A (lossless)
digital waveguide is defined as a
bidirectional delay line at some
wave impedance 
[
430,
433].
Figure
2.11 illustrates one digital waveguide.

Figure 2.11:
A digital waveguide
samples long at wave-impedance
.
 |
As before, each
delay line contains a sampled acoustic
traveling wave.
However, since we now have a
bidirectional delay line, we have
two traveling waves, one to the ``left'' and one to the
``right'', say. It has been known since 1747 [
100] that
the
vibration of an ideal string
can be described as the sum of two traveling waves going in opposite
directions. (See Appendix
C for a mathematical derivation of this
important fact.) Thus, while a single delay line can model an
acoustic
plane wave, a
bidirectional delay line (a digital
waveguide) can model any one-dimensional linear acoustic system such
as a violin string, clarinet bore, flute pipe, trumpet-valve pipe, or
the like. Of course, in real acoustic strings and bores, the 1D
waveguides exhibit some loss and
dispersion
3.4 so that we will need some
filtering in
the waveguide to obtain an accurate
physical model of such systems.
The
wave impedance 
(derived in Chapter
6) is
needed for connecting digital waveguides to other physical simulations
(such as another digital waveguide or
finite-difference model).
Physical Outputs
Physical variables (
force,
pressure,
velocity, ...) are obtained by
summing traveling-wave components, as shown in
Fig.
2.12, and more elaborated in
Fig.
2.13.
Figure:
More detailed diagram of
Fig.2.12.
 |
It is important to understand that the two
traveling waves in a
digital
waveguide are now
components of a more general acoustic
vibration. The physical wave vibration is obtained by
summing
the left- and right-going traveling waves. A traveling wave by itself
in one of the
delay lines is no longer regarded as ``physical'' unless
the signal in the opposite-going delay line is zero. Traveling waves
are efficient for simulation, but they are not easily estimated from
real-world measurements [
476], except when the
traveling-wave component in one direction can be arranged to be zero.
Note that traveling-wave components are not necessarily
unique.
For example, we can add a constant to the right-going wave and
subtract the same constant from the left-going wave without altering
the (physical) sum [
263]. However, as
derived in Appendix
C (§
C.3.6), 1D traveling-wave components
are uniquely specified by
two linearly independent physical
variables along the waveguide, such as position and velocity
(
vibrating strings) or pressure and velocity (acoustic tubes).
Physical Inputs
A
digital waveguide input signal corresponds to a
disturbance of the 1D
propagation medium. For example, a
vibrating string is
plucked or
bowed by such an external
disturbance. The result of the disturbance is wave propagation to the
left and right of the input point. By physical symmetry, the
amplitude of the left- and right-going propagating disturbances will
normally be equal.
3.5 If
the disturbance
superimposes with the waves already passing
through at that point (an idealized case), then it is purely an
additive input, as shown in Fig.
2.14.
Figure 2.14:
Summing a signal into a
digital waveguide corresponding to a superimposing disturbance at
one point. The original state is unaffected, i.e., the input signal
enters the waveguide in superposition with whatever is
already going on.
 |
Note that the superimposing input of
Fig.
2.14 is the graph-theoretic
transpose of the ideal output shown in
Fig.
2.13. In other words, the
superimposing input injects by means of two
transposed taps.
Transposed taps are discussed further in §
2.5.2 below.
In practical reality, physical driving inputs do not merely
superimpose with the current state of the driven system. Instead,
there is normally some amount of
interaction with the current
system state (when it is nonzero), as discussed further in the next
section. Note that there are similarly no ideal outputs as depicted
in Fig.
2.13. Real physical ouputs must
present some kind of
load on the system (energy must be
extracted). Superimposing inputs and non-loading outputs are ideals
that are often approximated in real-world systems. Of course, in the
virtual world, they are no problem at all--in fact, they are usually
easier to implement, and more efficient.
Interacting Physical Input
Figure
2.15 shows the general case of an
input
signal that interacts with the state of the system at one point
along the
waveguide. Since the interaction is physical, it only
depends on the ``incoming state'' (
traveling-wave components) and the
driving input signal.
Figure 2.15:
Driving a digital waveguide
with physical interaction between the driving input signal
and the current state of the waveguide.
 |
A less general but commonly encountered case is shown in
Fig.
2.16. This case requires the ``outgoing
disturbance'' to be distributed equally to the left and right, and it
sums with the incoming waves to produce the outgoing waves.
Figure 2.16:
Symmetric outgoing
disturbance in superposition with incoming waves.
 |
Figure
2.17 shows a further reduction in
generality--also commonly encountered--in which the interaction
depends only on the
amplitude of the simulated physical
variable (such as string
velocity or
displacement). The incoming
amplitude is formed as the sum of the incoming
traveling-wave
components. We will encounter examples of this nature in later
chapters (such as Chapter
9). It provides realistic models
of physical excitations such as a guitar plectra, violin bows, and
woodwind reeds.
Figure:
Same as Fig.2.15
for the case in which the interaction depends only upon incoming
amplitude.
 |
If an output signal is desired at this precise point, it can be
computed as the incoming amplitude plus twice the outgoing disturbance
signal (equivalent to summing the inputs of the two outgoing
delay
lines).
Note that the above examples all involve waveguide excitation at a
single spatial point. While this can give a sufficiently good
approximation to physical reality in many applications, one should
also consider excitations that are spread out over multiple spatial
samples (even just two).
We will develop the topic of
digital waveguide modeling more
systematically in Chapter
6 and Appendix
C, among other
places in this book. This section is intended only as a high-level
preview and overview. For the next several chapters, we will restrict
attention to normal signal processing structures in which signals may
have physical units (such as acoustic
pressure), and
delay lines hold
sampled acoustic waves propagating in one direction, but successive
processing blocks do not ``load each other down'' or connect
``bidirectionally'' (as every truly physical interaction must, by
Newton's third law
3.6).
Thus, when one processing block feeds a signal to a next block, an
``ideal output'' drives an ``ideal input''. This is typical in
digital signal processing: Loading effects and return
waves
3.7 are
neglected.
3.8
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