is intuitively reasonable that a passive reflection gain cannot
exceed at any frequency (i.e., the reflectance is a Schur filter,
as defined in Eq.
(C.79)). It is also reasonable that lossless
reflection would have a gain of 1 (i.e., it is allpass).
Note that reflection filters always have an equal number of poles and
zeros, as can be seen from Eq.(C.76) above. This property is
preserved by the bilinear transform, so it holds in both the
continuous- and discrete-time cases.
Reflectance and Transmittance of a Yielding String Termination
Consider the special case of a reflection and transmission at a
yielding termination, or ``bridge'', of an ideal vibrating
string on its right end, as shown in Fig.C.28. Denote the
incident and reflected velocity waves by and
,
respectively, and similarly denote the force-wave components by
and
. Finally, denote the velocity of the
termination itself by
, and its force-wave
reflectance by
![$\displaystyle \hat{\rho}_f(s) \isdefs \frac{F^{-}(s)}{F^{+}(s)} \eqsp -\frac{V^{-}(s)}{V^{+}(s)}
\eqsp \frac{R_b(s)-R_0}{R_b(s)+R_0},
$](http://www.dsprelated.com/josimages_new/pasp/img3754.png)
![$ R_0$](http://www.dsprelated.com/josimages_new/pasp/img141.png)
The bridge velocity is given by
![$\displaystyle V_b(s) \eqsp V^{+}(s) + V^{-}(s),
$](http://www.dsprelated.com/josimages_new/pasp/img3755.png)
![$\displaystyle \hat{\tau}_v(s) \isdefs \frac{V_b(s)}{V^{+}(s)}
\eqsp \frac{V^{+}(s)+V^{-}(s)}{V^{+}(s)}
\eqsp 1+\hat{\rho}_v(s)
\eqsp 1-\hat{\rho}_f(s),
$](http://www.dsprelated.com/josimages_new/pasp/img3756.png)
![$\displaystyle \hat{\tau}_f(s) \isdefs \frac{F_b(s)}{F^{+}(s)}
\eqsp \frac{R_0[V^{+}(s)-V^{-}(s)]}{R_0V^{+}(s)}
\eqsp 1+\hat{\rho}_f(s)
$](http://www.dsprelated.com/josimages_new/pasp/img3757.png)
![$ f = -Ky'$](http://www.dsprelated.com/josimages_new/pasp/img3758.png)
![$ K$](http://www.dsprelated.com/josimages_new/pasp/img185.png)
![$ y'$](http://www.dsprelated.com/josimages_new/pasp/img2259.png)
Power-Complementary Reflection and Transmission
We can show that the reflectance and transmittance of the yielding termination are power complementary. That is, the reflected and transmitted signal-power sum to yield the incident signal-power.
The average power incident at the bridge at frequency can be
expressed in the frequency domain as
.
The reflected power is then
. Removing the minus sign, which can be
associated with reversed direction of travel, we obtain that the
power reflection frequency response is
, which
generalizes by analytic continuation to
. The power
transmittance is given by
![$\displaystyle F_b\overline{V_b}
\eqsp (\hat{\tau}_fF^{+})\overline{(1-\hat{\rh...
...ine{V^{+}})
\eqsp (1-\left\vert\hat{\rho}_f\right\vert^2)F^{+}\overline{V^{+}}
$](http://www.dsprelated.com/josimages_new/pasp/img3763.png)
![$ s$](http://www.dsprelated.com/josimages_new/pasp/img144.png)
![$\displaystyle F_b(s)V_b(-s) = \left[1-\hat{\rho}_f(s)\hat{\rho}_f(-s)\right]F^{+}(s)V^{+}(-s)
$](http://www.dsprelated.com/josimages_new/pasp/img3764.png)
![$\displaystyle -F^{-}(s)V^{-}(-s) + F_b(s)V_b(-s) \eqsp F^{+}(s)V^{+}(-s)
$](http://www.dsprelated.com/josimages_new/pasp/img3765.png)
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Positive Real Functions
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Reflectance of an Impedance